Listed Volatility And Variance Derivatives A Python Based Guide
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Listed Volatility
and Variance
Derivatives
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Listed Volatility
and Variance
Derivatives
A Python-based Guide
DR. YVES J. HILPISCH
This edition first published 2017
© 2017 Yves Hilpisch
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Contents
Preface
xi
PART ONE
Introduction to Volatility and Variance
CHAPTER 1
Derivatives, Volatility and Variance
1.1
1.2
1.3
Option Pricing and Hedging
Notions of Volatility and Variance
Listed Volatility and Variance Derivatives
1.3.1 The US History
1.3.2 The European History
1.3.3 Volatility of Volatility Indexes
1.3.4 Products Covered in this Book
1.4 Volatility and Variance Trading
1.4.1 Volatility Trading
1.4.2 Variance Trading
1.5 Python as Our Tool of Choice
1.6 Quick Guide Through the Rest of the Book
CHAPTER 2
Introduction to Python
2.1
Python Basics
2.1.1 Data Types
2.1.2 Data Structures
2.1.3 Control Structures
2.1.4 Special Python Idioms
2.2 NumPy
2.3 matplotlib
2.4 pandas
2.4.1 pandas DataFrame class
2.4.2 Input-Output Operations
2.4.3 Financial Analytics Examples
2.5 Conclusions
3
3
6
7
7
8
9
10
11
11
13
14
14
17
17
17
20
22
23
28
34
38
39
45
47
53
v
vi
CONTENTS
CHAPTER 3
Model-Free Replication of Variance
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
Introduction
Spanning with Options
Log Contracts
Static Replication of Realized Variance and Variance Swaps
Constant Dollar Gamma Derivatives and Portfolios
Practical Replication of Realized Variance
VSTOXX as Volatility Index
Conclusions
55
55
56
57
57
58
59
65
67
PART TWO
Listed Volatility Derivatives
CHAPTER 4
Data Analysis and Strategies
4.1 Introduction
4.2 Retrieving Base Data
4.2.1 EURO STOXX 50 Data
4.2.2 VSTOXX Data
4.2.3 Combining the Data Sets
4.2.4 Saving the Data
4.3 Basic Data Analysis
4.4 Correlation Analysis
4.5 Constant Proportion Investment Strategies
4.6 Conclusions
CHAPTER 5
VSTOXX Index
5.1 Introduction
5.2 Collecting Option Data
5.3 Calculating the Sub-Indexes
5.3.1 The Algorithm
5.4 Calculating the VSTOXX Index
5.5 Conclusions
5.6 Python Scripts
5.6.1 index collect option_data.py
5.6.2 index_subindex_calculation.py
5.6.3 index_vstoxx_calculation.py
CHAPTER 6
Valuing Volatility Derivatives
6.1 Introduction
6.2 The Valuation Framework
6.3 The Futures Pricing Formula
71
71
71
71
74
76
78
78
83
87
93
95
95
95
105
106
114
118
118
118
123
127
129
129
129
130
vii
Contents
6.4
6.5
6.6
The Option Pricing Formula
Monte Carlo Simulation
Automated Monte Carlo Tests
6.6.1 The Automated Testing
6.6.2 The Storage Functions
6.6.3 The Results
Model Calibration
6.7.1 The Option Quotes
6.7.2 The Calibration Procedure
6.7.3 The Calibration Results
Conclusions
Python Scripts
6.9.1 srd_functions.py
6.9.2 srd simulation analysis.py
6.9.3 srd simulation results.py
6.9.4 srd model calibration.py
132
135
141
141
145
146
153
154
155
160
163
163
163
167
171
174
CHAPTER 7
Advanced Modeling of the VSTOXX Index
179
6.7
6.8
6.9
7.1
7.2
7.3
7.4
Introduction
Market Quotes for Call Options
The SRJD Model
Term Structure Calibration
7.4.1 Futures Term Structure
7.4.2 Shifted Volatility Process
7.5 Option Valuation by Monte Carlo Simulation
7.5.1 Monte Carlo Valuation
7.5.2 Technical Implementation
7.6 Model Calibration
7.6.1 The Python Code
7.6.2 Short Maturity
7.6.3 Two Maturities
7.6.4 Four Maturities
7.6.5 All Maturities
7.7 Conclusions
7.8 Python Scripts
7.8.1 srjd fwd calibration.py
7.8.2 srjd_simulation.py
7.8.3 srjd_model_calibration.py
CHAPTER 8
Terms of the VSTOXX and its Derivatives
8.1
8.2
8.3
8.4
8.5
The EURO STOXX 50 Index
The VSTOXX Index
VSTOXX Futures Contracts
VSTOXX Options Contracts
Conclusions
179
179
182
183
184
190
191
191
192
195
196
199
201
203
205
209
210
210
212
215
221
221
221
223
224
225
viii
CONTENTS
PART THREE
Listed Variance Derivatives
CHAPTER 9
Realized Variance and Variance Swaps
9.1 Introduction
9.2 Realized Variance
9.3 Variance Swaps
9.3.1 Definition of a Variance Swap
9.3.2 Numerical Example
9.3.3 Mark-to-Market
9.3.4 Vega Sensitivity
9.3.5 Variance Swap on the EURO STOXX 50
9.4 Variance vs. Volatility
9.4.1 Squared Variations
9.4.2 Additivity in Time
9.4.3 Static Hedges
9.4.4 Broad Measure of Risk
9.5 Conclusions
CHAPTER 10
Variance Futures at Eurex
10.1
10.2
10.3
10.4
10.5
Introduction
Variance Futures Concepts
10.2.1 Realized Variance
10.2.2 Net Present Value Concepts
10.2.3 Traded Variance Strike
10.2.4 Traded Futures Price
10.2.5 Number of Futures
10.2.6 Par Variance Strike
10.2.7 Futures Settlement Price
Example Calculation for a Variance Future
Comparison of Variance Swap and Future
Conclusions
CHAPTER 11
Trading and Settlement
11.1
11.2
11.3
11.4
11.5
11.6
11.7
11.8
Introduction
Overview of Variance Futures Terms
Intraday Trading
Trade Matching
Different Traded Volatilities
After the Trade Matching
Further Details
11.7.1 Interest Rate Calculation
11.7.2 Market Disruption Events
Conclusions
229
229
229
235
235
235
239
241
242
247
247
247
250
250
250
251
251
252
252
252
257
257
258
258
258
258
265
268
269
269
269
270
274
275
277
279
279
280
280
ix
Contents
PART FOUR
DX Analytics
CHAPTER 12
DX Analytics – An Overview
12.1
12.2
12.3
12.4
Introduction
Modeling Risk Factors
Modeling Derivatives
Derivatives Portfolios
12.4.1 Modeling Portfolios
12.4.2 Simulation and Valuation
12.4.3 Risk Reports
12.5 Conclusions
CHAPTER 13
DX Analytics – Square-Root Diffusion
13.1
13.2
13.3
13.4
13.5
13.6
283
283
284
287
290
292
293
294
296
297
Introduction
Data Import and Selection
Modeling the VSTOXX Options
Calibration of the VSTOXX Model
Conclusions
Python Scripts
13.6.1 dx srd calibration.py
297
297
301
303
308
308
308
CHAPTER 14
DX Analytics – Square-Root Jump Diffusion
315
14.1
14.2
14.3
14.4
14.5
14.6
Introduction
Modeling the VSTOXX Options
Calibration of the VSTOXX Model
Calibration Results
14.4.1 Calibration to One Maturity
14.4.2 Calibration to Two Maturities
14.4.3 Calibration to Five Maturities
14.4.4 Calibration without Penalties
Conclusions
Python Scripts
14.6.1 dx srjd calibration.py
315
315
320
325
325
325
325
331
332
334
334
Bibliography
345
Index
347
Preface
olatility and variance trading has evolved from something opaque to a standard tool in
today’s financial markets. The motives for trading volatility and variance as an asset class
of its own are numerous. Among others, it allows for effective option and equity portfolio hedging and risk management as well as straight out speculation on future volatility (index) movements. The potential benefits of volatility- and variance-based strategies are widely accepted
by researchers and practitioners alike.
With regard to products it mainly started out around 1993 with over-the-counter (OTC)
variance swaps. At about the same time, the Chicago Board Options Exchange introduced the
VIX volatility index. This index still serves today – after a significant change in its methodology – as the underlying risk factor for some of the most liquidly traded listed derivatives
in this area. The listing of such derivatives allows for a more standardized, cost efficient and
transparent approach to volatility and variance trading.
This book covers some of the most important listed volatility and variance derivatives
with a focus on products provided by Eurex. Larger parts of the content are based on the
Eurex Advanced Services tutorial series which use Python to illustrate the main concepts of
volatility and variance products. I am grateful that Eurex allowed me to use the contents of the
tutorial series freely for this book.
Python has become not only one of the most widely used programming languages but also
one of the major technology platforms in the financial industry. It is more like a platform since
the Python ecosystem provides a wealth of powerful libraries and packages useful for financial
analytics and application building. It also integrates well with many other technologies, like
the statistical programming language R, used in the financial industry. You can find links to
all Python resources under http://lvvd.tpq.io.
I thank Michael Schwed for providing parts of the Python code. I also thank my family
for all their love and support over the years, especially my wife Sandra and our children Lilli
and Henry. I dedicate this book to my beloved dog Jil. I miss you.
V
Yves
Voelklingen, Saarland, April 2016
xi
PART
One
Introduction to Volatility
and Variance
Listed Volatility and Variance Derivatives: A
Python-based Guide
By Dr. Yves J. Hilpisch
© 2017 Yves Hilpisch
CHAPTER
1
Derivatives, Volatility and Variance
he first chapter provides some background information for the rest of the book. It mainly
covers concepts and notions of importance for later chapters. In particular, it shows how the
delta hedging of options is connected with variance swaps and futures. It also discusses different notions of volatility and variance, the history of traded volatility and variance derivatives
as well as why Python is a good choice for the analysis of such instruments.
T
1.1 OPTION PRICING AND HEDGING
In the Black-Scholes-Merton (1973) benchmark model for option pricing, uncertainty with
regard to the single underlying risk factor S (stock price, index level, etc.) is driven by a geometric Brownian motion with stochastic differential equation (SDE)
dSt = 𝜇St dt + 𝜎St dZt
Throughout we may think of the risk factor as being a stock index paying no dividends. St is
then the level of the index at time t, 𝜇 the constant drift, 𝜎 the instantaneous volatility and Zt is
a standard Brownian motion. In a risk-neutral setting, the drift 𝜇 is replaced by the (constant)
risk-less short rate r
dSt = rSt dt + 𝜎St dZt
In addition to the index which is assumed to be directly tradable, there is also a risk-less bond
B available for trading. It satisfies the differential equation
dBt = rBt dt
In this model, it is possible to derive a closed pricing formula for a vanilla European call option
C maturing at some future date T with payoff max[ST − K, 0], K being the fixed strike price.
It is
C(S, K, t, T, r, 𝜎) = St ⋅ N(d1 ) − e−r(T−t) ⋅ K ⋅ N(d2 )
3
4
LISTED VOLATILITY AND VARIANCE DERIVATIVES
where
d
1 2
1
e− 2 x dx
N(d) = √
2𝜋 ∫−∞
)
(
2
S
log Kt + r + 𝜎2 (T − t)
d1 =
√
𝜎 T −t
)
(
2
S
log Kt + r − 𝜎2 (T − t)
d2 =
√
𝜎 T −t
The price of a vanilla European put option P with payoff max[K − ST , 0] is determined by
put-call parity as
Pt = Ct − St + e−r(T−t) K
There are multiple ways to derive this famous Black-Scholes-Merton formula. One way relies
on the construction of a portfolio comprised of the index and the risk-less bond that perfectly
replicates the option payoff at maturity. To avoid risk-less arbitrage, the value of the option
must equal the payoff of the replicating portfolio. Another method relies on calculating the
risk-neutral expectation of the option payoff at maturity and discounting it back to the present
by the risk-neutral short rate. For detailed explanations of these approaches refer, for example,
to Björk (2009).
Yet another way, which we want to look at in a bit more detail, is to perfectly hedge
the risk resulting from an option (e.g. from the point of view of a seller of the option) by
dynamically trading the index and the risk-less bond. This approach is usually called delta
hedging (see Sinclair (2008), ch. 1). The delta of a European call option is given by the first
𝛿C
partial derivative of the pricing formula with respect to the value of the risk factor, i.e. 𝛿t = 𝛿S t .
t
More specifically, we get
𝛿t =
𝛿Ct
= N(d1 )
𝛿St
When trading takes place continuously, the European call option position hedged by 𝛿t index
units short is risk-less:
dΠt ≡ dCt − 𝛿t St = 0
This is due to the fact that the only (instantaneous) risk results from changes in the index level
and all such (marginal) changes are compensated for by the delta short index position.
Continuous models and trading are a mathematically convenient description of the real
world. However, in practice trading and therefore hedging can only take place at discrete points
in time. This does not lead to a complete breakdown of the delta hedging approach, but it
5
Derivatives, Volatility and Variance
introduces hedge errors. If hedging takes place at every discrete time interval of length Δt, the
Profit-Loss (PL) for such a time interval is roughly (see Bossu (2014), p. 59)
PLΔt ≈
1
Γ ⋅ ΔS2 + Θ ⋅ Δt
2
Γ is the gamma of the option and measures how the delta (marginally) changes with the changing index level. ΔS is the change in the index level over the time interval Δt. It is given by
Γ=
N′ (d1 )
𝜕2 C
=
√
𝜕S2
S𝜎 T − t
Θ is the theta of the option and measures how the option value changes with the passage of
time. It is given approximately by (see Bossu (2014), p. 60)
1
Θ ≈ − ΓS2 𝜎 2
2
With this we get
1
1
Γ ⋅ ΔS2 − ΓS2 𝜎 2 ⋅ Δt
2
[ ( 2 )2 ( √ ) ]
2
ΔS
1
= Γ ⋅ S2
− 𝜎 ⋅ Δt
2
S
PLΔt ≈
The quantity 12 Γ ⋅ S2 is called the dollar gamma of the option and gives the second order change
)2 is the squared
in the option price induced by a (marginal) change in the index level. ( ΔS
S
realized return over the time interval Δt – it might be interpreted as the (instantaneously)
realized
√ variance if the time interval is short enough and the drift is close to zero. Finally,
(𝜎 ⋅ Δt)2 is the fixed, “theoretical” variance in the model for the time interval.
The above reasoning illustrates that the PL of a discretely delta hedged option position is
determined by the difference between the realized variance during the discrete hedge interval
and the theoretically expected variance given the model parameter for the volatility. The total
T
intervals is given by
hedge error over N = Δt
1∑
≈
Γ ⋅ S2
2 t=1 t−1 t−1
N
Cumulative PLΔt
[(
ΔSt
St−1
)2
(
− 𝜎⋅
√
)2
Δt
]
(1.1)
This little exercise in option hedging leads us to a result which is already quite close to a
product intensively discussed in this book: listed variance futures. Variance futures, and their
Over-the-Counter (OTC) relatives variance swaps, pay to the holder the difference between
realized variance over a certain period of time and a fixed variance strike.
6
LISTED VOLATILITY AND VARIANCE DERIVATIVES
1.2 NOTIONS OF VOLATILITY AND VARIANCE
The previous section already touches on different notions of volatility and variance. This
section provides formal definitions for these and other quantities of importance. For a more
detailed exposition refer to Sinclair (2008). In what follows we assume that a time series is
given with quotes Sn , n ∈ {0, … , N} (see Hilpisch (2015, ch. 3)). We do not assume any specific model that might generate the time series data. The log return for n > 0 is defined by
Rn ≡ log Sn − log Sn−1 = log
realized or historical volatility: this refers to the standard deviation of the log returns of
a financial time series; suppose we observe N (past) log returns Rn , n ∈ {1, … , N}, with
∑
mean return 𝜇̂ = N1 N
̂ is then given by
n=1 Rn ; the realized or historical volatility 𝜎
√
√
√
𝜎̂ = √
Sn
Sn−1
N
1 ∑
(R − 𝜇)
̂ 2
N − 1 n=1 n
instantaneous volatility: this refers to the volatility factor of a diffusion process; for
example, in the Black-Scholes-Merton model the instantaneous volatility 𝜎 is found in
the respective (risk-neutral) stochastic differential equation (SDE)
dSt = rSt dt + 𝜎St dZt
implied volatility: this is the volatility that, if put into the Black-Scholes-Merton option
pricing formula, gives the market-observed price of an option; suppose we observe today
a price of C0∗ for a European call option; the implied volatility 𝜎 imp is the quantity that
solves ceteris paribus the implicit equation
C0∗ = CBSM (S0 , K, t = 0, T, r, 𝜎 imp )
These volatilities all have squared counterparts which are then named variance, such as realized variance, instantenous variance or implied variance. We have already encountered realized
variance in the previous section. Let us revisit this quantity for a moment. Simply applying the
above definition of realized volatility and squaring it we get
𝜎̂ 2 =
N
1 ∑
(R − 𝜇)
̂ 2
N − 1 n=1 n
In practice, however, this definition usually gets adjusted to
𝜎̂ 2 =
N
1 ∑ 2
R
N n=1 n
7
Derivatives, Volatility and Variance
The drift of the process is assumed to be zero and only the log return terms get squared. It
is also common practice to use the definition for the uncorrected (biased) standard deviation
with factor N1 instead of the definition for the corrected (unbiased) standard deviation with
ΔS
1
factor N−1
. This explains why we call the term ( S t )2 from the delta hedge PL in the previous
t−1
section realized variance over the time interval Δt. In that case, however, the return is the
simple return instead of the log return.
Other adjustments in practice are to scale the value to an annual quantity by multiplying
it by 252 (trading days) and to introduce an additional scaling term (to get percent values
instead of decimal ones). One then usually ends up with (see chapter 9, Realized Variance and
Variance Swaps)
𝜎̂ 2 ≡ 10000 ⋅
N
252 ∑ 2
R
⋅
N n=1 n
Later on we will also drop the hat notation when there is no ambiguity.
1.3 LISTED VOLATILITY AND VARIANCE DERIVATIVES
Volatility is one of the most important notions and concepts in derivatives pricing and analytics.
Early research and financial practice considered volatility as a major input for pricing and
hedging. It is not that long ago that the market started thinking of volatility as an asset class
of its own and designed products to make it directly tradable.
The idea for a volatility index was conceived by Brenner and Galai in 1987 and published in the note Brenner and Galai (1989) in the Financial Analysts Journal. They write in
their note:
“While there are efficient tools for hedging against general changes in overall market
directions, so far there are no effective tools available for hedging against changes
in volatility. … We therefore propose the construction of three volatility indexes on
which cash-settled options and futures can be traded.”
In what follows, we focus on the US and European markets.
1.3.1 The US History
The Chicago Board Options Exchange (CBOE) introduced an equity volatility index, called
VIX, in 1993. It was based on a methodology developed by Fleming, Ostdiek and Whaley
(1995) – a working paper version of which was circulated in 1993 – and data from S&P 100
index options. The methodology was changed in 2003 to the now standard practice which uses
the robust, model free replication results for variance (see chapter 3 Model-Free Replication
of Variance) and data from S&P 500 index options (see CBOE (2003)). While the first version
represented a proxy for the 30 day at-the-money implied volatility, the current version is a
proxy for the 30 day variance swap rate, i.e. the fixed variance strike which gives a zero value
for a respective swap at inception.
8
LISTED VOLATILITY AND VARIANCE DERIVATIVES
FIGURE 1.1
Historical volume of traded VIX derivatives on a log scale. Data source:
http://cfe.cboe.com/data/historicaldata.aspx
Carr and Lee (2009) provide a brief history of both OTC and listed volatility and variance
products. They claim that the first OTC variance swap has been engineered and offered by
Union Bank of Switzerland (UBS) in 1993, at about the same time the CBOE announced the
VIX. These were also the first traded contracts to attract some liquidity in contrast to volatility
swaps which were also introduced shortly afterwards. One reason for this is that variance swaps
can be robustly hedged – as we will see in later chapters – while volatility swaps, in general,
cannot. It is more or less the same reasoning behind the change of methodology for the VIX
in 2003.
Trading in futures on the VIX started in 2004 while the first options on the index were
introduced in 2006. These instruments are already described in Whaley (1993), although their
market launch took more than 10 years after the introduction of the VIX. These were not the
first listed volatility derivatives but the first to attract significant liquidity and they are more
actively traded at the time of writing than ever. Those listed instruments introduced earlier,
such as volatility futures launched in 1998 by Deutsche Terminbörse (now Eurex), could not
attract enough liquidity and are now only a footnote in the financial history books.
The volume of traded contracts on the VIX has risen sharply on average over recent years
as Figure 1.1 illustrates. The volume varies rather erratically and is influenced inter alia by
seasonal effects and the general market environment (bullish or bearish sentiment).
In December 2012, the CBOE launched the S&P 500 variance futures contract – almost
20 years after their OTC counterparts started trading. After some early successes in building
liquidity in 2013, liquidity has dried up almost completely in 2014 and 2015.
1.3.2 The European History
Eurex – back then Deutsche Terminbörse – introduced in 1994 the VDAX volatility index, an
index representing the 45 day implied volatility of DAX index options. As mentioned before,
Derivatives, Volatility and Variance
9
FIGURE 1.2
Historical daily closing levels for the VIX and the VSTOXX volatility indexes. Data
source: Yahoo! Finance and http://stoxx.com
in 1998 Eurex introduced futures on the VDAX which could, however, not attract enough
liquidity and were later delisted. In 2005, the methodolgy for calculating the index was also
changed to the more robust, model-free replication approach for variance swaps. The index
was renamed VDAX-NEW and a new futures contract on this index was introduced.
In 2005, Eurex also launched futures on the VSTOXX volatility index which is based on
options on the EURO STOXX 50 index and uses the by now standard methodology for volatility index calculation as laid out in CBOE (2003). In 2009 they were re-launched as “Mini
VSTOXX Futures” with the symbol FVS. At the same time, Eurex stopped the trading of
other volatility futures, such as those on the VDAX-NEW and the VSMI.
Since the launch of the new VSTOXX futures, they have attracted significant liquidity and
are now actively traded. A major reason for this can be seen in the financial crisis of 2007–2009
when volatility indexes saw their highest levels ever. This is illustrated in Figure 1.2 where the
maximum values for the VIX and VSTOXX are observed towards the end of 2008. This led
to a higher sensitivity of market participants to the risks that spikes in volatility can bring and
thus increased the demand for products to hedge against such adverse market environments.
Observe also in Figure 1.2 that the two indexes are positively correlated in general, over the
period shown with about +0.55.
In March 2010, Eurex introduced options on the VSTOXX index. These instruments also
attracted some liquidity and are at the time of writing actively traded. In September 2014,
Eurex then launched a variance futures contract on the EURO STOXX 50 index.
1.3.3 Volatility of Volatility Indexes
Nowadays, we are already one step further on. There are now indexes available that measure
the volatility of volatility (vol-vol). The so-called VVIX of the CBOE was introduced in March
10
LISTED VOLATILITY AND VARIANCE DERIVATIVES
FIGURE 1.3
Historical daily closing levels for the VIX and the VVIX volatility (of volatility)
indexes. Data source: Thomson Reuters Eikon
2012. In October 2015 the index provider STOXX Limited introduced the V-VSTOXX indexes
which are described on www.stoxx.com as follows:
“The V-VSTOXX Indices are based on VSTOXX realtime options prices and are
designed to reflect the market expectations of near-term up to long-term volatilityof-volatility by measuring the square root of the implied variance across all options
of a given time to expiration.”
These new indexes and potential products written on them seem to be a beneficial addition to
the volatility asset class. Such products might be used, for example, to hedge options written
on the volatility index itself since the vol-vol is stochastic in nature rather than constant or
deterministic.
The VVIX index is generally on a much higher level than the VIX index as Figure 1.3
illustrates. This indicates a much higher volatility for the VIX index itself compared to the
S&P 500 volatility.
Over the period shown, the VVIX is highly positively correlated with the VIX at a level
of about +0.66. Figure 1.4 plots the time series data for the two indexes on two different scales
to show this stylized fact graphically.
1.3.4 Products Covered in this Book
There is quite a diverse spectrum of volatility and variance futures available. Out of all possible
products, the focus of this book is on the European market and these instruments:
VSTOXX as a volatility index
VSTOXX futures
VSTOXX options
Eurex Variance Futures.
Derivatives, Volatility and Variance
11
FIGURE 1.4
Different scalings for the VVIX and VIX indexes to illustrate the positive correlation.
Data Source: Thomson Reuters Eikon
The majority of the material has originally been developed as part of the Eurex Advanced
Services. Although the focus is on Europe, the methods and approaches presented can usually be tranferred easily to the American landscape, for instance. This results from the fact
that some methodological unification has taken place over the past few years with regard to
volatility and variance related indexes and products.
1.4 VOLATILITY AND VARIANCE TRADING
This section discusses motives and rationales for trading listed volatility and variance derivatives. It does not cover volatility (variance) trading strategies that can be implemented with,
for example, regular equity options (see Cohen (2005), ch. 4).
1.4.1 Volatility Trading
It is instructive to first list characteristics of volatility indexes. We focus on the VSTOXX
and distinguish between facts (which follow from construction) and stylized facts (which are
supported by empirical evidence):
market expection (fact): the VSTOXX represents a 30 day implied volatility average
from out-of-the money options, i.e. the market consensus with regard to the “to be realized” volatility over the next 30 days
non-tradable asset (fact): the VSTOXX itself is not directly tradable, only derivatives on
the VSTOXX can be traded
mean-reverting nature (fact): the VSTOXX index is mean-reverting, it does not show a
positive or negative drift over longer periods of time
12
LISTED VOLATILITY AND VARIANCE DERIVATIVES
FIGURE 1.5
Different scalings for the S&P 500 and VIX indexes to illustrate the negative
correlation. Data source: Yahoo! Finance
negative correlation (stylized fact): the VSTOXX is (on average) negatively correlated
with the respective equity index, the EURO STOXX 50
positive jumps (stylized fact): during times of stock market crisis, the VSTOXX can jump
to rather high levels; the mean reversion generally happens much more slowly
higher than realized volatility (stylized fact): on average, the VSTOXX index is higher
than the realized volatility over the next 30 days.
Figure 1.5 illustrates the negative correlation between the S&P 500 index and the VIX index
graphically. Over the period shown, the correlation is about −0.75.
The value of a VSTOXX future represents the (market) expectation with regard to the
future value of the VSTOXX at the maturity date of the future. Given this background, typical
volatility trading strategies involving futures include the following:
long VSTOXX future: such a position can be used to hedge equity positions (due to the
negative correlation with the EURO STOXX 50) or to increase returns of an equity portfolio (e.g. through a constant-proportion investment strategy, see chapter 4, Data Analysis
and Strategies); it can also be used to hedge a short realized volatility strategy
short VSTOXX future: such a trade can be entered, for example, when the VSTOXX
spikes and the expectation is that it will revert (fast enough) to its mean; it might also
serve to reduce vega exposure in long vega option portfolios
term structure arbitrage: for example, shorting the front month futures contract and
going long the nearby futures contract represents a typical term structure arbitrage strategy
when the term structure is in contango; this is due to different carries associated with
different futures contracts and maturities, respectively
Derivatives, Volatility and Variance
13
relative value arbitrage: VSTOXX futures can also be traded against other volatility/variance sensitive instruments and positions, like OTC variance swaps, equity options
portfolios, etc.
Similar and other strategies can be implemented involving VSTOXX options. With regard to
exercise they are European in nature and can be delta hedged by using VSTOXX futures which
is yet another motive for trading in futures. Typical trading strategies involving VSTOXX
options are:
long OTM calls: such a position might protect an equity portfolio from losses due to a
market crash (again due to the negative correlation between VSTOXX and EURO STOXX
50)
short ATM calls: writing ATM calls, and pocketing the option premium, might be attractive when the current implied volatility levels are relatively high
long ATM straddle: buying put and call options on the VSTOXX with same (ATM) strike
and maturity yields a profit when the VSTOXX moves fast enough in one direction; this
is typically to be expected when the volatility of volatility (vol-vol) is high.
For both VSTOXX futures and options many other strategies can be implemented that exploit
some special situation (e.g. contango or backwardation in the futures prices) or reflect a certain
expectation of the trader (e.g. that realized volatility will be lower/higher than the implied/
expected volatility).
1.4.2 Variance Trading
The motives and rationales for trading in EURO STOXX 50 variance futures are not too different from those involving VSTOXX derivatives (see Bossu et al. (2005)). Typical strategies
include:
long variance future: this position benefits when the realized variance is higher than
the variance strike (implied variance at inception); it might also be used to hedge equity
portfolio risks or short vega options positions
short variance future: this position benefits when the realized variance is below the variance strike which tends to be the case on average; it can also hedge a long vega options
position
forward volatility/variance trading: since variance is additive over time (which volatility is not), one can get a perfect exposure to forward implied volatility by, for example, shorting the September variance future and going long the October contract; this
gives an exposure to the forward implied volatility from September maturity to October
maturity
correlation trading: variance futures can be traded to exploit (statistical) arbitrage opportunities between, for example, the (implied) variance of an equity index and its components or the (implied) variance of one equity index versus another one; in both cases, the
rationale is generally based on the correlation of the different assets and their variance,
respectively.
14
LISTED VOLATILITY AND VARIANCE DERIVATIVES
1.5 PYTHON AS OUR TOOL OF CHOICE
There are some general reasons why Python is a good choice for computational finance and
financial data science these days. Among others, these are:
open source: Python is open source and can be used by students and big financial institutions alike for free
syntax: Python’s readable and concise syntax make it a good choice for presenting formal
concepts, like those in finance
ecosystem: compared to other languages Python has an excellent ecosystem of libraries
and packages that are useful for data analytics and scientific computing in general and
financial analytics in particular
performance: in recent years, the ecosystem of Python has grown especially in the
area of performance libraries, making it much easier to get to computing speeds more
than sufficient for the most computationally demanding algorithms, such as Monte Carlo
simulation
adoption: at the time of writing, Python has established itself as a core technology at
major financial institutions, be it leading investment banks, big hedge funds or more traditional asset management firms
career: given the widespread adoption of Python, learning and mastering the language
seems like a good career move for everybody working in the industry or planning to
do so.
In view of the scope and style of the book, one special feature is noteworthy:
interactivity: the majority of the code examples presented in this book can be executed
in interactive fashion within the Jupyter Notebook environment (see http://jupyter.org);
in this regard, Python has a major advantage as an interpreted language compared to a
compiled one with its typical edit-compile-run cycle.
Chapter 1 of Hilpisch (2014) provides a more detailed overview of aspects related to
Python for Finance. All the code presented in this book is available via resources listed
under http://lvvd.tpq.io, especially on the Quant Platform for which you can register under
http://lvvd.quant-platform.com.
1.6 QUICK GUIDE THROUGH THE REST OF THE BOOK
The remainder of this introductory part of the book is organized as follows:
chapter 2: this chapter introduces Python as a technology platform for (interactive) financial analytics; a more detailed account of Python for Finance is provided in Hilpisch
(2014)
chapter 3: this chapter presents the model-free replication approach for variance; it is
important for both volatility indexes and derivatives written on them as well as for variance
futures.
Derivatives, Volatility and Variance
15
The second part of the book is about the VSTOXX and listed volatility derivatives. It comprises
the following chapters:
chapter 4: as a starting point, chapter 4 uses Python to analyze historical data for the
VSTOXX and EURO STOXX 50 indexes; a focal point is the analysis of some simple
trading strategies involving the VSTOXX
chapter 5: using the model-free replication approach for variance, this chapter shows in
detail how the VSTOXX index is calculated and how to use Python to (re-)calculate it
using raw option data as input
chapter 6: Grünbichler and Longstaff (1996) were among the first to propose a parameterized model to value futures and options on volatility indexes; chapter 6 presents their
model which is based on a square-root diffusion process and shows how to simulate and
calibrate it to volatility option market quotes
chapter 7: building on chapter 6, chapter 7 presents a more sophisticated framework – a
deterministic shift square-root jump diffusion (SRJD) process – to model the VSTOXX
index and to better capture the implied volatility smiles and volatility term structure
observed in the market; the exposition is slightly more formal compared to the rest of
the book
chapter 8: this brief chapter discusses terms of the VSTOXX volatility index as well as
the futures and options traded on the VSTOXX.
Part three of the book is about the Eurex Variance Futures contract as listed in September 2014.
This part comprises three chapters:
chapter 9: listed variance futures are mainly based on the popular OTC variance swap
contracts with some differences introduced by intraday trading; chapter 9 therefore covers variance swaps in some detail and also discusses differences between variance and
volatility as an underlying asset
chapter 10: this chapter provides a detailed discussion of all concepts related to the listed
Eurex Variance Futures contract and shows how to (re-)calculate its value given historical
data; it also features a comparison between the futures contract and a respective OTC
variance swap contract
chapter 11: this chapter discusses all those special characteristics of the Eurex Variance
Futures when it comes to (intraday) trading and settlement.
Part four of the book focuses on the DX Analytics financial library (see http://dxanalytics.com) to model the VSTOXX index and to calibrate different models to VSTOXX
options quotes. It consists of three chapters:
chapter 12: this chapter introduces basic concepts and API elements of the DX Analytics
library
chapter 13: using the square-root diffusion model as introduced in chapter 6, this chapter
implements a calibration study to a single maturity of VSTOXX options over the first
quarter of 2014
chapter 14: chapter 14 replicates the same calibration study but in a more sophisticated
fashion; it calibrates the deterministic shift square-root jump diffusion process not only
to a single maturity of options but to as many as five simultaneously.
Listed Volatility and Variance Derivatives: A
Python-based Guide
By Dr. Yves J. Hilpisch
© 2017 Yves Hilpisch
CHAPTER
2
Introduction to Python
ython has become a powerful programming language and has developed a huge ecosystem
of helpful libraries over the last couple of years. This chapter provides a concise overview
of Python and two of the major pillars of the the so-called scientific stack:
P
NumPy (see http://numpy.scipy.org)
pandas (see http://pandas.pydata.org)
NumPy provides performant array operations on numerical data while pandas is specifically designed to handle more complex data analytics operations, e.g. on (financial) times
series data.
Such an introductory chapter – only addressing selected topics relevant to the contents
of this book – can of course not replace a thorough introduction to Python and the libraries
covered. However, if you are rather new to Python or programming in general you might get
a first overview and a feeling of what Python is all about. If you are already experienced in
another language typically used in quantitative finance (e.g. Matlab, R, C++, VBA), you see
what typical data structures, programming paradigms and idioms in Python look like.
For a comprehensive overview of Python applied to finance see Hilpisch (2014). Other,
more general introductions to the language with a scientific and data analysis focus are Haenel
et al. (2013), Langtangen (2009) and McKinney (2012).
This chapter and the rest of the book is based on Python 2.7 although the majority of the
code should be easily transformed to Python 3.5 after some minor modifications.
2.1 PYTHON BASICS
This section introduces basic Python data types and structures, control structures and some
Python idioms.
2.1.1 Data Types
It is noteworthy that Python is a dynamically typed system which means that types of objects
are inferred from their contexts. Let us start with numbers.
17
18
LISTED VOLATILITY AND VARIANCE DERIVATIVES
In [1]: a = 3
# defining an integer object
In [2]: type(a)
Out[2]: int
In [3]: a.bit_length()
Out[3]: 2
In [4]: b = 5.
# number of bits used
# defining a float object
In [5]: type(b)
Out[5]: float
Python can handle arbitrarily large integers which is quite beneficial for number of theoretical
applications, for instance:
In [6]: c = 10 ** 100
# googol number
In [7]: type(c)
Out[7]: long
In [8]: c
# long integer object
Out[8]: 1000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000L
In [9]: c.bit_length()
Out[9]: 333
# number of bits used
Arithmetic operations on these objects work as expected.
In [10]: 3 / 5.
Out[10]: 0.6
# division
In [11]: a * b
Out[11]: 15.0
# multiplication
In [12]: a - b
Out[12]: -2.0
# difference
In [13]: b + a
Out[13]: 8.0
# addition
In [14]: a ** b
# power
Out[14]: 243.0
19
Introduction to Python
However, be aware of the floor division which is standard in Python 2.7.
In [15]: 3 / 5
# int type inferred => floor division
Out[15]: 0
Many often used mathematical functions are found in the math module which is part of
Python’s standard library.
In [16]: import math
# importing the library into the namespace
In [17]: math.log(a) # natural logarithm
Out[17]: 1.0986122886681098
In [18]: math.exp(a) # exponential function
Out[18]: 20.085536923187668
In [19]: math.sin(b) # sine function
Out[19]: -0.9589242746631385
Another important basic data type is string objects.
In [20]: s = 'Listed Volatility and Variance Derivatives.'
In [21]: type(s)
Out[21]: str
This object type has multiple methods attached.
In [22]: s.lower() # converting to lower case characters
Out[22]: 'listed volatility and variance derivatives.'
In [23]: s.upper() # converting to upper case characters
Out[23]: 'LISTED VOLATILITY AND VARIANCE DERIVATIVES.'
String objects can be easily sliced. Note that Python has in general zero-based numbering and
indexing.
In [24]: s[0:6]
Out[24]: 'Listed'
20
LISTED VOLATILITY AND VARIANCE DERIVATIVES
Such objects can also be combined using the + operator. The index value -1 represents the last
character of a string (or last element of a sequence in general).
In [25]: st = s[0:6] + s[-13:-1]
In [26]: print st
Listed Derivatives
String replacements are often used to parametrize text output.
In [27]: repl = "My name is %s, I am %d years old and %4.2f m tall."
# replace %s by a string, %d by an integer and
# %4.2f by a float showing 2 decimal values
In [28]: print repl % ('Peter', 35, 1.88)
My name is Peter, I am 35 years old and 1.88 m tall.
A different way to reach the same goal is the following:
In [29]: repl = "My name is {:s}, I am {:d} years old and {:4.2f} m tall."
In [30]: print repl.format('Peter', 35, 1.88)
My name is Peter, I am 35 years old and 1.88 m tall.
2.1.2 Data Structures
A lightweight data structure is tuples. These are immutable collections of other objects and
are constucted by objects separated by commas – with or without parentheses.
In [31]: t1 = (a, b, st)
In [32]: t1
Out[32]: (3, 5.0, 'Listed Derivatives')
In [33]: type(t1)
Out[33]: tuple
In [34]: t2 = st, b, a
In [35]: t2
Out[35]: ('Listed Derivatives', 5.0, 3)
In [36]: type(t2)
Out[36]: tuple
21
Introduction to Python
Nested structures are also possible.
In [37]: t = (t1, t2)
In [38]: t
Out[38]: ((3, 5.0, 'Listed Derivatives'), ('Listed Derivatives', 5.0, 3))
In [39]: t[0][2] # take 3rd element of 1st element
Out[39]: 'Listed Derivatives'
List objects are mutable collections of other objects and are generally constructed by providing
a comma separated collection of objects in brackets.
In [40]: l = [a, b, st]
In [41]: l
Out[41]: [3, 5.0, 'Listed Derivatives']
In [42]: type(l)
Out[42]: list
In [43]: l.append(s.split()[3])
# append 4th word of string
In [44]: l
Out[44]: [3, 5.0, 'Listed Derivatives', 'Variance']
Sorting is a typical operation on list objects which can also be constructed using the list
constructor (here applied to a tuple object).
In [45]: l = list(('Z', 'Q', 'D', 'J', 'E', 'H', 5., a))
In [46]: l
Out[46]: ['Z', 'Q', 'D', 'J', 'E', 'H', 5.0, 3]
In [47]: l.sort()
# in-place sorting
In [48]: l
Out[48]: [3, 5.0, 'D', 'E', 'H', 'J', 'Q', 'Z']
Dictionary objects are so-called key-value stores and are generally constructed with curly
brackets.
22
LISTED VOLATILITY AND VARIANCE DERIVATIVES
In [49]: d = {'int_obj': a,
....:
'float_obj': b,
....:
'string_obj': st}
....:
In [50]: type(d)
Out[50]: dict
In [51]: d
Out[51]: {'float_obj': 5.0, 'int_obj': 3, 'string_obj': 'Listed Derivatives'}
In [52]: d['float_obj']
# look-up of value given key
Out[52]: 5.0
In [53]: d['long_obj'] = c / 10 ** 90
# adding new key value pair
In [54]: d
Out[54]:
{'float_obj': 5.0,
'int_obj': 3,
'long_obj': 10000000000L,
'string_obj': 'Listed Derivatives'}
Keys and values of a dictionary object can be retrieved as list objects.
In [55]: d.keys()
Out[55]: ['long_obj', 'int_obj', 'float_obj', 'string_obj']
In [56]: d.values()
Out[56]: [10000000000L, 3, 5.0, 'Listed Derivatives']
2.1.3 Control Structures
Iterations are very important operations in programming in general and financial analytics in
particular. Many Python objects are iterable which proves rather convenient in many circumstances. Consider the special list object constructor range.
In [57]: range(5) # all integers from zero to 5 excluded
Out[57]: [0, 1, 2, 3, 4]
In [58]: range(3, 15, 2) # start at 3, step with 2 until 15 excluded
Out[58]: [3, 5, 7, 9, 11, 13]
23
Introduction to Python
Such a list object constructor is often used in the context of a for loop.
In [59]: for i in range(5):
....:
....:
print i ** 2,
0 1 4 9 16
However, you can iterate over any sequence.
# iteration over list object
In [60]: for _ in l:
....:
print _,
....:
3 5.0 D E H J Q Z
# iteration over string object
In [61]: for c in st:
....:
....:
print c + '|',
L| i| s| t| e| d| | D| e| r| i| v| a| t| i| v| e| s|
while loops are similar to their counterparts in other languages.
In [62]: i = 0
# initialize counter
In [63]: while i < 5:
....:
print i ** 0.5, # output
....:
i += 1 # increase counter by 1
....:
0.0 1.0 1.41421356237 1.73205080757 2.0
The if-elif-else control structure is introduced below in the context of Python function
definitions.
2.1.4 Special Python Idioms
Python relies in many places on a number of special idioms. Let us start with a rather popular
one, the list comprehension.
In [64]: lc = [i ** 2 for i in range(10)]
In [65]: lc
Out[65]: [0, 1, 4, 9, 16, 25, 36, 49, 64, 81]
24
LISTED VOLATILITY AND VARIANCE DERIVATIVES
As the name suggests, the result is a list object.
In [66]: type(lc)
Out[66]: list
So-called lambda or anonymous functions are useful helpers in many places.
In [67]: f = lambda x: math.cos(x)
# returns cos of x
In [68]: f(5)
Out[68]: 0.28366218546322625
List comprehensions can be combined with lambda functions to achieve concise constructions
of list objects.
In [69]: lc = [f(x) for x in range(10)]
In [70]: lc
Out[70]:
[1.0,
0.5403023058681398,
-0.4161468365471424,
-0.9899924966004454,
-0.6536436208636119,
0.28366218546322625,
0.960170286650366,
0.7539022543433046,
-0.14550003380861354,
-0.9111302618846769]
However, there is an even more concise way of constructing the same list object – using functional programming approaches, in the following case with map.
In [71]: map(lambda x: math.cos(x), range(10))
Out[71]:
[1.0,
0.5403023058681398,
-0.4161468365471424,
-0.9899924966004454,
-0.6536436208636119,
0.28366218546322625,
0.960170286650366,
0.7539022543433046,
-0.14550003380861354,
-0.9111302618846769]
25
Introduction to Python
In general, one works with regular Python functions (as opposed to lambda functions) which
are constructed as follows:
In [72]: def f(x):
....:
return math.exp(x)
....:
The general construction looks like this:
In [73]: def f(*args): # multiple arguments
....:
for arg in args:
....:
....:
....:
print arg
return None # return result(s) (not necessary)
In [74]: f(l)
[3, 5.0, 'D', 'E', 'H', 'J', 'Q', 'Z']
Consider the following function definition which returns different values/strings based on an
if-elif-else control structure:
In [75]: import random
# import random number library
In [76]: a = random.randint(0, 999)
# draw random number between 0 and 999
In [77]: print "Random number is %d" % a
Random number is 627
In [78]: def number_decide(number):
....:
....:
....:
....:
....:
....:
if a < 10:
return "Number is single digit."
elif 10 <= a < 100:
return "Number is double digit."
else:
return "Number is triple digit."
....: number_decide(a)
....:
Out[78]: 'Number is triple digit.'
A specialty of Python is generator objects. One constructor for such objects that is commonly
used is xrange.
26
LISTED VOLATILITY AND VARIANCE DERIVATIVES
In [79]: g = xrange(10)
In [80]: type(g)
# object type
Out[80]: xrange
In [81]: g # object instance
Out[81]: xrange(10)
In [82]: for _ in g:
....:
print _,
....:
# integers are "generated" when needed
0 1 2 3 4 5 6 7 8 9
Generator objects can in many scenarios replace (typical) list objects and have the major advantage that they are in general much more memory efficient. Consider a financial algorithm that
requires 10 mn loops. Iterating over a list of integers from 0 to 9,999,999 is not efficient since
the algorithm (in general) does not need to have all these numbers available at the same time.
But this is what happens when using range for such a loop.
Consider the following construction of the respective list object containing all integers:
In [83]: %time r = range(10000000)
CPU times: user 87.4 ms, sys: 211 ms, total: 299 ms
Wall time: 299 ms
This object consumes 80 MB (!) of RAM (10 mn times 8 bytes).
In [84]: import sys
In [85]: sys.getsizeof(r)
Out[85]: 80000072
# size in bytes of object
On the other hand, consider the analogous construction based on a generator (xrange) object.
It is much, much faster since no memory has to be allocated, no list object has to be generated
up-front, etc.
In [86]: %time xr = xrange(10000000)
CPU times: user 3 us, sys: 1e+03 ns, total: 4 us
Wall time: 5.96 us
Memory consumption is also much, much more efficient – 40 bytes compared to 80 MB.
Introduction to Python
27
In [87]: sys.getsizeof(xr)
Out[87]: 40
However, in practical applications the two can be used often interchangeably such that one
should always resort to the more efficient alternative when possible. The following examples
calculate – using the functional programming operation reduce – the sum of all integers
from 0 to 9,999,999. Although in this case there is hardly a performance difference, the first
operation requires 80 MB of memory while the second might only require less than 100 bytes.
In [88]: %time reduce(lambda x, y: x + y, range(1000000))
CPU times: user 131 ms, sys: 15.9 ms, total: 147 ms
Wall time: 147 ms
Out[88]: 499999500000
In [89]: %time reduce(lambda x, y: x + y, xrange(1000000))
CPU times: user 118 ms, sys: 0 ns, total: 118 ms
Wall time: 117 ms
Out[89]: 499999500000
More Pythonic (and faster in general) is to calculate the sum using the built-in sum function
– in this case a significant performance advantage for the generator approach emerges.
In [90]: %timeit sum(range(1000000))
10 loops, best of 3: 26.6 ms per loop
In [91]: %timeit sum(xrange(1000000))
100 loops, best of 3: 12.9 ms per loop
There is also a way of indirectly constructing a generator object, i.e. by the use of parentheses.
The following code results in a generator object for the sine values of the numbers from 0 to
99:
In [92]: g = (math.sin(x) for x in xrange(100))
In [93]: g
Out[93]: at 0x2ab247901f50>
Such an object stores its internal state and yields the next value when the method next() is
called.
28
LISTED VOLATILITY AND VARIANCE DERIVATIVES
In [94]: g.next()
Out[94]: 0.0
In [95]: g.next()
Out[95]: 0.8414709848078965
Yet another way of constructing a generator object is by a definition style that resembles the
standard function definition closely. The difference is that instead of the return statement,
the yield statement is used.
In [96]: def g(start, end):
....:
while start <= end:
....:
yield start # yield "next" value
....:
....:
start += 1
# increase by one
Usage then might be as follows:
In [97]: go = g(15, 20)
In [98]: for _ in go:
....:
print _,
....:
15 16 17 18 19 20
2.2 NumPy
Many operations in computational finance take place over (large) arrays of numerical data.
NumPy is a Python library that allows the efficient handling of and operation on such data
structures. Although quite a mighty library with a wealth of functionality, it suffices for the
purposes of this book to cover the basics of NumPy.
In [99]: import numpy as np
The workhorse is the NumPy ndarray class which provides the data structure for ndimensional, immutable array objects. You can generate an ndarray object e.g. out of a
list object.
29
Introduction to Python
In [100]: a = np.array(range(24))
In [101]: a
Out[101]:
array([ 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16,
17, 18, 19, 20, 21, 22, 23])
The power of these objects lies in the management of n-dimensional data structures (e.g. matrices or cubes of data).
In [102]: b = a.reshape((4, 6))
In [103]: b
Out[103]:
array([[ 0,
1,
2,
3,
4,
5],
[ 6, 7, 8, 9, 10, 11],
[12, 13, 14, 15, 16, 17],
[18, 19, 20, 21, 22, 23]])
In [104]: c = a.reshape((2, 3, 4))
In [105]: c
Out[105]:
array([[[ 0, 1,
2,
3],
[ 4, 5, 6, 7],
[ 8, 9, 10, 11]],
[[12, 13, 14, 15],
[16, 17, 18, 19],
[20, 21, 22, 23]]])
So-called standard arrays (in contrast to e.g. structured arrays) have a single dtype (i.e.
NumPy data type). Consider the following operation which changes the dtype parameter of
the b object to float:
In [106]: b = np.array(b, dtype=np.float)
In [107]: b
Out[107]:
array([[
[
0.,
6.,
1.,
7.,
2.,
8.,
3., 4., 5.],
9., 10., 11.],
[ 12., 13., 14., 15., 16., 17.],
[ 18., 19., 20., 21., 22., 23.]])
30
LISTED VOLATILITY AND VARIANCE DERIVATIVES
A major strength of NumPy is vectorized operations.
In [108]: 2 * b
Out[108]:
array([[ 0., 2.,
4.,
6.,
8., 10.],
[ 12., 14., 16., 18., 20., 22.],
[ 24., 26., 28., 30., 32., 34.],
[ 36., 38., 40., 42., 44., 46.]])
In [109]: b ** 2
Out[109]:
array([[
[
0.,
36.,
1.,
49.,
4.,
64.,
9., 16., 25.],
81., 100., 121.],
[ 144., 169., 196., 225., 256., 289.],
[ 324., 361., 400., 441., 484., 529.]])
You can also pass ndarray objects to lambda or standard Python functions.
In [110]: f = lambda x: x ** 2 - 2 * x + 0.5
In [111]: f(a)
Out[111]:
array([
0.5, -0.5,
0.5,
3.5,
8.5, 15.5, 24.5, 35.5,
48.5, 63.5, 80.5, 99.5, 120.5, 143.5, 168.5, 195.5,
224.5, 255.5, 288.5, 323.5, 360.5, 399.5, 440.5, 483.5])
In many scenarios, only a (small) part of the data stored in an ndarray object is of interest.
NumPy supports basic and advanced slicing and other selection features.
In [112]: a[2:6] # 3rd to 6th element
Out[112]: array([2, 3, 4, 5])
In [113]: b[2, 4]
Out[113]: 16.0
# 3rd row, final (5th)
In [114]: b[1:3, 2:4]
Out[114]:
# middle square of numbers
array([[ 8., 9.],
[ 14., 15.]])
Boolean operations are also supported in many places.
31
Introduction to Python
# which numbers are larger than 10?
In [115]: b > 10
Out[115]:
array([[False, False, False, False, False, False],
[False, False, False, False, False, True],
[ True,
[ True,
True,
True,
True,
True,
True,
True,
True,
True,
True],
True]], dtype=bool)
# only those numbers (flat) that are larger than 10
In [116]: b[b > 10]
Out[116]:
array([ 11., 12., 13., 14., 15., 16., 17., 18., 19., 20., 21.,
22., 23.])
Furthermore, ndarray objects have multiple (convenience) methods already built in.
In [117]: a.sum()
Out[117]: 276
In [118]: b.mean()
Out[118]: 11.5
# sum of all elements
# mean of all elements
In [119]: b.mean(axis=0) # mean along 1st axis
Out[119]: array([ 9., 10., 11., 12., 13., 14.])
In [120]: b.mean(axis=1)
# mean along 2nd axis
Out[120]: array([ 2.5, 8.5, 14.5, 20.5])
In [121]: c.std()
# standard deviation for all elements
Out[121]: 6.9221865524317288
Similarly, there is a wealth of so-called universal functions that the NumPy library provides.
Universal in the sense that they can be applied in general to NumPy ndarray objects and to
standard numerical Python data types.
In [122]: np.sum(a)
# sum of all elements
Out[122]: 276
In [123]: np.mean(b, axis=0)
# mean along 1st axis
Out[123]: array([ 9., 10., 11., 12., 13., 14.])
In [124]: np.sin(b).round(2)
Out[124]:
array([[ 0.
# sine of all elements (rounded)
, 0.84, 0.91, 0.14, -0.76, -0.96],
[-0.28, 0.66, 0.99, 0.41, -0.54, -1.
],
32
LISTED VOLATILITY AND VARIANCE DERIVATIVES
[-0.54, 0.42, 0.99, 0.65, -0.29, -0.96],
[-0.75, 0.15, 0.91, 0.84, -0.01, -0.85]])
In [125]: np.sin(4.5) # sine of Python float object
Out[125]: -0.97753011766509701
However, you should be aware that applying NumPy universal functions to standard Python
data types generally comes with a significant performance burden.
In [126]: %time l = [np.sin(x) for x in xrange(100000)]
CPU times: user 188 ms, sys: 3.89 ms, total: 192 ms
Wall time: 192 ms
In [127]: import math
In [128]: %time l = [math.sin(x) for x in xrange(100000)]
CPU times: user 27.7 ms, sys: 0 ns, total: 27.7 ms
Wall time: 27.7 ms
Using the vectorized operations from NumPy is faster than both of the above alternatives which
result in list objects.
In [129]: %time np.sin(np.arange(100000))
CPU times: user 5.23 ms, sys: 0 ns, total: 5.23 ms
Wall time: 5.24 ms
Out[129]:
array([ 0.
, 0.84147098, 0.90929743, ..., 0.10563876,
0.89383946, 0.86024828])
Here, we use the ndarray object constructor arange which yields an ndarray object of
integers – below is a simple example:
In [130]: ai = np.arange(10)
In [131]: ai
Out[131]: array([0, 1, 2, 3, 4, 5, 6, 7, 8, 9])
In [132]: ai.dtype
Out[132]: dtype('int64')
Using this constructor, you can also generate ndarray objects with different dtype
attributes:
33
Introduction to Python
In [133]: af = np.arange(0.5, 9.5, 0.5)
# start, end, step size
In [134]: af
Out[134]:
array([ 0.5, 1. , 1.5, 2. , 2.5, 3. , 3.5, 4. , 4.5, 5. , 5.5,
6. , 6.5, 7. , 7.5, 8. , 8.5, 9. ])
In [135]: af.dtype
Out[135]: dtype('float64')
In this context the linspace operator is also useful, providing an ndarray object with
evenly spaced numbers.
In [136]: np.linspace(0, 10, 12) # start, end, number of elements
Out[136]:
array([ 0.
, 0.90909091, 1.81818182, 2.72727273,
3.63636364, 4.54545455, 5.45454545, 6.36363636,
7.27272727, 8.18181818, 9.09090909, 10.
])
In financial analytics one often needs (pseudo-)random numbers. NumPy provides many functions to sample from different distributions. Those needed in this book are the standard normal
distribution and the Poisson distribution. The respective functions are found in the sub-library
numpy.random.
In [137]: np.random.standard_normal(10)
Out[137]:
array([ 1.70352821, -1.30223997, -0.16846238, -0.33605234, 0.84842565,
-0.7012202 , 1.31232816, 1.34394536, -0.08358828, 1.53690444])
In [138]: np.random.poisson(0.5, 10)
Out[138]: array([0, 1, 0, 1, 0, 0, 1, 2, 2, 0])
Let us generate an ndarray object which is a bit more “realistic” and which we work with
in what follows:
In [139]: np.random.seed(1000)
# fix the rng seed value
In [140]: data = np.random.standard_normal(( 5, 100))
Although this is a slightly larger array one cannot expect that the 500 numbers are indeed
standard normally distributed in the sense that the first moment is 0 and the second moment is
1. However, at least this can be easily corrected.
34
LISTED VOLATILITY AND VARIANCE DERIVATIVES
In [141]: data.mean() # should be 0.0
Out[141]: -0.02714981205311327
In [142]: data.std() # should be 1.0
Out[142]: 1.0016799134894265
The correction is called moment matching and can be implemented with NumPy by vectorized
operations.
In [143]: data = data - data.mean()
In [144]: data = data / data.std()
In [145]: data.mean()
# correction for the 1st moment
# correction for the 2nd moment
# now really close to 0.0
Out[145]: 7.105427357601002e-18
In [146]: data.std()
# now really close to 1.0
Out[146]: 1.0
2.3 matplotlib
At this stage, it makes sense to introduce plotting with matplotlib, the plotting work horse in the
Python ecosystem. We use matplotlib (see http://matplotlib.org) with the settings of another
library throughout, namely seaborn (see http://stanford.edu/∼mwaskom/software/seaborn/) –
this results in a more modern plotting style.
In [147]: import matplotlib.pyplot as plt
In [148]: import seaborn as sns; sns.set()
# import main plotting library
# set seaborn standards
In [149]: import matplotlib
# set font to serif
In [150]: matplotlib.rcParams['font.family'] = 'serif'
A standard plot is the line plot. The result of the code below is shown as Figure 2.1.
In [151]: plt.figure(figsize=(10, 6));
In [152]: plt.plot(data.cumsum());
# size of figure
# cumulative sum over all elements
35
Introduction to Python
FIGURE 2.1
Standard line plot with matplotlib.
Multiple lines plots are also easy to generate (see Figure 2.2). The operator T stands for
the transpose of the ndarray object (“matrix”).
In [153]: plt.figure(figsize=(10, 6));
# size of figure
# plotting five cumulative sums as lines
In [154]: plt.plot(data.T.cumsum(axis=0), label='line');
In [155]: plt.legend(loc=0);
# legend in best location
In [156]: plt.xlabel('data point');
In [157]: plt.ylabel('value');
# x axis label
# y axis label
In [158]: plt.title('random series');
# figure title
Other important plotting types are histograms and bar charts. A histogram for all 500 values
of the data object is shown as Figure 2.3. In the code, the flatten() method is used to
generate a one-dimensional array from the two-dimensional one.
In [159]: plt.figure(figsize=(10, 6));
# size of figure
In [160]: plt.hist(data.flatten(), bins=30);
36
LISTED VOLATILITY AND VARIANCE DERIVATIVES
FIGURE 2.2
Multiple lines plot with matplotlib.
FIGURE 2.3
Histrogram with matplotlib.
37
Introduction to Python
FIGURE 2.4
Bar chart with matplotlib.
Finally, consider the bar chart presented in Figure 2.4.
In [161]: plt.figure(figsize=(10, 6));
# size of figure
In [162]: plt.bar(np.arange(1, 12) - 0.25, data[0, :11], width=0.5);
To conclude the introduction to matplotlib consider the ordinary least squares (OLS) regression of the sample data displayed in Figure 2.1. NumPy provides with the two functions
polyfit() and polyval() convenience functions to implement OLS based on simple monomials i.e. x, x2 , x3 , ..., xn . For illustration purposes consider linear, quadratic and
cubic OLS.
In [163]: x = np.arange(len(data.cumsum()))
In [164]: y = data.cumsum()
In [165]: rg1 = np.polyfit(x, y, 1)
# linear OLS
In [166]: rg2 = np.polyfit(x, y, 2)
# quadratic OLS
In [167]: rg3 = np.polyfit(x, y, 3)
# cubic OLS
Figure 2.5 illustrates the regression results graphically.
38
FIGURE 2.5
LISTED VOLATILITY AND VARIANCE DERIVATIVES
Linear, quadratic and cubic regression.
In [168]: plt.figure(figsize=(10, 6));
In [169]: plt.plot(x, y, 'r', label='data');
In [170]: plt.plot(x, np.polyval(rg1, x), 'b--', label='linear');
In [171]: plt.plot(x, np.polyval(rg2, x), 'b-.', label='quadratic');
In [172]: plt.plot(x, np.polyval(rg3, x), 'b:', label='cubic');
In [173]: plt.legend(loc=0);
2.4 pandas
pandas is a library with which you can manage and operate on time series data and other
tabular data structures. It allows implementation of even sophisticated data analytics tasks on
larger data sets. While the focus lies on in-memory operations, there are also multiple options
for out-of-memory (on-disk) operations. Although pandas provides a number of different data
structures, embodied by powerful classes, the structure most often used is the DataFrame
class which resembles a typical table of a relational (SQL) database and is used to manage,
for instance, financial time series data. This is what we focus on in this section.
39
Introduction to Python
2.4.1 pandas DataFrame class
In its most basic form, a DataFrame object is characterized by an index, column names and
tabular data. To make this more specific, consider the following sample data set:
In [174]: np.random.seed(1000)
In [175]: a = np.random.standard_normal(( 10, 3)).cumsum(axis=0)
Also, consider the following dates which shall be our index:
In [176]: index = ['2016-1-31', '2016-2-28', '2016-3-31',
.....:
'2016-4-30', '2016-5-31', '2016-6-30',
'2016-7-31', '2016-8-31', '2016-9-30',
'2016-10-31']
.....:
.....:
.....:
Finally, the column names:
In [177]: columns = ['no1', 'no2', 'no3']
The instantiation of a DataFrame object then looks as follows:
In [178]: import pandas as pd
In [179]: df = pd.DataFrame(a, index=index, columns=columns)
A look at the new object reveals its resemblance with a typical table from a relational database
(or e.g. an Excel spreadsheet).
In [180]: df
Out[180]:
2016-1-31
2016-2-28
no1
-0.804458
-0.160134
no2
0.320932
0.020135
no3
-0.025483
0.363992
2016-3-31
2016-4-30
2016-5-31
-0.267572
-0.732239
-1.928309
-0.459848
0.207433
-0.198527
0.959027
0.152912
-0.029466
2016-6-30
2016-7-31
2016-8-31
-1.825116
-0.553321
-0.652803
-0.336949
-1.323696
-0.916504
0.676227
0.341391
1.260779
2016-9-30 -0.340685
2016-10-31 -0.723832
0.616657
-0.206284
0.710605
2.310688
40
LISTED VOLATILITY AND VARIANCE DERIVATIVES
DataFrame objects have built in a multitude of basic, advanced and convenience methods, a
few of which are illustrated without much commentary below.
In [181]: df.head()
# first five rows
Out[181]:
no1
no2
no3
2016-1-31 -0.804458
0.320932
-0.025483
2016-2-28 -0.160134
0.020135
0.363992
2016-3-31 -0.267572 -0.459848
0.959027
2016-4-30 -0.732239
0.207433
0.152912
2016-5-31 -1.928309 -0.198527
-0.029466
In [182]: df.tail()
# last five rows
Out[182]:
no2
no3
2016-6-30
-1.825116 -0.336949
no1
0.676227
2016-7-31
-0.553321 -1.323696
0.341391
2016-8-31
-0.652803 -0.916504
1.260779
2016-9-30
-0.340685
0.616657
0.710605
2016-10-31 -0.723832 -0.206284
2.310688
In [183]: df.index
# index object
Out[183]:
Index([u'2016-1-31', u'2016-2-28', u'2016-3-31', u'2016-4-30', u'2016-5-31',
u'2016-6-30', u'2016-7-31', u'2016-8-31', u'2016-9-30', u'2016-10-31'],
dtype='object')
In [184]: df.columns
# column names
Out[184]: Index([u'no1', u'no2', u'no3'], dtype='object')
In [185]: df.info()
# meta information
Index: 10 entries, 2016-1-31 to 2016-10-31
Data columns (total 3 columns):
no1
10 non-null float64
no2
10 non-null float64
no3
10 non-null float64
dtypes: float64(3)
memory usage: 320.0+ bytes
In [186]: df.describe()
# typical statistics
Out[186]:
no1
no2
no3
count
10.000000
10.000000
10.000000
mean
0.672067
-0.798847
-0.227665
std
0.607430
0.578071
0.712430
min
-1.928309
-1.323696
-0.029466
41
Introduction to Python
25%
-0.786404
-0.429123
0.200031
50%
-0.688317
-0.202406
0.520109
75%
-0.393844
0.160609
0.896922
max
-0.160134
0.616657
2.310688
Numerical operations are in general as easy with DataFrame objects as with NumPy ndarray objects. They are also quite close in terms of syntax.
In [187]: df * 2
Out[187]:
# vectorized multiplication
no1
no2
no3
2016-1-31
2016-2-28
2016-3-31
-1.608917
-0.320269
-0.535144
0.641863
0.040270
-0.919696
-0.050966
0.727983
1.918054
2016-4-30
2016-5-31
2016-6-30
-1.464479
-3.856618
-3.650232
0.414866
-0.397054
-0.673898
0.305823
-0.058932
1.352453
2016-7-31
2016-8-31
-1.106642
-1.305605
-2.647393
-1.833009
0.682782
2.521557
2016-9-30
-0.681369
1.233314
1.421210
2016-10-31 -1.447664
-0.412568
4.621376
In [188]: df.std()
# standard deviation by column
Out[188]:
no1
0.607430
no2
0.578071
no3
0.712430
dtype: float64
In [189]: df.mean(axis=1)
Out[189]:
2016-1-31
2016-2-28
2016-3-31
-0.169670
0.074664
0.077202
2016-4-30
2016-5-31
-0.123965
-0.718767
2016-6-30
-0.495280
2016-7-31
2016-8-31
2016-9-30
-0.511875
-0.102843
0.328859
# mean by index value
2016-10-31
0.460191
dtype: float64
In [190]: np.mean(df)
Out[190]:
# mean via universal function
42
no1
LISTED VOLATILITY AND VARIANCE DERIVATIVES
-0.798847
no2
-0.227665
no3
0.672067
dtype: float64
Pieces of data can be looked up via different mechanisms.
In [191]: df['no2']
Out[191]:
2016-1-31
2016-2-28
2016-3-31
# 2nd column
0.320932
0.020135
-0.459848
2016-4-30
0.207433
2016-5-31
2016-6-30
-0.198527
-0.336949
2016-7-31
2016-8-31
2016-9-30
-1.323696
-0.916504
0.616657
2016-10-31
-0.206284
Name: no2, dtype: float64
In [192]: df.iloc[0]
# 1st row
Out[192]:
no1
-0.804458
no2
0.320932
no3
-0.025483
Name: 2016-1-31, dtype: float64
In [193]: df.iloc[2:4]
Out[193]:
no1
2016-3-31
2016-4-30
-0.267572
-0.732239
# 3rd & 4th row
no2
no3
-0.459848
0.207433
0.959027
0.152912
In [194]: df.iloc[2:4, 1]
Out[194]:
2016-3-31
2016-4-30
# 3rd & 4th row, 2nd column
-0.459848
0.207433
Name: no2, dtype: float64
In [195]: df.no3.iloc[3:7]
Out[195]:
2016-4-30
2016-5-31
2016-6-30
0.152912
-0.029466
0.676227
2016-7-31
0.341391
Name: no3, dtype: float64
# dot look-up for column name
43
Introduction to Python
In [196]: df.loc['2016-3-31']
Out[196]:
no1
no2
no3
# row given index value
-0.267572
-0.459848
0.959027
Name: 2016-3-31, dtype: float64
In [197]: df.loc['2016-5-31', 'no3']
Out[197]: -0.02946577492329111
In [198]: df['no1'] + 3 * df['no3']
Out[198]:
2016-1-31
-0.880907
2016-2-28
2016-3-31
2016-4-30
0.931841
2.609510
-0.273505
2016-5-31
2016-6-30
2016-7-31
-2.016706
0.203564
0.470852
2016-8-31
2016-9-30
2016-10-31
3.129533
1.791130
6.208233
# single data point
# vectorized arithmetic operations
dtype: float64
Data selections based on Boolean operations are also a strength of pandas.
In [199]: df['no3'] > 0.5
Out[199]:
2016-1-31
2016-2-28
False
False
2016-3-31
2016-4-30
2016-5-31
True
False
False
2016-6-30
2016-7-31
2016-8-31
True
False
True
2016-9-30
2016-10-31
True
True
Name: no3, dtype: bool
In [200]: df[df['no3'] > 0.5]
Out[200]:
2016-3-31
no1
-0.267572
no2
-0.459848
no3
0.959027
2016-6-30
-1.825116
-0.336949
0.676227
44
2016-8-31
LISTED VOLATILITY AND VARIANCE DERIVATIVES
-0.652803
-0.916504
1.260779
2016-9-30 -0.340685
2016-10-31 -0.723832
0.616657
-0.206284
0.710605
2.310688
In [201]: df[(df.no3 > 0.5) & (df.no2 > 0.25)]
Out[201]:
no1
no2
no3
2016-9-30
2016-10-31
-0.340685
-0.723832
0.616657
-0.206284
0.710605
2.310688
In [202]: df[df.index > '2016-4-30']
Out[202]:
no1
2016-5-31 -1.928309
2016-6-30 -1.825116
no2
-0.198527
-0.336949
no3
-0.029466
0.676227
2016-7-31 -0.553321
2016-8-31 -0.652803
2016-9-30 -0.340685
-1.323696
-0.916504
0.616657
0.341391
1.260779
0.710605
pandas is well integrated with the matplotlib library which makes it really convenient to
plot data stored in DataFrame objects. In general, a single method call does the trick (see
Figure 2.6).
In [203]: df.plot(figsize=(10, 6));
FIGURE 2.6
Line plot from pandas DataFrame.
45
Introduction to Python
FIGURE 2.7
Histograms from pandas DataFrame.
Histograms are also generated this way (see Figure 2.7). In both cases, pandas takes care of
the handling of the single columns and automatically generates single lines (with respective
legend entries) and generates respective sub-plots with three different histograms.
In [204]: df.hist(figsize=(10, 6));
2.4.2 Input-Output Operations
Another strength of pandas is the exporting and importing of data to and from diverse data
storage formats. Consider the case of comma separated value (CSV) files.
In [205]: df.to_csv('data.csv')
# exports to CSV file
Let us have a look at the file just saved with basic Python functionality.
In [206]: with open('data.csv') as f:
.....:
for l in f.readlines():
.....:
.....:
,no1,no2,no3
print l,
# open file
# iterate over all lines
# print line
46
LISTED VOLATILITY AND VARIANCE DERIVATIVES
2016-1-31,-0.804458303525,0.32093154709,-0.0254828804721
2016-2-28,-0.160134475098,0.0201348743028,0.363991673815
2016-3-31,-0.267571776789,-0.459848201058,0.959027175892
2016-4-30,-0.732239302984,0.20743310593,0.152911565449
2016-5-31,-1.92830913682,-0.19852705543,-0.0294657749233
2016-6-30,-1.82511624278,-0.336949044016,0.676226600036
2016-7-31,-0.553320966375,-1.32369637281,0.341391146824
2016-8-31,-0.652802664384,-0.916504272472,1.26077868603
2016-9-30,-0.340684654318,0.616656792886,0.710604821
2016-10-31,-0.723832065202,-0.206284170553,2.31068818906
Reading data from such files is also straightforward.
In [207]: from_csv = pd.read_csv('data.csv',
.....:
index_col=0,
# filename
# index column
parse_dates=True)
.....:
.....:
In [208]: from_csv.head()
Out[208]:
no1
2016-01-31 -0.804458
2016-02-28 -0.160134
no2
# date index
no3
0.320932 -0.025483
0.020135 0.363992
2016-03-31 -0.267572 -0.459848 0.959027
2016-04-30 -0.732239 0.207433 0.152912
2016-05-31 -1.928309 -0.198527 -0.029466
However, in general you would store DataFrame objects on disk in more efficient binary
formats like HDF5 (see http://hdfgroup.org). pandas in this case wraps the functionality of
the PyTables library (see http://pytables.org). The constructor function to be used is HDFStore().
In [209]: h5 = pd.HDFStore('data.h5', 'w')
In [210]: h5['df'] = df
# open for writing
# write object to database
In [211]: h5
Out[211]:
File path: data.h5
/df
frame
Data retrieval is as simple as writing.
(shape->[10,3])
47
Introduction to Python
In [212]: from_h5 = h5['df']
In [213]: h5.close()
# reading from database
# closing the database
In [214]: from_h5.tail()
Out[214]:
2016-6-30
no1
no2
-1.825116 -0.336949
no3
0.676227
2016-7-31
2016-8-31
2016-9-30
-0.553321 -1.323696
-0.652803 -0.916504
-0.340685 0.616657
0.341391
1.260779
0.710605
2016-10-31 -0.723832 -0.206284
2.310688
In [215]: !rm data.csv data.h5
# remove the objects from disk
2.4.3 Financial Analytics Examples
When it comes to financial data, there are useful data retrieval functions available that wrap
both the Yahoo! Finance and Google Finance financial data APIs. The following code reads
historical daily data for the S&P 500 index and the VIX volatility index:
In [216]: from pandas_datareader import data as web
In [217]: spx = web.DataReader('ˆGSPC', data_source='yahoo',
end='2015-12-31')
In [218]: vix = web.DataReader('ˆVIX', data_source='yahoo',
end='2015-12-31')
In [219]: spx.info()
DatetimeIndex: 1510 entries, 2010-01-04 to 2015-12-31
Data columns (total 6 columns):
Open
High
Low
1510 non-null float64
1510 non-null float64
1510 non-null float64
Close
Volume
Adj Close
1510 non-null float64
1510 non-null int64
1510 non-null float64
dtypes: float64(5), int64(1)
memory usage: 82.6 KB
In [220]: vix.info()
DatetimeIndex: 1510 entries, 2010-01-04 to 2015-12-31
48
LISTED VOLATILITY AND VARIANCE DERIVATIVES
Data columns (total 6 columns):
Open
1510 non-null float64
High
Low
Close
1510 non-null float64
1510 non-null float64
1510 non-null float64
Volume
1510 non-null int64
Adj Close
1510 non-null float64
dtypes: float64(5), int64(1)
memory usage: 82.6 KB
Let us combine the respective Close columns into a single DataFrame object. There are
multiple ways to accomplish this goal.
# construction via join
In [221]: spxvix = pd.DataFrame(spx['Close']).join(vix['Close'] ,
.....:
lsuffix='SPX', rsuffix='VIX')
.....:
In [222]: spxvix.info()
DatetimeIndex: 1510 entries, 2010-01-04 to 2015-12-31
Data columns (total 2 columns):
CloseSPX
1510 non-null float64
CloseVIX
1510 non-null float64
dtypes: float64(2)
memory usage: 35.4 KB
# construction via merge
In [223]: spxvix = pd.merge(pd.DataFrame(spx['Close']),
.....:
pd.DataFrame(vix['Close']),
.....:
left_index=True, # merge on left index
.....:
.....:
.....:
right_index=True, # merge on right index
suffixes=['SPX', 'VIX'])
In [224]: spxvix.info()
DatetimeIndex: 1510 entries, 2010-01-04 to 2015-12-31
Data columns (total 2 columns):
CloseSPX
1510 non-null float64
CloseVIX
1510 non-null float64
dtypes: float64(2)
memory usage: 35.4 KB
Introduction to Python
49
In a case like this, the approach via dictionary objects might be the best and most intuitive
way.
# construction via dictionary object
In [225]: spxvix = pd.DataFrame({'SPX': spx['Close'] ,
.....:
'VIX': vix['Close'] } ,
.....:
index=spx.index)
.....:
In [226]: spxvix.info()
DatetimeIndex: 1510 entries, 2010-01-04 to 2015-12-31
Data columns (total 2 columns):
SPX
1510 non-null float64
VIX
1510 non-null float64
dtypes: float64(2)
memory usage: 35.4 KB
Having available the merged data in a single object makes visual analysis straightforward (see
Figure 2.8).
In [227]: spxvix.plot(figsize=(10, 6), subplots=True, color='b');
FIGURE 2.8
Historical closing data for S&P 500 stock index and VIX volatility index.
50
LISTED VOLATILITY AND VARIANCE DERIVATIVES
pandas also allows vectorized operations on whole DataFrame objects. The following code
calculates the log returns over the two columns of the spxvix object simultaneously in vectorized fashion. The shift() method shifts the data set by the number of index values as
provided (in this particular case by one trading day).
In [228]: rets = np.log(spxvix / spxvix.shift(1))
In [229]: rets.head()
Out[229]:
SPX
VIX
Date
2010-01-04
2010-01-05
NaN
0.003111
NaN
-0.035038
2010-01-06
2010-01-07
2010-01-08
0.000545
0.003993
0.002878
-0.009868
-0.005233
-0.050024
There is one row of log returns missing at the very beginning. This row can be deleted via the
dropna() method.
In [230]: rets = rets.dropna()
In [231]: rets.head()
Out[231]:
SPX
VIX
Date
2010-01-05
0.003111
-0.035038
2010-01-06
2010-01-07
2010-01-08
0.000545
0.003993
0.002878
-0.009868
-0.005233
-0.050024
2010-01-11
0.001745
-0.032514
Consider the plot in Figure 2.9 showing the VIX log returns against the SPX log returns in a
scatter plot with a linear regression. It illustrates a strong negative correlation between the two
indexes. This is a central result that is replicated in chapter 4, Data Analysis and Strategies for
the EURO STOXX 50 stock index and the VSTOXX volatility index, respectively.
In [232]: rets.plot(kind='scatter', x='SPX', y='VIX',
.....:
style='.', figsize=(10, 6));
.....:
In [233]: rg = np.polyfit(rets['SPX'], rets['VIX'], 1)
In [234]: plt.plot(rets['SPX'], np.polyval(rg, rets['SPX']), 'r.-');
51
Introduction to Python
FIGURE 2.9
Daily log returns for S&P 500 stock index and VIX volatility index and regression line.
Having financial time series data stored in a pandas DataFrame object makes the calculation
of typical statistics straightforward.
In [235]: ret = rets.mean() * 252
# annualized return
In [236]: ret
Out[236]:
SPX
0.098532
VIX
-0.015992
dtype: float64
In [237]: vol = rets.std() * math.sqrt(252)
# annualized volatility
In [238]: vol
Out[238]:
SPX
0.159335
VIX
1.185408
dtype: float64
In [239]: (ret - 0.01) / vol
Out[239]:
SPX
0.555635
VIX
-0.021926
dtype:
float64
# Sharpe ratio with rf = 0.01
52
FIGURE 2.10
LISTED VOLATILITY AND VARIANCE DERIVATIVES
S&P 500 stock index and running maximum value.
The maximum drawdown, which we only calculate for the S&P 500 index, is a bit more
involved. For its calculation, we use the cummax() method which records the running, historical maximum of the time series up to a certain date. Consider the plot in Figure 2.10 which
shows the S&P 500 index and the running maximum.
In [240]: plt.figure(figsize=(10, 6));
In [241]: spxvix['SPX'].plot(label='S&P 500');
In [242]: spxvix['SPX'].cummax().plot(label='running maximum');
In [243]: plt.legend(loc=0);
The maximum drawdown is the largest difference between the running maximum and the
current index level – in our particular case it is 264.
In [244]: adrawdown = spxvix['SPX'].cummax() - spxvix['SPX']
In [245]: adrawdown.max()
Out[245]: 264.38000499999998
The relative maximum drawdown might sometimes be slightly more meaningful. Here it is a
drawdown of about 20%.
53
Introduction to Python
In [246]: rdrawdown = (spxvix['SPX'].cummax() - spxvix['SPX']) /
spxvix['SPX'].cummax()
In [247]: rdrawdown.max()
Out[247]: 0.19388242085950988
The longest drawdown period is calculated as follows. The code below selects all those data
points where the drawdown is zero. It then calculates the difference between two consecutive
index values (i.e. trading dates) for which the drawdown is zero and takes the maximum value.
Given the data set we are analyzing, the longest drawdown period is 301 days.
In [248]: temp = adrawdown[adrawdown == 0]
In [249]: periods_spx = (temp.index[1:].to_pydatetime()
.....:
- temp.index[:-1].to_pydatetime())
.....:
In [250]: periods_spx[50:60]
Out[250]:
# some selected data points
array([datetime.timedelta(67), datetime.timedelta(1),
datetime.timedelta(1), datetime.timedelta(1),
datetime.timedelta(301), datetime.timedelta(3),
datetime.timedelta(1), datetime.timedelta(2),
datetime.timedelta(12), datetime.timedelta(2)], dtype=object)
In [251]: max(periods_spx)
Out[251]: datetime.timedelta(301)
See Appendix C of Hilpisch (2014) for the handling of date-time information with Python,
NumPy and pandas.
2.5 CONCLUSIONS
This chapter introduces basic data types and structures as well as certain Python idioms needed
for analyses in later chapters of the book. In addition, NumPy and the ndarray class are introduced which allow the efficient handling of and operating on (numerical) data stored as arrays.
Some basic visualization techniques using the matplotlib library are also introduced. However,
working with pandas and its powerful DataFrame class for tabular and time series data makes
plotting a bit more convenient – in general only a single method call is needed. Using pandas
and the capabilities of the DataFrame class, the chapter also illustrates by means of some
basic financial examples how to implement typical interactive financial analytics tasks.
Listed Volatility and Variance Derivatives: A
Python-based Guide
By Dr. Yves J. Hilpisch
© 2017 Yves Hilpisch
CHAPTER
3
Model-Free Replication of Variance
3.1 INTRODUCTION
Although volatility derivatives, variance swaps and futures are considered second generation
derivatives (since they are defined on volatility and variance directly), some of the most elegant
and robust results of quantitative finance apply to these kinds of derivatives. In particular, it is
possible to statically replicate realized variance without using any kind of specific model. This,
for example, does not apply to most other important concepts in option pricing and trading,
such as implied volatility or delta hedging of options.
This chapter mainly covers the following topics:
spanning with options: Breeden and Litzenberger (1978) showed in the 1970s how to
replicate state-contingent payoffs (satisfying certain conditions) by using positions in
options
log contracts: an important piece in the replication of (realized) variance and the valuation
of variance swaps is the so-called log contract
static replication of variance: this relates to the central result of replicating realized
variance by a log contract as well as static positions in options
derivatives with constant dollar gamma: in the replication and valuation of variance
swaps, constant dollar gamma positions play a central role
practical replication of variance swaps: using the theoretical insights and results, the
chapter also illustrates the generation of constant dollar gamma positions and thus the
practical replication of variance swaps based on numerical examples
VSTOXX as volatility index: using the central result, the final section in this chapter
explains and justifies the definition of the VSTOXX volatility index (and the VIX index
to this end).
The theoretical parts of this chapter roughly follow the lines of chapter 11 of Gatheral (2006).
Since they are of importance for both volatility and variance derivatives we have placed them
up front and present them in a connected fashion.
55
56
LISTED VOLATILITY AND VARIANCE DERIVATIVES
3.2 SPANNING WITH OPTIONS
Breeden and Litzenberger (1978) use option prices to derive prices for elementary securities,
i.e. securities that pay exactly 1 unit of currency at a certain future date given a certain state
of the economy. In economics, Arrow-Debreu security is also a common term. The prices of
elementary securities are generally called state prices. Having state prices available, every
other contingent claim can be priced by multiplying the contingent claim’s state-dependent
payoff by the respective state prices.
Let t ≥ 0 denote the current date and T > t the future date of interest (e.g. maturity of a
derivative instrument). For simplicity, states of the economy should be distinguished by the
level of a stock index S only. Let p(ST , T; St , t) denote the state price given a stock index level
of St at t for a stock index level ST at T, then:
p(ST , T; St , t) =
𝜕 2 P(St , K, T) ||
𝜕 2 C(St , K, T) ||
=
|
|
2
|
|
𝜕K
𝜕K 2
|ST =K
|ST =K
Here, P and C represent prices (given by some pricing formula) of European put and call
options, respectively. Therefore, a state price can be derived by taking the second partial derivative of option pricing formula with respect to the strike of the option. Equipped with these state
prices, the value of a state-contingent claim with payoff function g(ST ) is
∞
Et (g(ST )|St ) =
g(K)p(K, T; St , t)dK
∫0
F
=
g(K)
∫0
∞
𝜕 2 P(St , K, T)
𝜕 2 C(St , K, T)
dK
+
g(K)
dK
∫F
𝜕K 2
𝜕K 2
with F being the T-forward price of the index at t (see Breeden and Litzenberger (1978)).
We now apply integration by parts. With u = u(x), v = v(x), du = u′ (x)dx, dv = v′ (x)dx,
integration by parts states that
∫
u(x)v′ (x)dx = u(x)v(x) −
∫
u′ (x)v(x)dx
Therefore, we get
Et (g(ST )|St ) = g(K)
F
𝜕P(St , K, T) ||F
𝜕P(St , K, T)
−
g′ (K)
dK
|
𝜕K
𝜕K
|0 ∫ 0
+ g(K)
∞
𝜕C(St , K, T) ||∞
𝜕C(St , K, T)
−
g′ (K)
dK
|
∫
𝜕K
𝜕K
|F
F
F
= g(F) −
∫0
g′ (K)
∞
𝜕P(St , K, T)
𝜕C(St , K, T)
g′ (K)
dK −
dK
∫F
𝜕K
𝜕K
57
Model-Free Replication of Variance
Applying integration by parts once again yields
Et (g(ST )|St ) = g(F) − g′ (K)P(St , K, T)||0 +
F
F
− g′ (K)C(St , K, T)||F +
F
∫0
g′′ (K)P(St , K, T)dK
∞
∞
= g(F) +
∫0
∫F
g′′ (K)C(St , K, T)dK
g′′ (K)P(St , K, T)dK +
∞
∫F
g′′ (K)C(St , K, T)dK
As is evident from this last equation, in this setting any twice continuously differentiable payoff
g due at T can be replicated by infinite strips of European put and call options maturing at T.
In other words, these options span the space of twice continuously differentiable payoffs.
3.3 LOG CONTRACTS
So-called log contracts are a kind of derivative instrument that plays an important role in the
valuation of variance swaps. Recall that a long position in a variance contract pays at maturity
the difference between the realized variance over the life time of the swap and an up-front
fixed variance strike.
S
Consider the payoff g(ST ) = log FT = log ST − log F. Then
g′ (ST ) =
1
ST
g′′ (ST ) = −
1
ST2
Valuing this contract by making use of the option spanning approach yields
(
)
F
∞
S |
1
1
F
E log T || St = log +
− 2 P(St , K, T)dK +
− 2 C(St , K, T)dK
∫F
F|
F ∫0
K
K
F
∞
dK
dK
P(St , K, T) 2 −
C(St , K, T) 2
=−
∫
∫0
K
K
F
As a consequence, the log contract can be replicated by (infinite strips of) European put and
call options on the underlying. Every option is weighted by the square of the strike.
3.4 STATIC REPLICATION OF REALIZED VARIANCE AND
VARIANCE SWAPS
Assume for simplicity zero short rates such that F = St . For t = 0, one has
log
ST
S
= log T
F
S0
T
=
d log(St )
∫0
T
=
∫0
T 𝜎 2 (S )
dSt
t
−
dt
∫0
St
2
58
LISTED VOLATILITY AND VARIANCE DERIVATIVES
Compare this result with equation 1.1. Such a comparison further illustrates the connection
between the (discrete) delta hedging of an option and (the valuation of) variance swaps. Above,
we use the fact that the total return over the interval [0, T] equals the integral over the marginal
returns of S for the same time interval. The second term in the last equation results from the
application of It̂o’s lemma. This term is equal to half of the total variance of S over the time
interval [0, T]. Taking the risk-neutral expectation of that expression gives
(
E
T
∫0
)
𝜎 (St )dt
2
(
S
= −2E log T
F
)
Combining the results with regard to the log contract replication and valuation with this last
insight shows that realized variance is given in a model-free manner through the prices of
European put and call options. This is due to the fact that realized variance can be replicated
by the use of a log contract.
3.5 CONSTANT DOLLAR GAMMA DERIVATIVES
AND PORTFOLIOS
Above, we establish the replication of realized variance by the log contract. The log contract
has, as a defining characteristic, a constant dollar gamma. The gamma of a derivative defined
on some underlying S with value ft (St ) is defined as
Γt =
𝜕 2 ft
𝜕St2
i.e. the second partial derivative of the pricing function with respect to the value of the underlying (assuming that the pricing function is indeed twice continuously differentiable).
Its dollar gamma is then defined by the product of the gamma and the square of the value
of the underlying (sometimes the factor 0.5 is added):
Γ$t ≡ Γt ⋅ St2
Omitting the time index, a constant dollar gamma implies, for some fixed value a, a gamma
of
Γ$ = Γ ⋅ S 2 ≡ a
a
⇔Γ= 2
S
We therefore have
𝜕2f
a
= 2
2
𝜕S
S
This partial differential equation has a solution of the form
f (S) = a log(S) + bS + c
59
Model-Free Replication of Variance
Indeed, the log contract fits trivially into the general solution by setting a = 1, b = 0, c = 0,
i.e.
f (S) = log(S)
illustrating its importance.
With 𝜎̂ 2 being the realized variance and 𝜎K2 being the fixed variance strike, a long position
in a variance swap (with notional of 1 currency unit), pays at expiry
𝜎̂ 2 − 𝜎K2
With this payoff, we get a replicating portfolio of
ft (St ) = 2 log(St ) + e−r(T−t) 𝜎K2
i.e. a = 2, b = 0, c = e−r(T−t) 𝜎K2 . A variance swap therefore also has a constant dollar gamma.
This insight is used in the next section.
3.6 PRACTICAL REPLICATION OF REALIZED VARIANCE
Consider the Black-Scholes-Merton (1973) model economy (no dividends) with the present
value of a European call option given by
C(S, K, t, T, r, 𝜎) = St ⋅ N(d1 ) − e−r(T−t) ⋅ K ⋅ N(d2 )
d
1 2
1
N(d) = √
e− 2 x dx
2𝜋 ∫−∞
(
)
S
𝜎2
log t + r +
(T − t)
K
2
d1 =
√
𝜎 T −t
(
)
St
𝜎2
log + r −
(T − t)
K
2
d2 =
√
𝜎 T −t
Here, St is the index level at date t, K is the option strike, T date of maturity (in year fractions),
r the constant, risk-less short rate and 𝜎 the instantaneous volatility.
The gamma of an option in this model is given as follows
Γt =
𝜕 2 Ct
𝜕St2
=
N′ (d1 )
√
St 𝜎 T − t
with
1 2
1
N′ (d) = √ e− 2 x
2𝜋
60
LISTED VOLATILITY AND VARIANCE DERIVATIVES
Our aim in this section is to build a portfolio of European call options with a constant dollar
gamma, i.e. ΓPortfolio ⋅ St2 ≡ a for some fixed value a.
The Python function dollar_gamma implements the dollar gamma formula for the
European call option.
In [1]: import math
In [2]: import numpy as np
In [3]: import scipy.stats as scs
In [4]: def dollar_gamma(St, K, t, T, r, sigma):
...:
''' Returns European call option dollar gamma. '''
...:
d1 = ((np.log(St / K) + (r + 0.5 * sigma ** 2) * (T - t))
...:
...:
/ sigma * math.sqrt(T- t))
gamma = scs.norm.pdf(d1) / (St * sigma * math.sqrt(T - t))
...:
...:
return gamma * St ** 2
Let us parametrize the financial model for t = 0, leaving the values for the initial index level
S0 and for the option strike K undefined for the moment.
In [5]: t
=
0.0
# current date in year fractions
In [6]: T
=
1.0
# maturity in year fractions
In [7]: r
=
0.01
In [8]: sigma
=
# constant risk-less short rate
0.2
# instantanous volatility
Next, we can calculate the dollar gamma values for different strikes of the European call
options over a range, respectively, of initial values for the index level.
In [9]: import pandas as pd
In [10]: import matplotlib.pyplot as plt
In [11]: import seaborn as sns; sns.set()
In [12]: import matplotlib
In [13]: matplotlib.rcParams['font.family'] = 'serif'
Model-Free Replication of Variance
61
In [14]: gammas = pd.DataFrame()
# 300 data points over the range form 0 to 300
In [15]: s_range = np.linspace(0.0001, 300, 300)
In [16]: strike_range = range(25, 226, 25)
In [17]: for K in strike_range:
....:
gammas['K=%d' % K] = dollar_gamma(s_range, K, t, T, r, sigma)
....:
For every strike level gamma values for 300 different initial values of the stock index have
been calculated and collected.
In [18]: gammas.info()
RangeIndex: 300 entries, 0 to 299
Data columns (total 9 columns):
K=25
300 non-null float64
K=50
300 non-null float64
K=75
K=100
K=125
300 non-null float64
300 non-null float64
300 non-null float64
K=150
K=175
K=200
300 non-null float64
300 non-null float64
300 non-null float64
K=225
300 non-null float64
dtypes: float64(9)
memory usage: 21.2 KB
Having the data stored in a pandas DataFrame object, the results are easily visualized. From
Figure 3.1, you can see that dollar gamma is more pronounced the higher the strike of the
option. A doubling of the strike leads to a doubling of the maximum dollar gamma value
which is always achieved at the ATM level (see strikes 25, 50, 100, 200).
In [19]: gammas.plot(figsize=(10, 5));
In [20]: plt.xlabel('index level');
In [21]: plt.ylabel('dollar gamma');
Let us check what happens when we add all dollar gamma values up by simply composing a
portfolio in which every option has weight 1 or equal weight.
62
LISTED VOLATILITY AND VARIANCE DERIVATIVES
FIGURE 3.1
Dollar gamma values different strikes and 300 different initial stock index values.
In [22]: gammas['sum'] = gammas.sum(axis=1)
In [23]: gammas.plot(figsize=(10, 5));
In [24]: plt.xlabel('index level');
In [25]: plt.ylabel('dollar gamma');
Obviously, given Figure 3.2 the dollar gamma of the portfolio of equally weighted options is
all but constant. Let us try a different weighting scheme that attaches a higher weight to smaller
strikes and a lower weight to higher strikes. To this end, we divide all dollar gamma values by
the strike K. This brings all maximum dollar gamma values in line (to two in this case).
In [26]: gammas_k = pd.DataFrame()
In [27]: for K in strike_range:
....:
gammas_k['K=%d' % K] = dollar_gamma(s_range, K, t, T, r, sigma) / K
....:
In [28]: gammas_k['sum'] = gammas_k.sum(axis=1)
In [29]: gammas_k.plot(figsize=(10, 5));
In [30]: plt.xlabel('index level');
In [31]: plt.ylabel('$K$ weighted dollar gamma');
Model-Free Replication of Variance
FIGURE 3.2
63
Dollar gamma values added over the range of 300 different initial index levels.
According to Figure 3.3 this seems to be slightly better in the sense that although we still do
not have a constant dollar gamma we do at least have a range where dollar gamma is linear
(mainly between strikes of 75 and 150). Therefore, let us weight the dollar gammas by the
square of the strike (something we have already seen in the replication result for log contract). In this case the highest dollar gamma values are observed for the lowest strikes and
vice versa.
FIGURE 3.3
Strike-weighted, added dollar gamma values.
64
LISTED VOLATILITY AND VARIANCE DERIVATIVES
In [32]: gammas_k2 = pd.DataFrame()
In [33]: for K in strike_range:
....:
gammas_k2['K=%d' % K] = dollar_gamma(s_range, K, t, T, r,
sigma) / K ** 2
....:
In [34]: gammas_k2['sum'] = gammas_k2.sum(axis=1)
In [35]: gammas_k2.plot(figsize=(10, 5));
In [36]: plt.xlabel('index level');
In [37]: plt.ylabel('$Kˆ2$ weighted dollar gamma');
As Figure 3.4 shows, this approach finally yields a constant dollar gamma value between
strikes of 75 and 150 at least. Let us have a final look at a more dense grid of option strikes
since the theoretical result is based on infinite strips of options. The graphical output is shown
in Figure 3.5.
# more dense strike range
In [38]: strike_range = range(10, 350, 5)
In [39]: gammas_k2 = pd.DataFrame()
In [40]: for K in strike_range:
....:
gammas_k2['K=%d' % K] = dollar_gamma(s_range, K, t, T, r,
sigma) / K ** 2
....:
In [41]: gammas_k2 ['sum'] = gammas_k2.sum(axis=1)
In [42]: gammas_k2.plot(figsize=(10, 5), legend=False);
In [43]: plt.xlabel('index level');
In [44]: plt.ylabel('$Kˆ2$ weighted dollar gamma');
This numerical example shows a constant dollar gamma over a much wider range from
about 25 to beyond 200. This further supports the theoretical result and the replication
approach.
Model-Free Replication of Variance
FIGURE 3.4
Squared strike-weighted, added dollar gamma values.
FIGURE 3.5
Strike-weighted, added dollar gamma values.
65
3.7 VSTOXX AS VOLATILITY INDEX
The VSTOXX volatility index measures the implied total variance across all options written
on the EURO STOXX 50 stock index for a given time-to-maturity. Another interpretation is
that the VSTOXX gives the fair variance swap rate for a variance swap with the respective
maturity. The major index with symbol V2TX has a fixed time-to-maturity of 30 days and is
calculated by the interpolation of two sub-indexes. Sub-indexes are calculated for a number of
fixed maturity dates.
66
LISTED VOLATILITY AND VARIANCE DERIVATIVES
Assume now the following setting. The discounted, i.e. time t, prices Ci , i = 0, ..., n, of a
series of European call options on the EURO STOXX 50 stock index are given, with fixed timeto-maturity T and strike prices Ki , i = 0, ..., n, as well as the discounted prices Pi , i = 0, ..., n,
of a series of European put options with the same time-to-maturity and strike prices. Let us
further hold Ki < Ki+1 for all i ∈ {0, ...., n − 1}.
The value of the VSTOXX (sub-)index V at t = 0 is defined by
V ≡ 100 ⋅
√
𝜎̂ 2
where
𝜎̂ 2 ≡
n
2 ∑ ΔKi rT
1
e Mi −
T i=0 Ki 2
T
(
)2
F
−1
K∗
with
⎧K − K ,
⎪ K 1 −K 0
ΔKi = ⎨ i+1 i−1 ,
2
⎪K − K ,
n
n−1
⎩
for i = 0
for i = 1, ..., n − 1
for i = n
F = Kj + erT |Cj − Pj |, where j =
K∗ =
max
min {|Ci − Pi |}
i∈{0,...,n}
{Ki < F}
Ki ,i∈{0,...,n}
⎧P ,
⎪ P i−C
Mi = ⎨ i i ,
2
⎪ Ci ,
⎩
for Ki < K∗
for Ki = K∗
for Ki > K∗
and r the constant risk-free short rate appropriate for time-to-maturity T.
We are aiming to show that the defining equation for 𝜎̂ 2 is indeed a valid approximation
for total variance. To this end, we combine the above equations for the log contract and the
total variance to obtain
(
𝜎̂ = −2 ⋅ −
2
⇔
F
∫0
dK
P(St , K, T) 2 −
∫F
K
F
𝜎̂ 2 T
dK
=
P(St , K, T) 2 +
∫0
∫F
2erT
K
K∗
=
F
+
∫K∗
∫0
∞
C(St , K, T)
dK
K2
dK
dK
+
C(St , K, T) 2
K 2 ∫K∗
K
[
] dK
P(St , K, T) − C(St , K, T)
∞
=
dK
C(St , K, T) 2
K
∞
P(St , K, T)
∫0
∞
F
M(St , K, T)
K2
dK
dK
+
e−rT [K − F] 2
2
∫
K
K
K∗
)
⋅ erT
67
Model-Free Replication of Variance
The last term follows from the observation that only the call option is in-the-money, i.e.
e−rT [F − K] > 0, given the integration boundaries. Then
𝜎̂ 2 =
2
T ∫0
∞
2
T ∫0
∞
=
F
erT M(St , K, T)
dK 2 1
+
[K − F] dK
K 2 T K∗2 ∫K∗
erT M(St , K, T)
2
dK 1 (K∗ − F)
−
K2 T
K∗2
Note that
K∗2 − 2K∗ F + F 2
(K∗ − F)2
=
K∗2
K∗2
F 2 2F
−
+1
K∗2 K∗
)2
(
F
−1
=
K∗
=
A possible discretization for the integral then is
𝜎̂ 2 ≈
n
2 ∑ ΔKi rT
1
e Mi −
T i=0 K 2
T
i
(
)2
F
−1
K∗
giving the defining equation for 𝜎̂ 2 .
The calculation of implied volatilities, or variances, naturally relies on some option pricing
model, like that one of Black-Scholes-Merton (1973). By contrast, the VSTOXX and VIX
volatility index calculation only takes as input market-observed options prices. This is possible
since realized variance at a given future date can be replicated by strips of European call and
put prices maturing at the same date. The realized volatility can then be extracted by taking
the square root of the realized variance. The calculation of the VSTOXX and VIX avoids a
“model-bias” and uses standard market practice for the valuation of important variance related
instruments, like variance swaps.
3.8 CONCLUSIONS
This chapter introduces the elegant theory of the model-free replication of realized variance
and variance swaps which dates back at least to Breeden and Litzenberger (1978). The role of
log contracts and constant dollar gamma positions is discussed. Numerical examples also illustrate how to construct constant dollar gamma options positions in practice. Finally, based on
this theory the definition of the VSTOXX and VIX volatility indexes is presented and justified.
Subsequent chapters draw in different ways on these cornerstones in the theory of volatility
and variance modeling, measuring and trading.
PART
Two
Listed Volatility Derivatives
Listed Volatility and Variance Derivatives: A
Python-based Guide
By Dr. Yves J. Hilpisch
© 2017 Yves Hilpisch
CHAPTER
4
Data Analysis and Strategies
4.1 INTRODUCTION
This chapter is about the analysis of data and investment strategies related to the EURO
STOXX 50 and VSTOXX indexes. It uses public data sources (“open data”) and draws heavily
on the capabilities of the Python library pandas for data analytics.
The chapter has two major goals. First, it reproduces the stylized fact that stock indexes
and volatility indexes in general are negatively correlated. This suggests that (products based
on) volatility indexes are a means to hedge market risk resulting from stock indexes. The question, however, is how to best exploit the negative correlation in asset allocation terms. Therefore, the second goal is to illustrate the benefits for equity investors resulting from constant
proportion investment strategies involving a volatility index like the VSTOXX. For simplicity, the respective analysis assumes that a direct investment in the VSTOXX is possible. This
replicates results as found, for example, in the study by Guobuzaite and Martellini (2012).
4.2 RETRIEVING BASE DATA
This section shows how to retrieve and store historical daily closing data for the EURO STOXX
50 index and the VSTOXX volatility index. We mainly work with pandas in the following:
In [1]: import numpy as np
In [2]: import pandas as pd
In [3]: path = './source/data/'
# path to data folder
4.2.1 EURO STOXX 50 Data
On the website http://stoxx.com of the index provider STOXX Limited, you find text files
containing historical closing data for the EURO STOXX 50 index and others. Such data files
are found in this folder:
In [4]: source = 'https://www.stoxx.com/document/Indices/Current/HistoricalData/'
71
72
LISTED VOLATILITY AND VARIANCE DERIVATIVES
The respective file for the EURO STOXX 50 index has the following name and is updated
daily after closing.
In [5]: es_url = source + 'hbrbcpe.txt'
Let us inspect the first few rows of the data file directly, i.e. without pandas. We use the requests
library to read the file from the web source and to print the first 1,000 characters of the string
object returned.
In [6]: import requests
In [7]: print requests.get(es_url).text[:1000].replace(' ', ' ')
PriceIndices–EUROCurrency
Date;Blue–Chip;Blue–Chip;Broad;Broad;ExUK;ExEuroZone;Blue–Chip;Broad
;Europe;Euro–Zone;Europe;Euro–Zone;;;Nordic;Nordic
;SX5P;SX5E;SXXP;SXXE;SXXF;SXXA;DK5F;DKXF
31.12.1986;775.00;900.82;82.76;98.58;98.06;69.06;645.26;65.56
01.01.1987;775.00;900.82;82.76;98.58;98.06;69.06;645.26;65.56
02.01.1987;770.89;891.78;82.57;97.80;97.43;69.37;647.62;65.81
05.01.1987;771.89;898.33;82.82;98.60;98.19;69.16;649.94;65.82
06.01.1987;775.92;902.32;83.28;99.19;98.83;69.50;652.49;66.06
07.01.1987;781.21;899.15;83.78;98.96;98.62;70.59;651.97;66.20
08.01.1987;777.62;887.37;83.52;97.87;97.68;71.01;645.57;65.62
09.01.1987;769.80;868.31;83.03;96.31;96.22;71.40;638.03;65.14
The file changes the format in which data rows are presented (at least at the time of writing)
such that we need to use a little trick to import the data correctly with pandas. Namely, we
have to add an additional column which we call DEL.
# new column names (without white space)
# adding column 'DEL' -- to be deleted after parsing
In [8]: columns = ['Date', 'SX5P', 'SX5E', 'SXXP', 'SXXE',
...:
'SXXF', 'SXXA', 'DK5F', 'DKXF', 'DEL']
...:
We also skip the first four rows which we do not need and set a few other parameters for the
importing procedure with pandas.
In [9]: es = pd.read_csv(es_url, # url/filename
...:
index_col=0, # index column (dates)
...:
parse_dates=True, # parse date information
...:
...:
dayfirst=True, # day before month
header=None, # ignore header information
73
Data Analysis and Strategies
...:
...:
skiprows=4, # ignore first 4 rows
names=columns, # use custom column names
...:
...:
sep=';')
# separator character
The additional helper column can be deleted after the import because it is no longer needed
(and empty in any case).
In [10]: del es['DEL']
# deletes helper column
Let us inspect the first few rows of the DataFrame object. Given the raw data from above,
we seem to have done everything right.
In [11]: es = es[es.index <= '2015-12-31']
# data until the end of 2015
In [12]: es.tail()
Out[12]:
SX5P
SX5E
SXXP
SXXE
SXXF
SXXA
DK5F
DKXF
Date
2015-12-24 3108.11 3284.47 366.28 346.05 433.43 375.39 9931.72 614.38
2015-12-28 3093.61 3256.49 364.49 343.54 431.26 374.32 9873.94 611.58
2015-12-29 3139.28 3314.28 369.68 349.29 438.43 378.86 10023.66 620.66
2015-12-30 3118.07 3287.98 367.70 347.02 435.82 377.20
2015-12-31 3100.26 3267.52 365.81 345.16 433.81 375.34
9956.22 617.48
9978.59 618.73
The single time series starts at the end of 1986 and goes to the last available trading day or in
our case the final trading day of 2015.
In [13]: es.info()
DatetimeIndex: 7476 entries, 1986-12-31 to 2015-12-31
Data columns (total 8 columns):
SX5P
7476 non–null float64
SX5E
SXXP
SXXE
7476 non–null float64
7476 non–null float64
7476 non–null float64
SXXF
SXXA
7476 non–null float64
7476 non–null float64
DK5F
7476 non–null float64
DKXF
7476 non–null float64
dtypes: float64(8)
memory usage: 525.7 KB
74
FIGURE 4.1
LISTED VOLATILITY AND VARIANCE DERIVATIVES
Historical EURO STOXX 50 index levels.
Before going on to the corresponding procedure for the VSTOXX data, the following visualizes the historical closing values for the EURO STOXX 50 index, i.e. for symbol SX5E (see
Figure 4.1):
In [14]: import seaborn as sns; sns.set()
In [15]: import matplotlib
In [16]: matplotlib.rcParams['font.family'] = 'serif'
# set serif font
In [17]: es['SX5E'].plot(grid=True, figsize=(10, 6));
4.2.2 VSTOXX Data
Reading the data from the same source for the VSTOXX index is slightly more straightforward
since the respective data file does not change its format.
In [18]: vs_url = source + 'h_vstoxx.txt'
In [19]: print(requests.get(vs_url).text[160:1040])
Date,V2TX,V6I1,V6I2,V6I3,V6I4,V6I5,V6I6,V6I7,V6I8
04.01.1999,18.2033,21.2458,17.5555,31.2179,33.3124,33.7327,33.2232,31.8535,23.8209
05.01.1999,29.6912,36.6400,28.4274,32.6922,33.7326,33.1724,32.8457,32.2904,25.0532
06.01.1999,25.1670,25.4107,25.1351,32.2186,32.6459,31.9673,32.9260,33.2871,26.0107
75
Data Analysis and Strategies
07.01.1999,32.5205,35.4410,32.2004,36.1265,34.5150,33.1095,33.2843,33.7269,26.2205
08.01.1999,33.2296,35.8846,33.0020,36.0813,36.3964,33.4658,33.4837,33.9227,26.3672
11.01.1999,36.8411,46.9742,36.4643,39.9139,38.0755,34.6165,34.4576,34.6615,26.6732
12.01.1999,37.5664,48.6277,37.2998,40.5525,38.7981,35.0575,37.5290,36.5965,28.0273
13.01.1999,39.7373,55.2934,39.5218,43.0083,40.1301,36.0443,38.3889,37.2219,28.1443
14.01.1999,39.1373,NA,39.9544,41.7597,40.5913,36.3330,38.6781,37.4249,28.1515
15.01.1999,38.6741,NA,39.0602,40.1232,39.2451,35.9745,37.7796,36.7660,28.0793
The parametrization for pandas to import the data to a DataFrame object is as follows:
In [20]: vs = pd.read_csv(vs_url,
# url/filename
....:
....:
....:
index_col=0, # index column (dates)
parse_dates=True, # parse date information
dayfirst=True, # day before month
....:
....:
header=2)
# header/column names
In this case, data is available from the beginning of 1999 until the last available trading day.
However, we again select only data until the last trading day of 2015.
In [21]: vs = vs[vs.index <= '2015-12-31']
In [22]: vs.info()
DatetimeIndex: 4327 entries, 1999-01-04 to 2015-12-30
Data columns (total 9 columns):
V2TX
4327 non–null float64
V6I1
3878 non–null float64
V6I2
V6I3
4327 non–null float64
4267 non–null float64
V6I4
V6I5
V6I6
4327 non–null float64
4327 non–null float64
4310 non–null float64
V6I7
4327 non–null float64
V6I8
4313 non–null float64
dtypes: float64(9)
memory usage: 338.0 KB
Figure 4.2 visualizes the times series data for the main volatility index with symbol V2TX.
Inspection of the figure reveals that the volatility index increases and even spikes when there
are crises in the markets – which is best seen towards the end of 2008.
In [23]: vs['V2TX'].plot(grid=True, figsize=(10, 6));
76
LISTED VOLATILITY AND VARIANCE DERIVATIVES
FIGURE 4.2
Historical VSTOXX index levels.
4.2.3 Combining the Data Sets
For what follows, we only need the time series data for the major indexes with symbols SX5E
and V2TX and only for those dates where data is available for both time series. To this end,
the following code generates a new DataFrame object out of the two original ones:
In [24]: import datetime as dt
In [25]: data = pd.DataFrame({'EUROSTOXX':
....:
es['SX5E'][es.index > dt.datetime(1999, 1, 1)],
....:
....:
....:
'VSTOXX':
vs['V2TX'][vs.index > dt.datetime(1999, 1, 1)]})
In [26]: data = data.dropna()
# deletes those rows with NaN values
The code below provides an overview of the new data set and excerpts from all three
DataFrame objects:
In [27]: data.info()
DatetimeIndex: 4326 entries, 1999-01-04 to 2015-12-30
Data columns (total 2 columns):
EUROSTOXX
4326 non–null float64
VSTOXX
4326 non–null float64
77
Data Analysis and Strategies
dtypes: float64(2)
memory usage: 101.4 KB
In [28]: es.iloc[–10:]
Out[28]:
SX5P
SX5E
SXXP
SXXE
SXXF
SXXA
DK5F
DKXF
Date
2015-12-17 3095.83 3306.47 364.90 347.46 433.96 371.43 9857.04 610.15
2015-12-18 3065.07 3260.72 361.23 343.23 429.32 368.38
2015-12-21 3022.51 3213.01 357.15 338.55 424.21 364.99
2015-12-22 3019.26 3214.32 356.87 338.77 423.95 364.23
9722.97 602.49
9707.52 601.18
9691.72 599.53
2015-12-23 3109.23 3286.68 366.39 346.14 433.58 375.53
2015-12-24 3108.11 3284.47 366.28 346.05 433.43 375.39
2015-12-28 3093.61 3256.49 364.49 343.54 431.26 374.32
9927.33 614.12
9931.72 614.38
9873.94 611.58
2015-12-29 3139.28 3314.28 369.68 349.29 438.43 378.86 10023.66 620.66
2015-12-30 3118.07 3287.98 367.70 347.02 435.82 377.20 9956.22 617.48
2015-12-31 3100.26 3267.52 365.81 345.16 433.81 375.34 9978.59 618.73
In [29]: np.round(vs, 2).iloc[-10:]
Out[29]:
V2TX
V6I1
V6I2
V6I3
V6I4
V6I5
V6I6
V6I7
V6I8
Date
2015-12-15 27.46 41.71 27.41 26.63 26.72 26.66 26.41 26.36 27.44
2015-12-16 25.44 49.12 25.44 25.21 25.34 25.57 25.54 25.77 26.65
2015-12-17 22.58
NaN 22.50 23.47 23.94 24.80 25.01 25.34 26.24
2015-12-18 23.90
NaN 23.76 24.78 25.06 25.80 25.75 25.91 26.54
2015-12-21 23.88 23.56 24.62 24.99 25.80 25.87 26.04 26.64
NaN
2015-12-22 22.54 21.92 23.64 24.14 25.25 25.31 25.64 26.04 26.05
2015-12-23 20.25 19.42 21.45 22.24 24.01 24.46 24.84 25.65 25.44
2015-12-28 22.45 21.43 23.08 23.65 24.70 24.91 25.25 25.46 25.34
2015-12-29 21.61 20.56 22.14 22.87 24.31 24.59 25.06 25.65 25.26
2015-12-30 22.17 21.13 22.64 23.33 24.64 24.72 25.15 25.48 25.16
In [30]: data.iloc[-10:]
Out[30]:
EUROSTOXX
VSTOXX
Date
2015-12-15
2015-12-16
3241.51 27.4591
3246.78 25.4362
2015-12-17
2015-12-18
2015-12-21
3306.47 22.5761
3260.72 23.8989
3213.01 23.8836
2015-12-22
3214.32 22.5361
2015-12-23
2015-12-28
3286.68 20.2504
3256.49 22.4544
2015-12-29
2015-12-30
3314.28 21.6067
3287.98 22.1745
78
LISTED VOLATILITY AND VARIANCE DERIVATIVES
4.2.4 Saving the Data
Finally, the DataFrame objects are saved to a HDFStore object on disk for later use. Something not really needed when an internet connection is available, but helpful when it is not.
In [31]: h5 = pd.HDFStore(path + 'es_vs_data.h5', 'w')
In [32]: h5['es'] = es
In [33]: h5['vs'] = vs
In [34]: h5['data'] = data
In [35]: h5
Out[35]:
File path: ./source/data/es_vs_data.h5
/data
/es
/vs
frame
frame
frame
(shape->[4326,2])
(shape->[7476,8])
(shape->[4327,9])
In [36]: h5.close()
4.3 BASIC DATA ANALYSIS
To illustrate how to read the data from the HDFStore object, consider the following code
which is almost “symmetric” to the code that writes the data to disk:
In [37]: h5 = pd. HDFStore(path + 'es_vs_data.h5', 'r')
In [38]: h5
Out[38]:
File path: ./source/data/es_vs_data.h5
/data
frame
(shape->[4326,2])
/es
/vs
frame
frame
In [39]: es = h5['es']
In [40]: vs = h5['vs']
In [41]: data = h5['data']
In [42]: h5.close()
(shape->[7476,8])
(shape->[4327,9])
79
Data Analysis and Strategies
As seen in chapter 2, Introduction to Python, pandas provides a wealth of options to analyze
data stored in DataFrame objects. A method which is sometimes helpful is describe()
which provides selected meta statistics for the single data sub-sets stored in DataFrame
columns.
In [43]: data.describe()
Out[43]:
count
EUROSTOXX
VSTOXX
4326.000000 4326.000000
mean
std
min
3265.935767
770.513845
1809.980000
25.209891
9.670981
11.596600
25%
50%
2692.385000
3089.965000
18.670200
23.162700
75%
max
3725.357500
5464.430000
28.310725
87.512700
Sub-plots of multiple data sub-sets are easily generated with pandas. The result of the following
plotting code is found as Figure 4.3:
In [44]: data.plot(subplots=True, # generate sub-plots per column
....:
figsize=(10, 6), # sizing of the figure
....:
color='blue'); # color to plot the data
....:
FIGURE 4.3
Historical EURO STOXX 50 and VSTOXX index levels.
80
LISTED VOLATILITY AND VARIANCE DERIVATIVES
FIGURE 4.4
Histograms of historical absolute daily differences of EURO STOXX 50 and
VSTOXX.
Similarly, you can visualize the absolute differences (over time) by using the diff method
and plotting them as a histogram (see Figure 4.4).
In [45]: data.diff().hist(figsize=(10, 6), # figure sizing
....:
color='blue', # color for the plotted data
....:
bins=100); # number of bins to be used
....:
In a similar spirit, we can calculate the relative or percentage changes for the time series data.
In [46]: data.pct_change().ix[-10:]
Out[46]:
EUROSTOXX
VSTOXX
Date
2015-12-15
2015-12-16
0.032578 -0.097067
0.001626 -0.073670
2015-12-17
2015-12-18
2015-12-21
0.018384 -0.112442
-0.013837 0.058593
-0.014632 -0.000640
2015-12-22
0.000408 -0.056419
81
Data Analysis and Strategies
FIGURE 4.5
Histograms of historical percentage changes of EURO STOXX 50 and VSTOXX.
2015-12-23
0.022512 -0.101424
2015-12-28
2015-12-29
2015-12-30
-0.009186 0.108837
0.017746 -0.037752
-0.007935 0.026279
The code to visualize this kind of result is the same as before (see Figure 4.5).
In [47]: data.pct_change().hist(figsize=(10, 6),
....:
....:
....:
color='blue',
bins=100);
The majority of statistical analysis approaches rely on (log) returns and not on absolute time
series data. The next few lines of code calculate the log returns for the two time series and
store them in yet another DataFrame object, called log_rets.
In [48]: import numpy as np
# fully vectorized calculation of log returns
In [49]: log_rets = np.log(data / data.shift(1))
82
LISTED VOLATILITY AND VARIANCE DERIVATIVES
In [50]: log_rets.ix[:10]
Out[50]:
Date
1999-01-04
1999-01-05
1999-01-06
EUROSTOXX
VSTOXX
NaN
NaN
0.017228 0.489248
0.022138 -0.165317
1999-01-07
1999-01-08
1999-01-11
-0.015723
-0.003120
-0.019593
0.256337
0.021570
0.103173
1999-01-12
1999-01-13
1999-01-14
-0.012490 0.019496
-0.048535 0.056180
0.008648 -0.015214
1999-01-15
0.017855 -0.011906
Figure 4.6 visualizes the log returns times series for both the EURO STOXX 50 and VSTOXX
indexes.
In [51]: log_rets.plot(subplots=True,
....:
....:
....:
figsize=(10, 6),
color='blue',
grid=True);
....:
FIGURE 4.6
Log returns of EURO STOXX 50 and VSTOXX over time.
Data Analysis and Strategies
83
4.4 CORRELATION ANALYSIS
Equipped with the log returns, a thorough investigation of the correlation between the
EURO STOXX 50 and VSTOXX indexes is straightforward. For example, pandas provides
the corr() method to calculate correlations between time series data stored in different
DataFrame columns.
In [52]: log_rets.corr()
Out[52]:
EUROSTOXX
VSTOXX
EUROSTOXX 1.000000 -0.735032
VSTOXX
-0.735032 1.000000
Similar results are obtained by calculating the correlation of the data stored in one pandas
Series object with another data set in another Series object.
In [53]: log_rets['EUROSTOXX'].corr(log_rets['VSTOXX'])
Out[53]: 0.73503163966378138
The plot method of pandas DataFrame objects allows for different types of plots. For
example, scatter plots are helpful to visualize return data of two different time series (see
Figure 4.7).
FIGURE 4.7
Scatter plot of log returns of EURO STOXX 50 and VSTOXX.
84
LISTED VOLATILITY AND VARIANCE DERIVATIVES
FIGURE 4.8
Scatter plot of log returns of EURO STOXX 50 and VSTOXX with dates.
# plot log returns as scatter plot
In [54]: log_rets.plot(x=' EUROSTOXX', y=' VSTOXX',
....:
kind=' scatter', figsize=(10, 6));
....:
Figure 4.8 adds the time dimension to the data through shades of gray to illustrate the relation
between the two time series over time (during different “regimes”).
In [55]: log_rets = log_rets.dropna()
# delete NaN values
In [56]: import matplotlib as mpl
In [57]: import matplotlib.pyplot as plt
In [58]: plt.ioff()
# turn off interactive mode
In [59]: plt.set_cmap(mpl.cm.gray);
# set color map
In [60]: mpl_dates = mpl.dates.date2num(log_rets.index.to_pydatetime())
# conversion
In [61]: plt.figure(figsize=(10, 6));
85
Data Analysis and Strategies
In [62]: plt.scatter(log_rets['EUROSTOXX'], log_rets['VSTOXX'],
....:
c=mpl_dates, marker='o'); # the actual plot
....:
In [63]: plt.xlabel('EUROSTOXX');
In [64]: plt.ylabel('VSTOXX');
In [65]: plt.colorbar(ticks=mpl.dates.DayLocator(interval=250),
....:
format=mpl.dates.DateFormatter('%d %b %y')); # adding bar
....:
Although the negative correlation between the two indexes is almost evident from Figures 4.7
and 4.8, let us formally calculate and represent the negative correlation by a linear regression
line. Figure 4.9 adds such a regression line to the raw log returns in the scatter plot. Negative
correlation translates into a negative slope of the regression line.
# conduct linear regression
In [66]: p = np.polyfit(log_rets['EUROSTOXX'].values,
....:
log_rets['VSTOXX'].values,
....:
deg=1)
# the regression
....:
FIGURE 4.9
Scatter plot of log returns of EURO STOXX 50 and VSTOXX with regression line.
86
LISTED VOLATILITY AND VARIANCE DERIVATIVES
In [67]: log_rets.plot(x='EUROSTOXX', y='VSTOXX',
....:
kind='scatter', figsize=(10, 6));
# the actual plotting
....:
# plot the regression line
In [68]: plt.plot(log_rets['EUROSTOXX'], np.polyval(p, log_rets['EUROSTOXX']),
....:
'r');
# adding the regression line
....:
The seaborn plotting library is specifically developed with statistical applications in mind.
It therefore provides multiple useful, high level plotting capabilities. One of these is the
jointplot() function. The result of applying this function to the log return data is displayed as Figure 4.10. This is definitely the richest and most insightful presentation so far for
our purposes.
In [69]: sns.jointplot(x=log_rets['EUROSTOXX'], y=log_rets['VSTOXX'],
....:
kind='reg', size=7);
....:
FIGURE 4.10
Scatter plot of log returns of EURO STOXX 50 and VSTOXX with
regression line and histograms.
87
Data Analysis and Strategies
FIGURE 4.11
Rolling yearly correlation of EURO STOXX 50 and VSTOXX.
The final analysis in this section considers correlation over time. To this end, pandas provides
the rolling() method which allows, among others, the vectorized calculation of correlation
for moving time windows via corr(). We chose a window size of 252 trading days which
represents roughly one year.
In [70]: data['CORR'] = log_rets['EUROSTOXX'].rolling(
....:
window=252).corr(log_rets['VSTOXX'])
....:
Figure 4.11 plots the rolling correlation data and illustrates well that correlation fluctuates for
different yearly windows but that it is negative for any chosen yearly window.
In [71]: to_plot = data.dropna()
# drop NaN values
In [72]: plt.figure(figsize=(10, 6));
In [73]: plt.plot(data.index, data.CORR);
In [74]: plt.ylabel('rolling yearly correlation');
4.5 CONSTANT PROPORTION INVESTMENT STRATEGIES
One way to make use of the fact that the VSTOXX index is negatively correlated with the
EURO STOXX 50 index is to implement a constant proportion investment strategy. Respective
88
LISTED VOLATILITY AND VARIANCE DERIVATIVES
results are found, for example, in the study by Guobuzaite and Martellini (2012). Basically,
a constant (dollar) proportion investment strategy keeps the proportion of money invested in
securities of a portfolio over time constant by dynamic re-balancings given the movements in
the prices of the single securities.
In the example in this section, we assume that direct investments both in the EURO
STOXX 50 and the VSTOXX indexes are possible and that no transaction costs apply.
Although this might not be realistic, it simplifies the anaylsis and illustrates the basic idea
pretty well. This application allows the use of, for example, VSTOXX futures with different
roll-over strategies and the inclusion of transaction costs.
To begin with, let us adjust the original data sets for the EURO STOXX 50 and VSTOXX
indexes by mainly normalizing both time series to starting values of 100.
In [75]: del data['CORR']
# delete correlation data
In [76]: data = data.dropna()
# drop NaN values
In [77]: data = data / data.iloc[0] * 100
# normalization
In [78]: data.head()
Out[78]:
EUROSTOXX
VSTOXX
Date
1999-01-04
100.000000
100.000000
1999-01-05
1999-01-06
101.737744
104.015128
163.108887
138.255152
1999-01-07
1999-01-08
102.392538
102.073608
178.651673
182.547121
First, we analyze a typical passive investment strategy allocating 30% of a portfolio to the
VSTOXX index and the remaining 70% to the EURO STOXX 50.
In [79]: invest = 100
# initial investment
In [80]: cratio = 0.3
# VSTOXX ratio in the beginning
# number of EURO STOXX (fictional) securities
In [81]: data['Equity'] = (1 - cratio) * invest / data['EUROSTOXX'][0]
# number of VSTOXX (fictional) securities
In [82]: data['Volatility'] = cratio * invest / data['VSTOXX'][0]
In vectorized fashion, we calculate next the absolute values of such a portfolio over time, i.e.
as a time series.
89
Data Analysis and Strategies
FIGURE 4.12
Passive investment strategy (hypothetical) with EURO STOXX 50 and VSTOXX.
In [83]: data['Passive'] = (data['Equity'] * data['EUROSTOXX']
....:
+ data['Volatility'] * data['VSTOXX'])
....:
Figure 4.12 illustrates the performance of this particular passive investment strategy in comparison to a passive strategy purely investing in the EURO STOXX 50 index. In times of crisis,
for example at the end of 2008, the strategy peforms better. However, over the whole period
there is hardly any difference in the end result.
In [84]: data[['EUROSTOXX', 'Passive']].plot(figsize=(10, 6), style=['—-', '–']);
Second, the active constant proportion investment strategy, which keeps the dollar proportion
invested in the VSTOXX index constant at 30% over time through daily re-balancings of the
portfolio. In this case, the single calculations are done step-by-step through looping over the
single historical trading dates.
In [85]: for i in range(1, len(data)):
# daily re-balancing
....:
evalue = data['Equity'][i - 1] * data['EUROSTOXX'][i]
....:
vvalue = data['Volatility'][i - 1] * data['VSTOXX'][i]
....:
tvalue = evalue + vvalue
....:
data['Equity'][i] = (1 - cratio) * tvalue / data['EUROSTOXX'][i]
....:
data['Volatility'][i] = cratio * tvalue / data['VSTOXX'][i]
....:
# total wealth
90
LISTED VOLATILITY AND VARIANCE DERIVATIVES
Based on the results, calculate the absolute performance of this active strategy as before. Over
the first few trading days no major performance differences arise.
In [86]: data['Active'] = (data['Equity'] * data['EUROSTOXX']
+ data['Volatility'] * data['VSTOXX'])
....:
....:
In [87]: np.round(data.head(), 2)
Out[87]:
EUROSTOXX
VSTOXX
Equity
Volatility
Passive
Active
Date
1999-01-04
100.00
100.00
0.70
0.30
100.00
100.00
1999-01-05
1999-01-06
101.74
104.02
163.11
138.26
0.83
0.78
0.22
0.25
120.15
114.29
120.15
116.54
1999-01-07
102.39
178.65
0.86
0.21
125.27
125.48
1999-01-08
102.07
182.55
0.86
0.21
126.22
126.03
Over the whole time period, however, the active strategy – showing a 800+% gain – significantly outperforms the passive one.
In [88]: np.round(data.tail(), 2)
Out[88]:
EUROSTOXX
VSTOXX
Equity
Volatility
Passive
Active
Date
2015-12-22
90.72
123.80
7.22
2.27
100.65
936.01
2015-12-23
2015-12-28
92.76
91.91
111.25
123.35
6.96
7.21
2.49
2.30
98.31
101.34
922.28
946.46
2015-12-29
2015-12-30
93.54
92.80
118.70
121.82
7.09
7.16
2.39
2.34
101.09
101.50
947.50
949.71
Let us briefly verify whether the above implementation indeed yields constant proportions for
the two (fictional) securities in the portfolio.
In [89]: (data['Volatility'] * data['VSTOXX'] / data['Active'])[:5]
Out[89]:
Date
1999-01-04
0.3
1999-01-05
1999-01-06
1999-01-07
0.3
0.3
0.3
1999-01-08
0.3
dtype: float64
Data Analysis and Strategies
91
FIGURE 4.13
Active, constant proportion investment strategy (hypothetical) with EURO STOXX
50 and VSTOXX.
In [90]: (data['Equity'] * data['EUROSTOXX'] / data['Active'])[:5]
Out[90]:
Date
1999-01-04
1999-01-05
0.7
0.7
1999-01-06
1999-01-07
1999-01-08
0.7
0.7
0.7
dtype: float64
Being assured that we have indeed implemented a constant proportion trading strategy, have a
look at Figure 4.13 which impressively illustrates the outperformance of the active approach
over a passive investment in the stock index itself. However, bear in mind that all this rests on
a number of simplifying assumptions.
In [91]: data[['EUROSTOXX', 'Active']].plot(figsize=(10, 6), style=['—-', '–']);
The assumption of 30% invested in the VSTOXX index might seem a bit ad hoc. Therefore,
the following derives the optimal allocation for the given time period and data sets by a brute
force approach. First, we clean up the DataFrame object to reduce it again to the original
time series data for the two indexes.
92
LISTED VOLATILITY AND VARIANCE DERIVATIVES
# re-initialize DataFrame
In [92]: data = data[['EUROSTOXX', 'VSTOXX']]
The function below calculates the performance of a constant proportion investment strategy for
different VSTOXX dollar proportions and different starting and ending dates (both defaulting
to the data sets’ start and end dates).
In [93]: from copy import deepcopy
In [94]: def vstoxx_strategy(cratio, start=data.index[0], end=data.index[-1]):
....:
base = deepcopy(data[(data.index >= start) & (data.index <= end)])
....:
invest = 100
....:
base['Equity'] = (1 - cratio) * invest / base['EUROSTOXX'][0]
....:
base['Volatility'] = cratio * invest / base['VSTOXX'][0]
....:
for i in range(1, len(base)):
# initial invest
# daily re–balancing
....:
evalue = base['Equity'][i - 1] * base['EUROSTOXX'][i]
....:
vvalue = base['Volatility'][i - 1] * base['VSTOXX'][i]
....:
tvalue = evalue + vvalue
....:
base['Equity'][i] = (1 - cratio) * tvalue / base['EUROSTOXX'][i]
....:
....:
....:
....:
....:
base['Volatility'][i] = cratio * tvalue / base['VSTOXX'][i]
base['Active'] = (base['Equity'] * base['EUROSTOXX']
+ base['Volatility'] * base['VSTOXX'])
# wealth position
print("A con. VSTOXX ratio of %.2f yields a net perform. of %6.1f %%.")
\
% (cratio, (base['Active'][-1] / base['Active'][0] - 1) * 100)
....:
Equipped with this function, let us calculate the net perfomance for a VSTOXX dollar ratio of
30% as before.
In [95]: vstoxx_strategy(0.3)
A con. VSTOXX ratio of 0.30 yields a net perform.of 849.7%.
We do the same for 40% which obviously yields an even better result.
In [96]: vstoxx_strategy(0.4)
A con. VSTOXX ratio of 0.40 yields a net perform.of 1231.9%.
Applying an approach which is slightly more systematic, we get the following results for different constant dollar proportion assumptions. Over the whole period for which data is available
a 50:50 investment strategy seems to be optimal with a net performance of more than 1,400%.
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Data Analysis and Strategies
In [97]: for cratio in np.arange(0, 1.01, 0.1):
....:
vstoxx_strategy(cratio)
....:
A con. VSTOXX ratio of 0.00 yields a net perform.of
A con. VSTOXX ratio of 0.10 yields a net perform.of
-7.2%.
151.5%.
A con. VSTOXX ratio of 0.20 yields a net perform.of 445.4%.
A con. VSTOXX ratio of 0.30 yields a net perform.of 849.7%.
A con. VSTOXX ratio of 0.40 yields a net perform.of 1231.9%.
A con. VSTOXX ratio of 0.50 yields a net perform.of 1408.7%.
A con. VSTOXX ratio of 0.60 yields a net perform.of 1283.2%.
A con. VSTOXX ratio of 0.70 yields a net perform.of 928.4%.
A con. VSTOXX ratio of 0.80 yields a net perform.of
A con. VSTOXX ratio of 0.90 yields a net perform.of
521.0%.
204.9%.
A con. VSTOXX ratio of 1.00 yields a net perform.of
21.8%.
Let us implement the same analysis for the time period beginning in January 2013 and ending
with the last quarter of 2015. In this case, a constant dollar proportion invested in the VSTOXX
of again about 50% seems optimal and yields a net performance of close to 110%.
In [98]: for cratio in np.arange(0, 1.01, 0.1):
....:
vstoxx_strategy(cratio, start='2013-1-1', end=' 2015-12-31')
....:
A con. VSTOXX ratio of 0.00 yields a net perform.of
21.3%.
A con. VSTOXX ratio of 0.10 yields a net perform.of
A con. VSTOXX ratio of 0.20 yields a net perform.of
A con. VSTOXX ratio of 0.30 yields a net perform.of
47.9%.
72.4%.
92.2%.
A con. VSTOXX ratio of 0.40 yields a net perform.of
A con. VSTOXX ratio of 0.50 yields a net perform.of
A con. VSTOXX ratio of 0.60 yields a net perform.of
105.0%.
109.2%.
104.3%.
A con. VSTOXX ratio of 0.70 yields a net perform.of
A con. VSTOXX ratio of 0.80 yields a net perform.of
91.1%.
71.2%.
A con. VSTOXX ratio of 0.90 yields a net perform.of
A con. VSTOXX ratio of 1.00 yields a net perform.of
46.9%.
20.8%.
4.6 CONCLUSIONS
This chapter is about the retrieval and analysis of EURO STOXX 50 and VSTOXX historical data. It shows how to use Python and pandas to retrieve and clean up historical data sets
from the index provider’s website http://stoxx.com with pandas. It also shows how to implement Python code to replicate central stylized facts about stock and volatility indexes, namely
their highly negative correlation and the benefits of constant (dollar) proportion investment
strategies involving (products based on) equity and volatility indexes.
Listed Volatility and Variance Derivatives: A
Python-based Guide
By Dr. Yves J. Hilpisch
© 2017 Yves Hilpisch
CHAPTER
5
VSTOXX Index
5.1 INTRODUCTION
This chapter is about the (re-)calculation of the VSTOXX index, the volatility index based on
EURO STOXX 50 index options. The goal is to achieve a good understanding of the processes
and underlying mechanics of calculating the VSTOXX index. You will find all the background
information as well as Python code that will enable you to recalculate both historical VSTOXX
index values and current ones in (almost) real-time. Chapter 3, Model-Free Replication of
Variance provides the theoretical background for the concepts presented in this chapter.
The (main) VSTOXX index itself is based on two sub-indexes, which themselves are
derived from Eurex option series for both European puts and calls on the EURO STOXX 50
index. The algorithm, and therefore this chapter as well, are comprised of three main parts:
collect and clean-up the data of the necessary option series
compute the sub-indexes from the option data
compute the VSTOXX index from the relevant sub-indexes.
A few remarks about the option series and sub-indexes used and their expiry dates and time
horizons, respectively, seem in order. There are eight sub-indexes of the VSTOXX which each
measure the implied volatility of an option series with fixed expiry. For example, the VSTOXX
1M sub-index starts with the option series that has one month expiry and is calculated up to
two days prior to the fixed maturity date of the relevant option series. The VSTOXX index
measures the implied volatility of an “imaginary” options series with a fixed time to expiry of
30 days. This is achieved through linear interpolation of the two nearest sub-indexes, generally
VSTOXX 1M and VSTOXX 2M. On the two days before VSTOXX 1M expiry, the VSTOXX
2M and VSTOXX 3M are used instead and an extrapolation takes place.
Table 5.1 lists all the sub-indexes and provides additional information.
5.2 COLLECTING OPTION DATA
As already pointed out, the VSTOXX is based on two sub-indexes, generally the VSTOXX
1M and VSTOXX 2M, sometimes the VSTOXX 2M and VSTOXX 3M. The sub-indexes
95
96
LISTED VOLATILITY AND VARIANCE DERIVATIVES
TABLE 5.1
The VSTOXX sub-indexes.
Sub-index
Code
ISIN
Settlement date of the option series used
VSTOXX 1M
VSTOXX 2M
VSTOXX 3M
VSTOXX 6M
VSTOXX 9M
VSTOXX 12M
VSTOXX 18M
VSTOXX 24M
V6I1
V6I2
V6I3
V6I4
V6I5
V6I6
V6I7
V6I8
DE000A0G87B2
DE000A0G87C0
DE000A0G87D8
DE000A0G87E6
DE000A0G87F3
DE000A0G87G1
DE000A0G87H9
DE000A0G87J5
The last available within 1 month
The last available within 2 months
The last available within 3 months
The last available within 6 months
The last available within 9 months
The last available within 12 months
The last available within 18 months
The last available within 24 months
themselves are based on the option series on the EURO STOXX 50 index with respective
time to expiry. We therefore need the prices of all options with maturities up to 3 months.
We use historical data as provided by Eurex itself as the data source. See the website
http://bit.ly/1GY5KCI.
The code to collect the data can be found in the module index_collect_option_
data.py (see sub-section 5.6.1, index_collect_option_data.py for the complete script). As
usual, the module starts with some imports and parameter definitions.
import requests
import datetime as dt
import pandas as pd
import numpy as np
from StringIO import *
from index_date_functions import *
#
# The URL template
#
url1 = 'http://www.eurexchange.com/action/exchange-en/'
url2 = '180106-180102/180102/onlineStats.do?productGroupId=846'
url3 = '&productId=19068&viewType=3&cp=%s&month=%s&year=%s&busDate=%s'
URL = url1 + url2 + url3
In addition, the module contains six functions. The first is collect_option_series():
def collect_option_series(month, year, start):
''' Collects daily option data from web source.
Parameters
==========
month: int
maturity month
year: int
maturity year
97
VSTOXX Index
start: datetime object
starting date
Returns
=======
dataset: pandas DataFrame object
object containing the collected data
'''
end = dt.datetime.today()
delta = (end - start).days
dataset = pd.DataFrame()
for t in range(0, delta): # runs from start to today
date = start + dt.timedelta(t)
dummy = get_data(month, year, date)
if len(dummy) != 0:
if len(dataset) == 0:
# get data for one day
dataset = dummy
else:
dataset = pd.concat((dataset, dummy))
return dataset
# add data
This function collects the data of the option series with maturity in the month month and year
year. It is called by the function start_collecting() and calls the function
get_data() for every single day from the date start to today. It returns a complete set of
prices (both puts and calls) for that series.
The second function is get_data().
def get_data(month, year, date):
''' Get the data for an option series.
Parameters
==========
month: int
maturity month
year: int
maturity year
date: datetime object
the date for which the data is collected
Returns
=======
dataset: pandas DataFrame object
object containing call & put option data
'''
98
LISTED VOLATILITY AND VARIANCE DERIVATIVES
date_string = date.strftime('%Y%m%d')
# loads the call data from the web
data = get_data_from_www('Call', month, year, date_string)
calls = parse_data(data, date) # parse the raw data
calls = calls.rename(columns={'Daily settlem.price': 'Call_Price'})
calls = pd.DataFrame(calls.pop('Call_Price').astype(float))
# the same for puts
data = get_data_from_www('Put', month, year, date_string)
puts = parse_data(data, date)
puts = puts.rename(columns={'Daily settlem.price': 'Put_Price'})
puts = pd.DataFrame(puts.pop('Put_Price').astype(float))
dataset = merge_and_filter(puts, calls)
# merges the two time series
return dataset
This one is called by the function collect_option_series() and calls itself the
functions get_data_from_www(), parse_data(data, date) and merge_and_
filter(). It returns the prices of the option series with expiry date in month month and year
year for the day date.
The third function is get_data_from_www().
def get_data_from_www(oType, matMonth, matYear, date):
''' Retrieves the raw data of an option series from the web.
Parameters
==========
oType: string
either 'Put' or 'Call'
matMonth: int
maturity month
matYear: int
maturity year
date: string
expiry in the format 'YYYYMMDD'
Returns
=======
a: string
raw text with option data
'''
url = URL % (oType, matMonth, matYear, date)
a = requests.get(url).text
return a
# parametrizes the URL
99
VSTOXX Index
The function collects the prices of an option series for a single day (defined by date) from the
web. The option series is defined by the date of its expiry, given by matMonth and matYear;
the type of the options is given by oType which can be either Put or Call. It returns a
complete HTML file.
merge_and_filter() is the fourth function.
def merge_and_filter(puts, calls):
''' Gets two pandas time series for the puts and calls
(from the same option series), merges them, filters out
all options with prices smaller than 0.5 and
returns the resulting DataFrame object.
Parameters
==========
puts: pandas DataFrame object
put option data
calls: pandas DataFrame object
call option data
Returns
=======
df: pandas DataFrame object
merged & filtered options data
'''
df = calls.join(puts, how='inner')
# merges the two time series
# filters all prices which are too small
df = df[(df.Put_Price >= 0.5) & (df.Call_Price >= 0.5)]
return df
This one gets two time series puts and calls (typically of the same option series), merges
them, filters out all options with prices below 0.5 and returns the resulting pandas DataFrame
object.
parse_data() is the fifth function.
def parse_data(data, date):
''' Parses the HTML table and transforms it into a CSV compatible
format. The result can be directly imported into a pandas DataFrame.
Parameters
==========
data: string
document containing the Web content
date: datetime object
date for which the data is parsed
100
LISTED VOLATILITY AND VARIANCE DERIVATIVES
Returns
=======
dataset: pandas DataFrame object
transformed option raw data
'''
parts = data.split('Total')
table = parts3[0]
# the html table containing the data
table = table.replace('class="dataTable">', 'Pricing day')
# replace tags by commas and newlines
table = table.replace('', '\n')
table = table.replace(',', '')
table = table.replace('', ',')
table = table.replace(' ', '')
table = table.replace('', ',')
table = table.replace(' ', '')
table = table.replace(' ', '\n')
# the resulting string looks like a CSV file
date_string = date.strftime('%d.%m.%Y')
table = table.replace('', date_string)
string = StringIO(table)
# mask the string as file
dataset = pd.read_csv(string, parse_dates=[0], index_col=(0, 1),
dayfirst=True)
# read the 'file' as pandas DataFrame object
return dataset
It gets the string data which contains the HTML text delivered by function
get_data_from_www(), parses that string to a pandas DataFrame object with
double index date and strike price and returns that object.
The sixth and final function is data_collection().
def data_collection(path):
''' Main function which saves data into the HDF5 file
'index_option_series.h5' for later use.
Parameters
==========
path: string
path to store the data
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VSTOXX Index
'''
# file to store data
store = pd.HDFStore(path + 'index_option_series.h5', 'a')
today = dt.datetime.today()
start = today – dt.timedelta(31)
# the last 31 days
day = start.day
month = start.month
year = start.year
for i in range(4):
# iterates over the next 4 months
dummy_month = month + i
dummy_year = year
if dummy_month > 12:
dummy_month
-= 12
dummy_year += 1
# collect daily data beginning 31 days ago (start) for
# option series with expiry dummy_month, dummy_year
dataset = collect_option_series(dummy_month, dummy_year, start)
dummy_date = dt.datetime(dummy_year, dummy_month, day)
# abbreviation for expiry date (for example Oct14)
series_name = dummy_date.strftime('%b%y')
if series_name in store.keys():
# if data for that series exists
index_old = store[series_name].index
index_new = dataset.index
if len(index_new – index_old) > 0:
dummy = pd.concat((store[series_name],
dataset.ix[index_new - index_old]))
# add the new data
store[series_name] = dummy
else:
if len(dataset) > 0:
# if series is new, write whole data set into data store
store[series_name] = dataset
store.close()
This function is to initiate and finalize the collection of all relevant option series data sets. It
saves the resulting data in a file named index_option_series.h5.
In [1]: path = './source/data/'
102
LISTED VOLATILITY AND VARIANCE DERIVATIVES
Let us collect option data since all other steps depend on this data. We import the module as
icod.
In [2]: import numpy as np
In [3]: import pandas as pd
In [4]: import datetime as dt
In [5]: import index_collect_option_data as icod
Next, fix a target day relative to today such that you hit a business day for which closing data
is available.
In [6]: today = dt.datetime.now()
# make sure to hit a business day
In [7]: target_day = today - dt.timedelta(days=1)
In [8]: ds = target_day.strftime('%Y%m%d')
In [9]: ds
Out[9]: '20160201'
Then, for example, collect option data for puts and calls with a maturity defined by the parameters as follows:
# adjust maturity parameters if necessary
In [10]: call_data = icod.get_data_from_www(oType='Call', matMonth=3,
....:
....:
matYear=2016, date=ds)
In [11]: put_data = icod.get_data_from_www(oType='Put', matMonth=3,
....:
matYear=2016, date=ds)
....:
The return objects need to be parsed.
# parse the raw data
In [12]: calls = icod.parse_data(call_data, target_day)
In [13]: puts = icod.parse_data(put_data, target_day)
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VSTOXX Index
Let us have a look at some meta information about the call options data.
In [14]: calls.info()
MultiIndex: 114 entries, (2016-02-01 00:00:00, 600.0) to
(2016-02-01 00:00:00, 6000.0)
Data columns (total 8 columns):
Version number
114 non-null float64
Opening price
114 non-null float64
Daily high
Daily low
Underlying closing price
114 non-null float64
114 non-null float64
114 non-null float64
Daily settlem. price
Traded contracts
Open interest (adj.)*
114 non-null float64
114 non-null int64
114 non-null int64
dtypes: float64(6), int64(2)
memory usage: 8.0+ KB
And about the put options data.
In [15]: puts.info()
MultiIndex: 114 entries, (2016-02-01 00:00:00, 600.0) to
(2016-02-01 00:00:00, 6000.0)
Data columns (total 8 columns):
Version number
114 non-null float64
Opening price
114 non-null float64
Daily high
Daily low
Underlying closing price
114 non-null float64
114 non-null float64
114 non-null float64
Daily settlem. price
Traded contracts
Open interest (adj.)*
114 non-null float64
114 non-null int64
114 non-null int64
dtypes: float64(6), int64(2)
memory usage: 8.0+ KB
In a next step, we take out the daily settlement prices for both the puts and calls and define two
new DataFrame objects.
In [16]: calls = pd.DataFrame(calls.rename(
....:
columns={'Daily settlem. price': 'Call_Price'}
....:
).pop('Call_Price').astype(float))
....:
104
LISTED VOLATILITY AND VARIANCE DERIVATIVES
In [17]: puts = pd.DataFrame(puts.rename(
....:
columns={'Daily settlem. price': 'Put_Price'}
....:
).pop('Put_Price').astype(float))
....:
These two are then merged via the function merge_and_filter() into another new
DataFrame object.
In [18]: dataset = icod.merge_and_filter(puts, calls)
In [19]: dataset.info()
MultiIndex: 67 entries, (2016-02-01 00:00:00, 1900.0) to
(2016-02-01 00:00:00, 3625.0)
Data columns (total 2 columns):
Call_Price
67 non-null float64
Put_Price
67 non-null float64
dtypes: float64(2)
memory usage: 1.6+ KB
This whole procedure is implemented in the function collect_option_series() which
yields the same result.
In [20]: os = icod.collect_option_series(3, 2016, target_day)
In [21]: os.info()
MultiIndex: 67 entries, (2016-02-01 00:00:00, 1900.0) to
(2016-02-01 00:00:00, 3625.0)
Data columns (total 2 columns):
Call_Price
67 non-null float64
Put_Price
67 non-null float64
dtypes: float64(2)
memory usage: 1.6+ KB
The function data_collection() repeats this procedure for all those dates for which
option data is available and writes (appends) the results in a HDF5 database file.
# uncomment to initiate the process (takes a while)
# %time icod.data_collection(path)
For the further analyses, we open this HDF5 database file.
In [22]: store = pd.HDFStore(path + 'index_option_series.h5', 'r')
In [23]: store
105
VSTOXX Index
Out[23]:
File path: ./source/data/index_option_series.h5
/Feb16
/Jan16
/Mar16
frame
frame
frame
(shape->[1398,2])
(shape->[638,2])
(shape->[1385,2])
The collected option series data is easily read from the HDF5 database file in monthly chunks.
In [24]: Mar16 = store['Mar16']
In [25]: Mar16.info()
MultiIndex: 1385 entries, (2015-12-18 00:00:00, 1500.0) to (2016-01-15
00:00:00, 3650.0)
Data columns (total 2 columns):
Call_Price
1385 non-null float64
Put_Price
1385 non-null float64
dtypes: float64(2)
memory usage: 32.5+ KB
In [26]: store.close()
Some selected option prices from the large data set:
In [27]: Mar16.ix[25:35]
Out[27]:
Call_Price Put_Price
Pricing day Strike price
2015-12-18
2675.0
2700.0
600.4
577.3
21.9
23.8
2725.0
2750.0
2775.0
554.3
531.6
509.0
25.9
28.1
30.6
2800.0
2825.0
486.6
464.5
33.2
36.1
2850.0
2875.0
2900.0
442.6
421.0
399.6
39.3
42.6
46.3
5.3 CALCULATING THE SUB-INDEXES
In this section, we use the data file created in the previous one. For all dates
of the data file, the Python module index_subindex_calculation.py (see
106
LISTED VOLATILITY AND VARIANCE DERIVATIVES
index_subindex_calculation.py for the complete script) used in this section decides whether
the VSTOXX 1M sub-index is defined or not (remember that the sub-index is not defined at the
final settlement day and one day before). If it is defined, the script computes the value of the
sub-indexes VSTOXX 1M and VSTOXX 2M; if not, it computes the values of the sub-indexes
VSTOXX 2M and VSTOXX 3M, respectively. Finally, it returns a pandas DataFrame object
with the three time series.
5.3.1 The Algorithm
First, we focus on the computation of the value of a single sub-index for a given date. The
prices Ci , i ∈ {0, ..., n}, of a series of European call options on the EURO STOXX 50 with
fixed maturity date T and exercise prices Ki , i ∈ {0, ..., n} are given, as well as the prices Pi , i ∈
{0, ..., n}, of a series of European put options on EURO STOXX 50 with the same maturity
date T and exercise prices Ki . Let us further hold that Ki < Ki+1 for all i ∈ {0, ...., n − 1}.
Then, the value of the relevant sub-index V is as follows (see chapter 3, Model-Free Replication of Variance):
V = 100 ⋅
√
𝜎̂ 2
with
𝜎̂ 2 =
n
2 ∑ ΔKi rT
1
e Mi −
T i=0 Ki 2
T
(
)2
F
−1
K∗
where
ΔKi
⎧K − K
for i = 0
0
⎪ 1
⎪ K − Ki−1
= ⎨ i+1
for i = 1, ..., n − 1
2
⎪
⎪ Kn − Kn−1 for i = n
⎩
r
= constant risk-free short rate appropriate for maturity T
F
= Kj + erT |Cj − Pj |, where j =
K∗
=
Mi
⎧P
⎪ i
⎪ P − Ci
=⎨ i
⎪ 2
⎪ Ci
⎩
min {|Ci − Pi |}
i∈{0,...,n}
max {Ki < F},
Ki|i∈{0,...,n}
for Ki < K∗
for Ki = K∗
for Ki > K∗
We implement a function to compute one value of a single sub-index. Thereafter, we extend
that function to compute time series for both VSTOXX 1M and VSTOXX 2M indexes as well
as parts of the VSTOXX 3M index. Imports again make up the beginning of the script.
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VSTOXX Index
import math
import numpy as np
import pandas as pd
import datetime as dt
import index_date_functions as idf
A core function of the script is compute_subindex().
def compute_subindex(data, delta_T, R):
''' Computes a sub-index for given option series data.
Parameters
==========
data: pandas.DataFrame object
contains the option data
delta_T: float
time interval
R: float
discount factor
Returns
=======
subVSTOXX: float
sub-index value
'''
# difference between put and call option with same strike
data['Diff_Put_Call']
=
np.abs(data.Put_Price – data.Call_Price)
# converts the strike price which serves as index so far
# to a regular data column
data = data.reset_index()
data['delta_K'] = None
# differences between the different strikes of the series
data['delta_K'].iloc[1:–1] = [(data['Strike price'][i + 1]
– data['Strike price'][i – 1]) / 2
for i in data.index[1:–1]]
# where possible, for the i-th entry it is
# half of the difference between the (i-1)-th
# and (i+1)-th price
# for i=0 it is just the difference to the next strike
data['delta_K'].iloc[0] = data['Strike price'][1] – data['Strike price'][0]
data['delta_K'].iloc[data.index[–1:]] = float(data['Strike price'][–1:])
– float(data['Strike price'][-2:-1])
# for the last entry, it is just the difference
# between the second but last strike and the last strike price
# find the smallest difference between put and call price
min_index = data. Diff_Put_Call.argmin()
\
108
LISTED VOLATILITY AND VARIANCE DERIVATIVES
# the forward price of that option
forward_price = data['Strike price'][min_index]
\
+ R * data.Diff_Put_Call[min_index]
K_0 = data['Strike price'][forward_price data['Strike price'] > 0].max()
# the index of the ATM strike
K_0_index = data.index[data['Strike price'] == K_0][0]
# selects the OTM options
data['M'] = pd.concat((data.Put_Price[0:K_0_index],
data.Call_Price[K_0_index:]))
# ATM we take the average of put and call price
data['M'].iloc[K_0_index] = (data['Call_Price'][K_0_index]
+ data['Put_Price'][K_0_index]) / 2
# the single OTM values
data['MFactor'] = (R * (data['delta_K'] * data['M'])
/ (data['Strike price']) ** 2)
# the forward term
fterm = 1. / delta_T *(forward_price / K_0 - 1) ** 2
# summing up
sigma = 2 / delta_T * np.sum(data.MFactor) – fterm
subVSTOXX = 100 * math.sqrt(sigma)
return subVSTOXX
This script calculates a single index value. It implements mainly the following steps:
the calculation of ΔKi
the computation of the forward price and the index of K∗
the selection of the at-the-money option and the out-of-the-money options
the combination of the results of the other three steps.
The next step is the derivation of time series data for the VSTOXX 1M and VSTOXX 2M as
well as parts of the VSTOXX 3M indexes and storage of the results in a pandas DataFrame
object. As data source we use the file created in the last section. Remember that this file contains a dictionary-like HDFStore object with one entry for every options series. The keys for
the entries are three letter abbreviations of the respective month’s name plus the actual year
represented by two numbers, for example Mar16, Jun16 and so on. The value of an entry is
a pandas DataFrame object with a pandas MultiIndex (date, strike price) and prices for
the put and call options for the dates and strike prices.
All this is implemented as function make_subindex().
def make_subindex(path):
''' Depending on the content of the file 'index_option_series.h5',
the function computes the sub-indexes V6I1, V6I2 and parts
of V6I3 and returns a pandas DataFrame object with the results.
109
VSTOXX Index
Parameters
==========
path: string
string with path of data file
Returns
=======
df: pandas DataFrame object
sub-index data as computed by the function
'''
# the data source, created with index_collect_option_data.py
datastore = pd.HDFStore(path + 'index_option_series.h5', 'r')
max_date = dt.datetime.today()
# find the latest date in the source
for series in datastore.keys():
dummy_date = datastore[series].index.get_level_values(0)[0]
dummy_date = dummy_date.to_pydatetime()
if dummy_date > max_date:
max_date = dummy_date
start_date = dt.datetime.today()
# find the earliest date in the source
for series in datastore.keys():
dummy_date = datastore[series].index.get_level_values(0)[0]
dummy_data = dummy_date.to_pydatetime()
if dummy_date < start_date:
start_date = dummy_date
V1 = dict()
# dicts to store the values, V stands for the sub-indices,
# T for their expiry
V2 = dict()
V3 = dict()
T1 = dict()
T2 = dict()
T3 = dict()
# from start_date to max_date, but only weekdays
for day in pd.bdate_range(start=start_date.date(), end=max_date.date()):
# is V6I1 defined?
is_V1_defined = idf.not_a_day_before_expiry(day)
# the settlement date
settlement_date = idf.first_settlement_day(day)
# abbreviation for the expiry date, like Oct14
key = settlement_date.strftime('%b%y')
# days until maturity
delta_T = idf.compute_delta(day, settlement_date)
try:
# data of the option series for that date
data = datastore[key].ix[day]
except:
continue
110
LISTED VOLATILITY AND VARIANCE DERIVATIVES
if
is_V1_defined:
# if V6I1 is defined
# compute its value
V1[day] = compute_subindex(data, delta_T,
math.exp(0.0015 * delta_T))
T1[day] = settlement_date
else:
# compute the value of V6I2 instead
V2[day] = compute_subindex(data,
delta_T,
math.exp(0.0015 * delta_T))
T2[day] = settlement_date
settlement_date_2 = idf.second_settlement_day(day)
# the same for the next index
key_2 = settlement_date_2.strftime('%b%y')
delta_T_2 = idf.compute_delta(day, settlement_date_2)
data_2 = datastore[key_2].ix[day]
if is_V1_defined:
V2[day] = compute_subindex(data_2, delta_T_2,
math.exp(0.001 * delta_T_2))
T2[day] = settlement_date_2
else:
V3[day] = compute_subindex(data_2, delta_T_2,
math.exp(0.001 * delta_T_2))
T3[day] =
settlement_date_2
datastore.close()
# create the pandas DataFrame object and return it
df = pd.DataFrame(data={'V6I1': V1, 'Expiry V6I1': T1, 'V6I2': V2,
'Expiry V6I2': T2, 'V6I3': V3, 'Expiry V6I3': T3})
return df
This function uses the collected option series data and selects those data sub-sets needed for
the calculation at hand. It generates sub-index values for all those days for which option data
is available. The result is a pandas DataFrame object.
Let us see how it works. To this end, we first import the module as isc.
In [28]: import index_subindex_calculation as isc
In [29]: si = isc.make_subindex(path)
In [30]: si
Out[30]:
2015-12-18
2015-12-21
2015-12-22
2015-12-23
Expiry V6I1 Expiry V6I2
NaT 2016-01-15
2016-01-15 2016-02-19
2016-01-15
2016-01-15
2016-02-19
2016-02-19
Expiry V6I3
2016-02-19
NaT
V6I1
NaN
23.616427
V6I2
23.757165
24.665801
NaT
NaT
21.908727
19.398907
23.629867
21.488879
\
111
VSTOXX Index
2015-12-24
2016-01-15 2016-02-19
NaT
19.839390
21.677283
2015-12-28
2015-12-29
2015-12-30
2016-01-15 2016-02-19
2016-01-15 2016-02-19
2016-01-15 2016-02-19
NaT
NaT
NaT
21.493148
20.611811
21.522201
23.080926
22.199482
22.788559
2015-12-31
2016-01-04
2016-01-05
2016-01-15 2016-02-19
2016-01-15 2016-02-19
2016-01-15 2016-02-19
NaT
NaT
NaT
22.238762
28.926899
27.510874
23.016324
26.646698
25.983436
2016-01-06
2016-01-07
2016-01-08
2016-01-15 2016-02-19
2016-01-15 2016-02-19
2016-01-15 2016-02-19
NaT
NaT
NaT
27.451164
33.536380
33.400592
26.101145
29.502767
30.060601
2016-01-11
2016-01-12
2016-01-15 2016-02-19
2016-01-15 2016-02-19
NaT
NaT
35.084411
33.266282
30.005212
28.420558
2016-01-13
2016-01-14
2016-01-15
2016-01-15 2016-02-19
NaT 2016-02-19
NaT 2016-02-19
NaT
2016-03-18
2016-03-18
34.566710
NaN
NaN
28.058019
29.864680
34.195075
2015-12-18
V6I3
24.740060
2015-12-21
2015-12-22
2015-12-23
NaN
NaN
NaN
2015-12-24
2015-12-28
2015-12-29
NaN
NaN
NaN
2015-12-30
2015-12-31
2016-01-04
NaN
NaN
NaN
2016-01-05
2016-01-06
NaN
NaN
2016-01-07
2016-01-08
2016-01-11
NaN
NaN
NaN
2016-01-12
2016-01-13
2016-01-14
NaN
NaN
29.495976
2016-01-15
32.797564
For comparison, we retrieve the “real” historical VSTOXX (sub-)index values from the official
website.
In [31]: vs_url = 'https://www.stoxx.com/document/'
In [32]: vs_url += 'Indices/Current/HistoricalData/h_vstoxx.txt'
In [33]: vs = pd.read_csv(vs_url, # filename
....:
index_col=0, # index column (dates)
....:
parse_dates=True, # parse date information
112
LISTED VOLATILITY AND VARIANCE DERIVATIVES
dayfirst=True, # day before month
header=2) # header/column names
....:
....:
....:
In [34]: vs.to_csv(path + 'vs.csv')
# write as CSV file
In [35]: vs = vs[vs.index <= '2015-12-31']
In [36]: vs.info()
DatetimeIndex: 4327 entries, 1999-01-04 to 2015-12-30
Data columns (total 9 columns):
V2TX
4327 non-null float64
V6I1
3878 non-null float64
V6I2
V6I3
V6I4
4327 non-null float64
4267 non-null float64
4327 non-null float64
V6I5
V6I6
V6I7
4327 non-null float64
4310 non-null float64
4327 non-null float64
V6I8
4313 non-null float64
dtypes: float64(9)
memory usage: 338.0 KB
Next, combine the re-calculated VSTOXX 2M values with the historical ones into a new
DataFrame object and add a new column with the absolute differences.
In [37]: comp = pd.concat((si['V6I2'], vs['V6I2']),
axis=1, join='inner')
....:
....:
In [38]: comp.index = comp.index.normalize()
In [39]: comp.columns = ['CALC', 'REAL']
In [40]: comp['DIFF'] = comp['CALC'] - comp['REAL']
In [41]: comp
Out[41]:
CALC
REAL
DIFF
2015-12-18
2015-12-21
2015-12-22
23.757165
24.665801
23.629867
23.7604 -0.003235
24.6237 0.042101
23.6439 -0.014033
2015-12-23
2015-12-28
2015-12-29
21.488879
23.080926
22.199482
21.4523
23.0757
22.1429
0.036579
0.005226
0.056582
2015-12-30
22.788559
22.6419
0.146659
113
VSTOXX Index
FIGURE 5.1
Calculated VSTOXX 2M sub-index values vs. real ones.
Figure 5.1 shows the two time series in direct comparison.
In [42]: import seaborn as sns; sns.set()
In [43]: import matplotlib
In [44]: matplotlib.rcParams['font.family'] = 'serif'
# set serif font
In [45]: comp[['CALC', 'REAL']].plot(style=['ro', 'b'], figsize=(10, 6));
Figure 5.2 shows the point-wise differences between the two time series.
In [46]: import matplotlib.pyplot as plt
In [47]: plt.figure(figsize=(10, 6));
In [48]: plt.bar(comp.index, comp['DIFF']);
In [49]: plt.gcf().autofmt_xdate();
114
LISTED VOLATILITY AND VARIANCE DERIVATIVES
FIGURE 5.2
Differences of calculated VSTOXX 2M index values and real ones.
5.4 CALCULATING THE VSTOXX INDEX
If the values for the sub-indexes VSTOXX 1M and VSTOXX 2M, V1 and V2 say, are given,
then the value for the VSTOXX index itself, V say, is calculated by the linear interpolation of
V1 and V2 :
√(
(
(
)
))
√
√
NT2 − N30
N30 − NT1
N
√
2
2
T1 ⋅ V1 ⋅
+ T2 ⋅ V2 ⋅
⋅ 365
V=
NT2 − NT1
NT2 − NT1
N30
where
NT1 = time to expiry of V1 ’s options series in seconds
NT2 = time to expiry of V2 ’s options series in seconds
N30 = 30 days in seconds
N365 = time for a standard year in seconds
T1 = NT1 ∕N365
T2 = NT2 ∕N365 .
Recall that the sub-index VSTOXX 1M is not defined on the final settlement day of the underlying option series and the day before. For these dates, we use VSTOXX 2M and VSTOXX
3M as V1 and V2 , respectively.
The Python module index_vstoxx_calculation.py (see sub-section 5.6.3,
index_vstoxx_calculation.py for the module in its entirety) implements the VSTOXX index
calculation routine – given the respective sub-index time series data sets. The module starts as
usual with some imports.
115
VSTOXX Index
import pandas as pd
import numpy as np
import matplotlib.pyplot as plt
from index_date_functions import *
The function calculate_vstoxx() is the core of the module.
def calculate_vstoxx(path):
''' Function to calculate the VSTOXX volatility index given time series
of the relevant sub-indexes.
Parameters
==========
path: string
string with path of data files
Returns
=======
data: pandas DataFrame object
results of index calculation
'''
# constant parameters
seconds_year = 365 * 24 * 3600.
seconds_30_days = 30 * 24 * 3600.
# import historical VSTOXX data
data = pd.read_csv(path + 'vs.csv', index_col=0, parse_dates=True)
# determine the settlement dates for the two underlying option series
data['Settlement date 1'] = [first_settlement_day(a) for a in data.index]
data['Settlement date 2'] = [second_settlement_day(a) for a in data.index]
# deduce the life time (in seconds) from current date to
# final settlement Date
data['Life time 1'] = [(data['Settlement date 1'][i] - i).days
* 24 * 60 * 60 for i in data.index]
data['Life time 2'] = [(data['Settlement date 2'][i] - i).days
* 24 * 60 * 60 for i in data.index]
data['Use V6I2'] = data['V6I1'].notnull()
# where V6I1 is not defined
data['Subindex to use 1'] = [data['V6I1'][i]
if data['Use V6I2'][i]
else data['V6I2'][i] for i in data.index]
# if V6I1 is defined, use V6I1 and V6I2 as data set
data['Subindex to use 2'] = [data['V6I2'][i]
if data['Use V6I2'][i]
else data['V6I3'][i] for i in data.index]
# else use V6I2 and V6I3
# the linear interpolation of the VSTOXX value
# from the two relevant sub-indexes
116
LISTED VOLATILITY AND VARIANCE DERIVATIVES
data['Part 1'] = data['Life time 1'] / seconds_year
\
* data['Subindex to use 1'] ** 2
\
* ((data['Life time 2'] - seconds_30_days)
/ (data['Life time 2'] - data['Life time 1']))
data['Part 2'] = data['Life time 2'] / seconds_year
\
* data['Subindex to use 2'] ** 2
\
* ((seconds_30_days - data['Life time 1'])
/ (data['Life time 2'] - data['Life time 1']))
data['VSTOXX'] = np.sqrt((data['Part 1'] + data['Part 2'])
*
seconds_year / seconds_30_days)
# difference between original VSTOXX data and re-calculated values
data['Difference'] = data['V2TX'] – data['VSTOXX']
return data
As its single argument, the function takes the path to a CSV file containing historical VSTOXX
data for the index itself and the sub-indexes. The re-calculation of it then is as straightforward
as follows:
In [50]: import index_vstoxx_calculation as ivc
In [51]: %time data = ivc.calculate_vstoxx(path)
CPU times: user 1.19 s, sys: 38 us, total: 1.19 s
Wall time: 1.2 s
Let us inspect the pandas DataFrame which now contains the results of the previous code
execution:
In [52]: data.info()
DatetimeIndex: 4357 entries, 1999-01-04 to 2016-02-12
Data columns (total 20 columns):
V2TX
V6I1
V6I2
4357 non-null float64
3906 non-null float64
4357 non-null float64
V6I3
V6I4
4296 non-null float64
4357 non-null float64
V6I5
V6I6
V6I7
4357 non-null float64
4340 non-null float64
4357 non-null float64
V6I8
Settlement date 1
Settlement date 2
4343 non-null float64
4357 non-null datetime64[ns]
4357 non-null datetime64[ns]
Life time 1
4357 non-null int64
117
VSTOXX Index
Life time 2
Use V6I2
4357 non-null int64
4357 non-null bool
Subindex to use 1
Subindex to use 2
Part 1
4357 non-null float64
4357 non-null float64
4357 non-null float64
Part 2
VSTOXX
Difference
4357 non-null float64
4357 non-null float64
4357 non-null float64
dtypes: bool(1), datetime64[ns](2), float64(15), int64(2)
memory usage: 685.0 KB
A brief look at the absolute average error of the re-calculation reveals that the implementation
yields quite accurate results.
# output: average error of re-calculation
In [53]: data['Difference'].mean()
Out[53]: 0.0012799470095575995
Figure 5.3 compares the original V2TX time series with the re-calculated values.
# original vs. re-calculated VSTOXX index
In [54]: data[['V2TX', 'VSTOXX']].plot(subplots=True, figsize=(10, 6),
....:
style="blue", grid=True);
....:
FIGURE 5.3
Historical VSTOXX index values re-calculated vs. real ones.
118
LISTED VOLATILITY AND VARIANCE DERIVATIVES
FIGURE 5.4
Differences of historical VSTOXX index values re-calculated vs. real ones.
Finally, Figure 5.4 presents the absolute differences. The figure shows that the differences are
in general marginal with a few outliers observed here and there.
# differences between single values
In [55]: data['Difference'].plot(figsize=(10, 6), style="r", grid=True,
....:
ylim=(-1, 1));
....:
5.5 CONCLUSIONS
This chapter (re-)calculates the VSTOXX volatility index based on historical sub-index values
and based on the volatility index definition as derived in chapter 3, Model-Free Replication of
Variance. The chapter also shows how to calculate the sub-index values themselves based on
EURO STOXX 50 options data. Python code is provided to automatically collect such data
from the Eurex website.
5.6 PYTHON SCRIPTS
5.6.1 index collect option data.py
#
# Module to collect option series data
# from the web
119
VSTOXX Index
# Source: www.eurexchange.com
# Data is needed to calculate the VSTOXX
# and its sub-indexes
#
# (c) Dr. Yves J. Hilpisch
# Listed Volatility and Variance Derivatives
#
import requests
import datetime as dt
import pandas as pd
import numpy as np
from StringIO import *
from index_date_functions import *
#
# The URL template
#
url1
=
'http://www.eurexchange.com/action/exchange-en/'
url2
=
'180106-180102/180102/onlineStats.do?productGroupId=846'
url3
=
'&productId=19068&viewType=3&cp=%s &month=%s &year=%s &busDate=%s'
URL = url1 + url2 + url3
def collect_option_series(month, year, start):
''' Collects daily option data from web source.
Parameters
==========
month: int
maturity month
year: int
maturity year
start: datetime object
starting date
Returns
=======
dataset: pandas DataFrame object
object containing the collected data
'''
end = dt.datetime.today()
delta = (end – start).days
dataset = pd.DataFrame()
for t in range(0, delta):
# runs from start to today
date = start + dt.timedelta(t)
dummy = get_data(month,
if len(dummy) != 0:
year,
date)
# get data for one day
120
LISTED VOLATILITY AND VARIANCE DERIVATIVES
if len(dataset) == 0:
dataset = dummy
else:
dataset = pd.concat((dataset, dummy))
# add data
return dataset
def get_data(month, year, date):
''' Get the data for an option series.
Parameters
==========
month: int
maturity month
year: int
maturity year
date: datetime object
the date for which the data is collected
Returns
=======
dataset: pandas DataFrame object
object containing call & put option data
'''
date_string = date.strftime('%Y%m%d')
# loads the call data from the web
data = get_data_from_www('Call', month,
calls = parse_data(data,
date)
year, date_string)
# parse the raw data
calls = calls.rename(columns={'Daily settlem.price': 'Call_Price'})
calls = pd.DataFrame(calls.pop('Call_Price').astype(float))
# the same for puts
data = get_data_from_www('Put',
puts = parse_data(data,
month,
year, date_string)
date)
puts = puts.rename(columns={'Daily settlem.price': 'Put_Price'})
puts = pd.DataFrame(puts.pop('Put_Price').astype(float))
dataset = merge_and_filter(puts, calls)
# merges the two time series
return dataset
def get_data_from_www(oType, matMonth, matYear, date):
''' Retrieves the raw data of an option series from the web.
Parameters
==========
oType: string
either 'Put' or 'Call'
121
VSTOXX Index
matMonth: int
maturity month
matYear: int
maturity year
date: string
expiry in the format 'YYYYMMDD'
Returns
=======
a: string
raw text with option data
'''
url = URL % (oType, matMonth, matYear, date)
# parametrizes the URL
a = requests.get(url).text
return a
def merge_and_filter(puts, calls):
''' Gets two pandas time series for the puts and calls
(from the same option series), merges them, filters out
all options with prices smaller than 0.5 and
returns the resulting DataFrame object.
Parameters
==========
puts: pandas DataFrame object
put option data
calls: pandas DataFrame object
call option data
Returns
=======
df: pandas DataFrame object
merged & filtered options data
'''
df = calls.join(puts,
how=' inner')
# merges the two time series
# filters all prices which are too small
df = df[(df. Put_Price
>= 0.5)
& (df.Call_Price
>=
0.5)]
return df
def parse_data(data, date):
''' Parses the HTML table and transforms it into a CSV compatible
format. The result can be directly imported into a pandas DataFrame.
Parameters
==========
122
LISTED VOLATILITY AND VARIANCE DERIVATIVES
data: string
document containing the Web content
date: datetime object
date for which the data is parsed
Returns
=======
dataset: pandas DataFrame object
transformed option raw data
'''
parts = data.split('Total')
table = parts3[0]
# the html table containing the data
table = table.replace('class="dataTable">', 'Pricing day')
# replace tags by commas and newlines
table = table.replace('', '\n')
table = table.replace(',', '')
table = table.replace('', ',')
table = table.replace(' ', '')
table = table.replace('', ',')
table = table.replace(' ', '')
table = table.replace(' ', '\n')
# the resulting string looks like a CSV file
date_string = date.strftime('%d .%m.%Y')
table = table.replace('',
string = StringIO(table)
date_string)
# mask the string as file
dataset = pd.read_csv(string, parse_dates=[0], index_col=(0, 1),
dayfirst=True)
# read the 'file' as pandas DataFrame object
return dataset
def data_collection(path):
''' Main function which saves data into the HDF5 file
'index_option_series.h5' for later use.
Parameters
==========
path: string
path to store the data
'''
# file to store data
store = pd.HDFStore(path + 'index_option_series.h5', 'a')
123
VSTOXX Index
today = dt.datetime.today()
start = today - dt.timedelta(31)
# the last 31 days
day = start.day
month = start.month
year = start.year
for i in range(4):
# iterates over the next 4 months
dummy_month = month + i
dummy_year = year
if dummy_month > 12:
dummy_month -= 12
dummy_year += 1
# collect daily data beginning 31 days ago (start) for
# option series with expiry dummy_month, dummy_year
dataset = collect_option_series(dummy_month, dummy_year, start)
dummy_date = dt.datetime(dummy_year, dummy_month, day)
# abbreviation for expiry date (for example Oct14)
series_name = dummy_date.strftime('%b%y')
if series_name in store.keys():
# if data for that series exists
index_old = store[series_name].index
index_new = dataset.index
if len(index_new - index_old) > 0:
dummy = pd.concat((store[series_name],
dataset.ix[index_new - index_old]))
# add the new data
store[series_name] = dummy
else:
if len(dataset) > 0:
# if series is new, write whole data set into data store
store[series_name]
=
dataset
store.close()
5.6.2 index subindex calculation.py
#
# Module with functions to compute VSTOXX sub-indexes
#
# Data as generated by the script index_collect_option_data.py
# is needed for the calculations in this module
#
# (c) Dr. Yves J. Hilpisch
124
LISTED VOLATILITY AND VARIANCE DERIVATIVES
# Listed Volatility and Variance Derivatives
#
import math
import numpy as np
import pandas as pd
import datetime as dt
import index_date_functions as idf
def compute_subindex(data, delta_T, R):
''' Computes a sub-index for given option series data.
Parameters
==========
data: pandas.DataFrame object
contains the option data
delta_T: float
time interval
R: float
discount factor
Returns
=======
subVSTOXX: float
sub-index value
'''
# difference between put and call option with same strike
data['Diff_Put_Call'] = np.abs(data. Put_Price - data.Call_Price)
# converts the strike price which serves as index so far
# to a regular data column
data = data.reset_index()
data['delta_K'] = None
# differences between the different strikes of the series
data['delta_K'].iloc[1:-1] = [(data['Strike price'][i + 1]
- data['Strike price'][i - 1]) / 2 for i in data.index[1:-1]]
# where possible, for the i-th entry it is
# half of the difference between the (i-1)-th
# and (i+1)-th price
# for i=0 it is just the difference to the next strike
data['delta_K'].iloc[0] = data['Strike price'][1] – data['Strike price'][0]
data['delta_K'].iloc[data.index[-1:]] = float(data['Strike price'][-1:])
- float(data['Strike price'][-2: -1])
# for the last entry, it is just the difference
# between the second but last strike and the last strike price
# find the smallest difference between put and call price
min_index = data. Diff_Put_Call.argmin()
# the forward price of that option
forward_price = data['Strike price'][min_index]
\
+ R * data.Diff_Put_Call[min_index]
\
125
VSTOXX Index
K_0 = data['Strike price'][forward_price data['Strike price'] > 0].max()
# the index of the ATM strike
K_0_index = data.index[data['Strike price'] == K_0][0]
# selects the OTM options
data['M'] = pd.concat((data.Put_Price[0:K_0_index],
data.Call_Price[K_0_index:]))
# ATM we take the average of put and call price
data['M'].iloc[K_0_index] = (data['Call_Price'][K_0_index]
+ data['Put_Price'][K_0_index]) / 2
# the single OTM values
data['MFactor'] = (R * (data['delta_K'] * data['M'])
/ (data['Strike price']) ** 2)
# the forward term
fterm = 1. / delta_T * (forward_price / K_0 - 1) ** 2
# summing up
sigma = 2 / delta_T * np.sum(data.MFactor) - fterm
subVSTOXX = 100 * math.sqrt(sigma)
return subVSTOXX
def make_subindex(path):
''' Depending on the content of the file 'index_option_series.h5',
the function computes the sub-indexes V6I1, V6I2 and parts
of V6I3 and returns a pandas DataFrame object with the results.
Parameters
==========
path: string
string with path of data file
Returns
=======
df: pandas DataFrame object
sub-index data as computed by the function
'''
# the data source, created with index_collect_option_data.py
datastore = pd.HDFStore(path + 'index_option_series.h5', 'r')
max_date = dt.datetime.today()
# find the latest date in the source
for series in datastore.keys():
dummy_date = datastore[series].index.get_level_values(0)[0]
dummy_date = dummy_date.to_pydatetime()
if dummy_date > max_date:
max_date = dummy_date
start_date = dt.datetime.today()
for series in datastore.keys():
# find the earliest date in the source
126
LISTED VOLATILITY AND VARIANCE DERIVATIVES
dummy_date = datastore[series].index.get_level_values(0)[0]
dummy_data = dummy_date.to_pydatetime()
if dummy_date < start_date:
start_date = dummy_date
V1 = dict()
# dicts to store the values, V stands for the sub-indices,
# T for their expiry
V2 = dict()
V3 = dict()
T1 = dict()
T2 = dict()
T3 = dict()
# from start_date to max_date, but only weekdays
for day in pd.bdate_range(start=start_date.date(), end=max_date.date()):
# is V6I1 defined?
is_V1_defined = idf.not_a_day_before_expiry(day)
# the settlement date
settlement_date = idf.first_settlement_day(day)
# abbreviation for the expiry date, like Oct14
key = settlement_date.strftime('%b%y')
# days until maturity
delta_T = idf.compute_delta(day, settlement_date)
try:
# data of the option series for that date
data = datastore[key].ix[day]
except:
continue
if is_V1_defined:
# if V6I1 is defined
# compute its value
V1[day] = compute_subindex(data, delta_T,
math.exp(0.0015 * delta_T))
T1[day] = settlement_date
else:
# compute the value of V6I2 instead
V2[day] = compute_subindex(data, delta_T,
math.exp(0.0015 * delta_T))
T2[day] = settlement_date
settlement_date_2 = idf.second_settlement_day(day)
# the same for the next index
key_2 = settlement_date_2.strftime('%b%y')
delta_T_2 = idf.compute_delta(day, settlement_date_2)
data_2 = datastore[key_2].ix[day]
if is_V1_defined:
V2[day] = compute_subindex(data_2, delta_T_2,
math.exp(0.001 * delta_T_2))
T2[day] = settlement_date_2
127
VSTOXX Index
else:
V3[day] = compute_subindex(data_2, delta_T_2,
math.exp(0.001 * delta_T_2))
T3[day] = settlement_date_2
datastore.close()
# create the pandas DataFrame object and return it
df = pd.DataFrame(data={'V6I1': V1, 'Expiry V6I1': T1, 'V6I2': V2,
'Expiry V6I2': T2, 'V6I3': V3, 'Expiry V6I3': T3})
return df
5.6.3 index vstoxx calculation.py
#
# Module to compute VSTOXX index values
# given the values for the relevant sub-indexes
# as generated by the module index_subindex_calculation.py
#
# (c) Dr. Yves J. Hilpisch
# Listed Volatility and Variance Derivatives
#
import pandas as pd
import numpy as np
import matplotlib.pyplot as plt
from index_date_functions import *
def calculate_vstoxx(path):
''' Function to calculate the VSTOXX volatility index given time series
of the relevant sub-indexes.
Parameters
==========
path: string
string with path of data files
Returns
=======
data: pandas DataFrame object
results of index calculation
'''
# constant parameters
seconds_year = 365 * 24 * 3600.
seconds_30_days = 30 * 24 * 3600.
# import historical VSTOXX data
data = pd.read_csv(path + 'vs.csv', index_col=0, parse_dates=True)
128
LISTED VOLATILITY AND VARIANCE DERIVATIVES
# determine the settlement dates for the two underlying option series
data['Settlement date 1'] = [first_settlement_day(a) for a in data.index]
data['Settlement date 2'] = [second_settlement_day(a) for a in data.index]
# deduce the life time (in seconds) from current date to
# final settlement Date
data['Life time 1'] = [(data['Settlement date 1'][i] - i).days
* 24 * 60 * 60 for i in data.index]
data['Life time 2'] = [(data['Settlement date 2'][i] - i).days
* 24 * 60 * 60 for i in data.index]
data['Use V6I2'] = data['V6I1'].notnull()
# where V6I1 is not defined
data['Subindex to use 1'] = [data['V6I1'][i] if data['Use V6I2'][i]
else data['V6I2'][i] for i in data.index]
# if V6I1 is defined, use V6I1 and V6I2 as data set
data['Subindex to use 2'] = [data['V6I2'][i] if data['Use V6I2'][i]
else data['V6I3'][i] for i in data.index]
# else use V6I2 and V6I3
# the linear interpolation of the VSTOXX value
# from the two relevant sub-indexes
data['Part 1'] = data['Life time 1'] / seconds_year
\
\
* data['Subindex to use 1'] ** 2
* ((data['Life time 2'] - seconds_30_days)
/ (data['Life time 2'] - data['Life time 1']))
data['Part 2'] = data['Life time 2'] / seconds_year
\
\
* data['Subindex to use 2'] ** 2
* ((seconds_30_days - data['Life time 1'])
/ (data['Life time 2'] - data['Life time 1']))
data['VSTOXX'] = np.sqrt((data['Part 1'] + data['Part 2']) *
seconds_year / seconds_30_days)
# difference between original VSTOXX data and re-calculated values
data['Difference'] = data['V2TX'] - data['VSTOXX']
return data
\
Listed Volatility and Variance Derivatives: A
Python-based Guide
By Dr. Yves J. Hilpisch
© 2017 Yves Hilpisch
CHAPTER
6
Valuing Volatility Derivatives
6.1 INTRODUCTION
This chapter illustrates the valuation of volatility futures and options according to Grünbichler
and Longstaff (1996), abbreviated in the following by GL96. They derive a semi-analytical
(“closed”) pricing formula for European volatility call options which is as easy to use as
the famous Black-Scholes-Merton formula for equity options pricing. They model volatility directly and make the assumption that volatility follows a square-root diffusion process.
The model is quite simple and parsimonious, such that it lends itself pretty well to serve as a
starting point.
This chapter introduces the financial model, the futures and option pricing formulas of
Grünbichler and Longstaff as well as a discretization of the model for Monte Carlo simulation
purposes. For both the formulas and the Monte Carlo simulation approach, Python implementations are presented. Finally, the chapter shows in detail how to calibrate the GL96 model to
market quotes for European call options on the VSTOXX volatility index.
6.2 THE VALUATION FRAMEWORK
Grünbichler and Longstaff (1996) model the volatility process (e.g. the process of a volatility
index) in direct fashion by a square-root diffusion or CIR process – named after the authors
John C. Cox, Jonathan E. Ingersoll and Stephen A. Ross, who first introduced this type of
stochastic process to finance; see Cox et al. (1985). The stochastic differential equation describing the evolution of the volatility over time takes the form
√
dvt = 𝜅(𝜃 − vt )dt + 𝜎 vdZt
where
vt is the time t value of the volatility index, for example the VSTOXX
𝜃 is the long-run mean of the volatility index
𝜅 is the rate at which vt reverts to 𝜃
𝜎 is the volatility of the volatility (“vol-vol”)
𝜃, 𝜅, 𝜎 are assumed to be constant and positive
Zt is a standard Brownian motion.
129
130
LISTED VOLATILITY AND VARIANCE DERIVATIVES
This process is known to exhibit convenient and realistic features for volatility modeling, such
as positivity and mean-reversion.
For the numerical examples in this chapter we use a parametrization as follows:
In [1]: import math
# model parameters
In [2]: v0 = 17.5 # initial level of volatility index
In [3]: kappa = 0.1
# speed of mean reversion
In [4]: theta = 20.0
# long-term index level
In [5]: sigma = 2.0
# volatility of volatility
In [6]: zeta = 0.0
In [7]: r = 0.01
# factor of the expected volatility risk premium
# risk-free short rate
# option parameters
In [8]: K = 20.0
In [9]: T = 1.0
# strike
# time horizon in year fractions
6.3 THE FUTURES PRICING FORMULA
Denote by F(v, T) the futures price of a futures contract on the volatility index v with maturity
T. The pricing of such a contract is done by taking expectations of the index’s value at maturity
F(v, T) = E(vT ). One obtains the following formula for the futures price:
F(v0 , T) = (1 − e−𝜅T ) ⋅ 𝜃 + e−𝜅T ⋅ v0
It is an exponentially weighted average of the long-run mean and the current value of the
volatility index. The function futures_price implements this formula in Python, in this
instance for a general volatility risk premium 𝜁 ≥ 0.
def futures_price(v0, kappa, theta, zeta, T):
''' Futures pricing formula in GL96 model.
Parameters
==========
v0: float (positive)
current volatility level
131
Valuing Volatility Derivatives
kappa: float (positive)
mean-reversion factor
theta: float (positive)
long-run mean of volatility
zeta: float (positive)
volatility risk premium
T: float (positive)
time-to-maturity
Returns
=======
future: float
price of a future
'''
alpha = kappa * theta
beta = kappa + zeta
future =(alpha / beta * (1 - math.exp(-beta * T))
+
math.exp(-beta * T) * v0)
return future
Application of the formula and function is straightforward. However, we have to first import
the function from the module in which it is stored.
In [10]: from srd_functions import *
In [11]: futures_price(v0, kappa, theta, zeta, T)
Out[11]: 17.7379064549101
We obtain a futures term structure by calculating the futures prices for different maturities (see
Figure 6.1).
In [12]: import numpy as np
In [13]: import seaborn as sns; sns.set()
In [14]: import matplotlib
In [15]: matplotlib.rcParams['font.family'] = 'serif'
In [16]: import matplotlib.pyplot as plt
In [17]: maturities = np.linspace(0, 2, 24)
132
FIGURE 6.1
LISTED VOLATILITY AND VARIANCE DERIVATIVES
Futures prices from today to a maturity of 24 months.
In [18]: futures_prices = [futures_price(v0, kappa, theta, zeta, T)
....:
for T in maturities]
....:
In [19]: plt.figure(figsize=(10, 6));
In [20]: plt.plot(maturities, futures_prices);
In [21]: plt.xlabel('maturity');
In [22]: plt.ylabel('futures price');
6.4 THE OPTION PRICING FORMULA
A European call option on the underlying V has a payoff function max(VT − K, 0) at maturity T
where K is the strike price of the option. Grünbichler and Longstaff (1996) derive the following
pricing formula for such a call option:
C(v0 , K, T) = e−rT ⋅ e−𝛽T ⋅ V0 ⋅ Q (𝛾 ⋅ K; 𝜈 + 4, 𝜆)
)
𝛼 (
+ e−rT ⋅ ( ) ⋅ 1 − e−𝛽T ⋅ Q (𝛾 ⋅ K; 𝜈 + 2, 𝜆)
𝛽
−rT
−e
⋅ K ⋅ Q (𝛾 ⋅ K; 𝜈, 𝜆)
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Valuing Volatility Derivatives
where
𝛼 = 𝜅𝜃
𝛽 = 𝜅+𝜁
4𝛽
𝜎 2 (1 − e−𝛽T )
4𝛼
𝜈= 2
𝜎
𝜆 = 𝛾 ⋅ e−𝛽T ⋅ v0
𝛾=
and e−rT as the discount factor for a fixed short rate r. The parameter 𝜁 denotes as before the
expected premium for volatility risk. In the following, we assume 𝜁 = 0.
Q(⋅; 𝜈, 𝜆) is the complementary non-central 𝜒 2 distribution with 𝜈 degrees of freedom and
non-centrality parameter 𝜆.
In our Python implementation srd_functions.py (see sub-section 6.9.1,
srd_functions.py for the complete module), we use the function ncx2.cdf() from
the scipy.stats sub-library for the non-central 𝜒 2 distribution. We implement two
different functions, a helper function and the pricing function itself:
cx(): this function returns values for the complementary distribution of the non-central
chi-squared density
call_price(): the implementation of the valuation formula for European calls on the
volatility index.
The function cx() is as follows:
def cx(K, gamma, nu, lamb):
''' Complementary distribution function of non-central chi-squared density.
Parameters
==========
K: float (positive)
strike price
gamma: float (positive)
as defined in the GL96 model
nu: float (positive)
degrees of freedom
lamb: float (positive)
non-centrality parameter
Returns
=======
complementary distribution of nc cs density
'''
return 1 - scs.ncx2.cdf(gamma * K, nu, lamb)
134
LISTED VOLATILITY AND VARIANCE DERIVATIVES
The function cx() is used in the valuation function call_price().
def call_price(v0, kappa, theta, sigma, zeta, T, r, K):
''' Call option pricing formula in GL96 Model
Parameters
==========
v0: float (positive)
current volatility level
kappa: float (positive)
mean-reversion factor
theta: float (positive)
long-run mean of volatility
sigma: float (positive)
volatility of volatility
zeta: float (positive)
volatility risk premium
T: float (positive)
time-to-maturity
r: float (positive)
risk-free short rate
K: float(positive)
strike price of the option
Returns
=======
call: float
present value of European call option
'''
D = math.exp(-r * T) # discount factor
alpha = kappa * theta
beta = kappa + zeta
gamma = 4 * beta / (sigma ** 2 * (1 - math.exp(-beta * T)))
nu = 4 * alpha / sigma ** 2
lamb = gamma * math.exp(-beta * T) * v0
# the pricing formula
call = (D * math.exp(-beta * T) * v0 * cx(K, gamma, nu + 4, lamb)
+ D * (alpha / beta) * (1 - math.exp(-beta * T))
* cx(K, gamma, nu + 2, lamb)
- D * K * cx(K, gamma, nu, lamb))
return call
As before, being equipped with such a Python function makes applying the pricing formula
straightforward.
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Valuing Volatility Derivatives
In [23]: call_price(v0, kappa, theta, sigma, zeta, T, r, K)
Out[23]: 3.3682878822902369
Let us calculate European call option prices over a wider range of strikes.
In [24]: import numpy as np
In [25]: import pandas as pd
In [26]: option_values = []
In [27]: strikes = np.linspace(15, 25)
In [28]: option_values = [call_price(v0, kappa, theta, sigma, zeta, T, r, k)
....:
for k in strikes]
....:
In [29]: data = pd.DataFrame(option_values, index=strikes,
....:
columns=['call values', ])
....:
In [30]: data.index.name = 'strike'
As Figure 6.2 shows, the European call option prices behave comparably to equity options:
the higher the strike price the lower the option present value ceteris paribus.
In [31]: data.plot(figsize=(10, 6));
6.5 MONTE CARLO SIMULATION
In this section, we solve the pricing problem for a European call option via Monte Carlo simulation, i.e. based on a large number of simulated volatility index paths. To this end, we have
to discretize the stochastic differential equation to obtain a difference equation which we can
evaluate numerically.
We use the same numerical parameters as before, but we have to define some additional
parameters which will determine the discretization interval and the number of simulated paths:
# simulation parameters
In [32]: M = 50 # time steps
In [33]: I = 20000
# number of MCS paths
136
FIGURE 6.2
LISTED VOLATILITY AND VARIANCE DERIVATIVES
European call option prices for different volatility strikes in the GL96 model.
There is a large literature available about the discretization of the square-root diffusion (see
Hilpisch (2015), ch. 10). In the following, we will implement an exact discretization scheme.
First, we divide the time interval [0, T] into M time intervals with M + 1 discrete points
in time: t ∈ {0, Δt, 2Δt, … , T}. We have:
vt =
𝜎 2 (1 − e−𝜅Δt ) 2
4−𝜅Δt
𝜒̃𝜈,𝜆 2
⋅ vs
4𝜅
𝜎 (1 − e−𝜅Δt )
with s = t − Δt.
2 denotes a non-central chi-squared distributed random variable with
𝜒̃𝜈,𝜆
𝜈=
4𝜃𝜅
𝜎2
degrees of freedom and non-centrality parameter
𝜆=
4𝜅e−𝜅Δt
⋅ vs
𝜎 2 (1 − e−𝜅Δt )
As it may be more convenient to sample a chi-squared random variable instead of non-central
chi-squared one, we use the following equations:
2
= (z +
𝜈 > 1 : 𝜒̃𝜈,𝜆
√
2
2
𝜈 ≤ 1 : 𝜒̃𝜈,𝜆
= 𝜒𝜈+2P
2
𝜆)2 + 𝜒𝜈−1
Valuing Volatility Derivatives
137
where z is an independent standard normally distributed random variable and P is a Poisson
distributed random variable with intensity 𝜆2 .
We again divide the implementation in Python into two functions:
generate_paths() generates simulated volatility level paths
call_estimator() simulates the volatility process and calculates the Monte Carlo
estimator for the European call option value.
The function to generate simulated volatility paths takes on a form as follows:
def generate_paths(x0, kappa, theta, sigma, T, M, I):
''' Simulation of square-root diffusion with exact discretization
Parameters
==========
x0: float (positive)
starting value
kappa: float (positive)
mean-reversion factor
theta: float (positive)
long-run mean
sigma: float (positive)
volatility (of volatility)
T: float (positive)
time-to-maturity
M: int
number of time intervals
I: int
number of simulation paths
Returns
=======
x: NumPy ndarray object
simulated paths
'''
dt = float(T) / M
x = np.zeros((M + 1, I), dtype=np.float)
x[0, :] = x0
# matrix filled with standard normal distributed rv
ran = np.random.standard_normal((M + 1, I))
d = 4 * kappa * theta / sigma ** 2
# constant factor in the integrated process of x
c = (sigma ** 2 * (1 - math.exp(-kappa * dt))) / (4 * kappa)
if d > 1:
for t in range(1, M + 1):
# non-centrality parameter
l = x[t - 1, :] * math.exp(-kappa * dt) / c
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LISTED VOLATILITY AND VARIANCE DERIVATIVES
# matrix with chi-squared distributed rv
chi = np.random.chisquare(d - 1, I)
x[t, :] = c * ((ran[t] + np.sqrt(l)) ** 2 + chi)
else:
for t in range(1, M + 1):
l = x[t - 1, :] * math.exp(-kappa * dt) / c
N = np.random.poisson(l / 2, I)
chi = np.random.chisquare(d + 2 * N, I)
x[t, :] = c * chi
return x
An application of this function might look as follows:
In [34]: paths = generate_paths(v0, kappa, theta, sigma, T, M, I)
Figure 6.3 shows the first 10 simulated volatility paths.
In [35]: pd.DataFrame(paths[:, :10]).plot(legend=False, figsize=(10, 6));
FIGURE 6.3
10 simulated volatility paths for GL96 model.
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Valuing Volatility Derivatives
The Monte Carlo estimator for a European call option is defined by
1∑
max(viT − K, 0)
I i=1
I
C(v0 , K, T) = e−rT
with max(vT − K, 0) being the payoff function of the European call option. Here, viT is the i-th
simulated value for the volatility index at maturity. The function to calculate the Monte Carlo
estimator for the value of the European call option is presented below.
def call_estimator(v0, kappa, theta, sigma, T, r, K, M, I):
''' Estimation of European call option price in GL96 Model
via Monte Carlo simulation
Parameters
==========
v0: float (positive)
current volatility level
kappa: float (positive)
mean-reversion factor
theta: float (positive)
long-run mean of volatility
sigma: float (positive)
volatility of volatility
T: float (positive)
time-to-maturity
r: float (positive)
risk-free short rate
K: float (positive)
strike price of the option
M: int
number of time intervals
I: int
number of simulation paths
Returns
=======
callvalue: float
MCS estimator for European call option
'''
V = generate_paths(v0, kappa, theta, sigma, T, M, I)
callvalue = math.exp(-r * T) * np.sum(np.maximum(V[-1] - K, 0)) / I
return callvalue
140
LISTED VOLATILITY AND VARIANCE DERIVATIVES
In [36]: call_estimator(v0, kappa, theta, sigma, T, r, K, M, I)
Out[36]: 3.3751776084056191
Again, let us calculate option prices over a wider range of strikes.
In [37]: %%time
....: estimates = []
....: for k in strikes:
....:
estimates.append(call_estimator(v0, kappa, theta, sigma,
....:
....: data['estimates'] = estimates
T, r, k, M, I))
....:
CPU times: user 11.4 s, sys: 38 us, total: 11.4 s
Wall time: 11.4 s
Figure 6.4 compares the Monte Carlo estimator values given the above parametrization with
the European call option values obtained from the formula by Grünbichler and Longstaff
(1996).
In [38]: data.plot(style=['b', 'r.'], figsize=(10, 6));
FIGURE 6.4
Monte Carlo estimates vs. European call values from formula in the GL96 model.
Valuing Volatility Derivatives
141
6.6 AUTOMATED MONTE CARLO TESTS
This section illustrates how to automate Monte Carlo-based valuation tests. Even if semianalytical option pricing formulas are available – as in the Grünbichler and Longstaff (1996)
model – in general one needs numerical methods as well. One of the most flexible and powerful methods when it comes to the valuation of more complex, exotic instruments is Monte
Carlo simulation (MCS).
When implementing MCS algorithms it is generally advised to benchmark the results from
these algorithms against results from other valuation methods which are known to deliver exact
values. Therefore, this section implements MCS for the Grünbichler and Longstaff (1996)
model and benchmarks the results against the semi-analytical formula for European volatility
call options as presented earlier in this chapter.
In the previous section, MCS was introduced and some benchmark results presented.
However, this section enhances the analysis and adds functionalities for the structured storage and systematic analysis of valuation results. In this regard, the Python library pandas is
used in combination with the storage capabilities of PyTables, the Python wrapper for the
HDF5 database file format.
6.6.1 The Automated Testing
The following test procedures are based on the Python script srd_ simulation_analysis.py (see sub-section 6.9.2, srd simulation analysis.py for the complete
code). To automate the valuation for a significant number of options, a set of parameter values
and lists has to be specified. First, some imports and the parametrization of the financial model.
import time
import math
import numpy as np
from datetime import datetime
from srd_functions import generate_paths, call_price
from srd_simulation_results import *
# Model Parameters
v0 = 20.0 # initial volatility
kappa = 3.0 # speed of mean reversion
theta = 20.0
sigma = 3.2
# long-term volatility
# standard deviation coefficient
zeta = 0.0 # factor of the expected volatility risk premium
r = 0.01 # risk-free short rate
The parameter values chosen are not too unrealistic, for example, for the VSTOXX volatility
index. Second, the following simulation parameters are assumed:
# General Simulation Parameters
write = True
var_red = [(False, False), (False, True), (True, False), (True, True)]
142
LISTED VOLATILITY AND VARIANCE DERIVATIVES
# 1st = mo_match -- random number correction (std + mean + drift)
# 2nd = anti_paths -- antithetic paths for variance reduction
# number of time steps
steps_list = [25, 50, 75, 100]
# number of paths per valuation
paths_list = [2500, 50000, 75000, 100000, 125000, 150000]
SEED = 100000 # seed value
runs = 3 # number of simulation runs
PY1 = 0.010 # performance yardstick 1: abs. error in currency units
PY2 = 0.010 # performance yardstick 2: rel. error in decimals
maturity_list = [1.0 / 12, 1.0 / 4, 1.0 / 2, 1.0] # maturity list
The single parameters and list objects have the following meanings:
write is just a flag to indicate that the results should be stored on disk
var_red contains the configuration for the variance reduction techniques, i.e. moment
matching of pseudo-random numbers and antithetic variates
steps_list is the list with the different numbers for the time steps to be used for the
discretization
paths_list contains the different numbers for the MCS paths for the volatility process
runs is a parameter which says how often the same option is valued for a single parameter
combination
SEED is a seed value for NumPy’s pseudo-random number generator; it is fixed to make
it easier to compare results
PY1 is the performance yardstick for absolute differences
PY2 is the performance yardstick for relative differences
maturity_list is the list with times-to-maturity
strike_list finally is the list with the different strike prices.
For details on such Monte Carlo simulation analyses and the respective MCS background
information refer to Hilpisch (2015, ch. 10). For a detailed exposition of Monte Carlo methods
implemented in Python see also Hilpisch (2014, ch. 10).
The script srd_simulation_analysis.py used in this section relies on the MCS
function from the previous section. In addition, it uses the following function which implements two variance reduction techniques, moment matching and antithetic variates:
def randoms(M, I):
''' Function to generate pseudo-random numbers with variance reduction.
Parameters
==========
M: int
number of discrete time intervals
I: int
number of simulated paths
143
Valuing Volatility Derivatives
Returns
=======
rand: Numpy ndarray object
object with pseudo-random numbers
'''
if anti_paths is True:
rand_ = np.random.standard_normal((M + 1, I / 2))
rand = np.concatenate((rand_, -rand_), 1)
else:
rand = np.random.standard_normal((M + 1, I))
if mo_match is True:
rand = rand / np.std(rand)
rand = rand - np.mean(rand)
return rand
The main valuation and testing loop is implemented through the following code. Important
passages are commented inline.
t0 = time.time()
sim_results = pd.DataFrame()
for vr in var_red:
# variance reduction techniques
mo_match, anti_paths = vr
for M in steps_list:
# number of time steps
for I in paths_list:
# number of paths
t1 = time.time()
d1 = datetime.now()
abs_errors = []
rel_errors = []
l = 0.0
errors = 0
# name of the simulation setup
name = ('Call_' + str(runs) + '_'
+ str(M) + '_' + str(I / 1000)
+ '_' + str(mo_match)[0] + str(anti_paths)[0] +
'_' + str(PY1 * 100) + '_' + str(PY2 * 100))
np.random.seed(SEED)
# RNG seed value
for run in range(runs):
# simulation runs
print "\nSimulation Run %d of %d" % (run + 1, runs)
print "-----------------------------------------------------------"
print ("Elapsed Time in Minutes %8.2f"
% ((time.time() - t0) / 60))
print " -----------------------------------------------------------"
z = 0
for T in maturity_list:
dt = T / M
# time-to-maturity
# time interval in year fractions
V = generate_paths(v0, kappa, theta, sigma, T, M, I)
# volatility process paths
print "\n
Results for Time-to-Maturity %6.3f" % T
144
LISTED VOLATILITY AND VARIANCE DERIVATIVES
print "
----------------------------------------------------"
for K in strike_list:
# Strikes
h = np.maximum(V[-1] - K, 0)
# inner value matrix
# MCS estimator
call_estimate = math.exp(-r * T) * np.sum(h) / I * 100
# BSM analytical value
callalue = call_price(v0, kappa, theta, sigma,
zeta, T, r, K) * 100
# errors
diff = call_estimate - callalue
rdiff = diff / callalue
abs_errors.append(diff)
rel_errors.append(rdiff * 100)
# output
br = "
-----------------------------------------------------"
print "\n
print ("
Results for Strike %4.2f\n" % K
European Op. Value MCS
%8.4f" %
call_estimate)
print ("
European Op. Value Closed %8.4f" %
callalue)
print "
Valuation Error (abs)
%8.4f" % diff
print "
Valuation Error (rel)
%8.4f" % rdiff
if abs(diff) < PY1 or abs(diff) / callalue < PY2:
print "
Accuracy ok!\n" + br
CORR = True
else:
print "
Accuracy NOT ok!\n" + br
CORR = False
errors = errors + 1
print "
%d Errors, %d Values, %.1f Min."
\
% (errors, len(abs_errors),
float((time.time() - t1) / 60))
%d Time Intervals, %d Paths"
print("
% (M, I))
z += 1
l += 1
t2 = time.time()
d2 = datetime.now()
if write is True:
# append simulation results
sim_results = write_results(sim_results, name, SEED,
runs, M, I, mo_match, anti_paths,
l, PY1, PY2, errors,
float(errors) / l, np.array(abs_errors),
np.array(rel_errors), t2 - t1, (t2 - t1) / 60, d1, d2)
if write is True:
# write/append DataFrame to HDFStore object
write_to_database(sim_results)
Valuing Volatility Derivatives
145
6.6.2 The Storage Functions
The script srd_simulation_analysis.py from the previous sub-section generates a
number of results and a lot of printed output during execution. However, one generally wants
to store such results in a structured manner and maybe analyze it later on. To this end, we use
the pandas library which provides convenient data storage and analysis features. In combination with the PyTables library, simulation results are easily stored on disk and read from disk
later on.
Consider now the Python module srd_simulation_results.py with several functions to store and analyze the data generated through the other script (see sub-section 6.9.3,
srd_simulation_results.py for the complete script). We will go through this script part by part.
Of course, there are some necessary imports and a filename for the database is also specified:
import numpy as np
import pandas as pd
import datetime as dt
import matplotlib.pyplot as plt
# filname for HDFStore to save results
filename = "../data/simulation_results.h5"
Having a DataFrame object instantiated (with results already stored or not), the following function allows us to add a set of valuation results to the DataFrame as an additional row:
def write_results(sim_results, name, SEED, runs, steps, paths, mo_match,
anti_paths, l, PY1, PY2, errors, error_ratio,
abs_errors, rel_errors, t1, t2, d1, d2):
''' Appends simulation results to pandas DataFrame df and returns it.
Parameters
==========
see srd_simulation_analysis.py
Returns
=======
df: pandas DataFrame object
updated results object
'''
results = {
'sim_name': name,
'seed': SEED,
'runs': runs,
'time_steps': steps,
'paths': paths,
'mo_match': mo_match,
146
LISTED VOLATILITY AND VARIANCE DERIVATIVES
'anti_paths': anti_paths,
'opt_prices': l,
'abs_tol': PY1,
'rel_tol': PY2,
'errors': errors,
'error_ratio': error_ratio,
'aval_err': sum(abs_errors) / l,
'abal_err': sum(abs(rel_errors)) / l,
'time_sec': t1,
'time_min': t2,
'time_opt': t1 / l,
'start_date': d1,
'end_date': d2
}
df = pd.concat([sim_results,
pd.DataFrame([results])],
ignore_index=True)
return df
Once all the single simulation results have been added to the DataFrame sim_results,
this should be stored on disk in a HDFStore file object. This is what the following function
accomplishes:
def write_to_database(sim_results, filename=filename):
''' Write pandas DataFrame sim_results to HDFStore object.
Parameters
==========
sim_results: pandas DataFrame object
object with simulation results
filename: string
name of the file for storage
'''
h5 = pd. HDFStore(filename, 'a')
h5.append('sim_results', sim_results, min_itemsize={'values': 30},
ignore_index=True)
h5.close()
6.6.3 The Results
Let us now turn to the analysis of valuation results. To this end, we work with the Python module srd_simulation_results.py (see sub-section 6.9.3, srd_simulation_results.py for
147
Valuing Volatility Derivatives
the code). The function print_results() prints simulation results stored in the HDFStore file object with name filename. If not otherwise parametrized, the function prints the
first 50 results stored in the database (if that many exist).
def print_results(filename=filename, idl=0, idh=50):
''' Prints valuation results in detailed form.
Parameters
==========
filename: string
HDFStore with pandas.DataFrame with results
idl: int
start index value
idh: int
stop index value
'''
h5 = pd. HDFStore(filename, 'r')
sim_results = h5['sim_results']
br = "-----------------------------------------------------------"
for i in range(idl, min(len(sim_results), idh + 1)):
row = sim_results.iloc[i]
print br
print "Start Calculations
print "ID Number
%32s" % row['start_date'] + "\n" + br
%32d" % i
print "Name of Simulation
print "Seed Value for RNG
print "Number of Runs
%32s" % row['sim_name']
%32d" % row['seed']
%32d" % row['runs']
print "Time Steps
print "Paths
print "Moment Matching
%32d" % row['time_steps']
%32d" % row['paths']
%32s" % row['mo_match']
print "Antithetic Paths
print "Option Prices
print "Absolute Tolerance
%32s" % row['anti_paths'] + "\n"
%32d" % row['opt_prices']
%32.4f" % row['abs_tol']
print
print
print
print
print
print
"Relative Tolerance
"Errors
"Error Ratio
"Aver Val Error
"Aver Abs Val Error
"Time in Seconds
%32.4f"
%32d" %
%32.4f"
%32.4f"
%32.4f"
%32.4f"
% row['rel_tol']
row['errors']
% row['error_ratio'] + "\n"
% row['aval_err']
% row['abal_err']
% row['time_sec']
print "Time in Minutes
%32.4f" % row['time_min']
print "Time per Option
%32.4f" % row['time_opt'] + "\n" + br
print "End Calculations
%32s" % row['end_date']\
+ "\n" + br + "\n"
print "Total number of rows in table %d" % len(sim_results)
h5.close()
148
LISTED VOLATILITY AND VARIANCE DERIVATIVES
Let us inspect selected valuation results from running the respective Python script
srd_simulation_analysis.py:
-----------------------------------------------------------------Start Calculations
2016-01-11 11:02:11.159464
-----------------------------------------------------------------ID Number
2
Name of Simulation
Call_3_25_75_FF_1.0_1.0
Seed Value for RNG
Number of Runs
Time Steps
Paths
Moment Matching
Antithetic Paths
100000
3
25
75000
False
False
Option Prices
Absolute Tolerance
60
0.0100
Relative Tolerance
Errors
Error Ratio
0.0100
9
0.1500
Aver Val Error
Aver Abs Val Error
Time in Seconds
Time in Minutes
-0.1963
0.4773
4.1964
0.0699
Time per Option
0.0699
-----------------------------------------------------------------End Calculations
2016-01-11 11:02:15.355906
----------------------------------------------------------------------------------------------------------------------------------Start Calculations
2016-01-11 11:02:15.362043
-----------------------------------------------------------------ID Number
3
Name of Simulation
Seed Value for RNG
Number of Runs
Time Steps
Paths
Moment Matching
Antithetic Paths
Option Prices
Absolute Tolerance
Relative Tolerance
Errors
Call_3_25_100_FF_1.0_1.0
100000
3
25
100000
False
False
60
0.0100
0.0100
5
149
Valuing Volatility Derivatives
Error Ratio
Aver Val Error
Aver Abs Val Error
Time in Seconds
0.0833
-0.1645
0.3589
5.7146
Time in Minutes
0.0952
Time per Option
0.0952
-----------------------------------------------------------------End Calculations
2016-01-11 11:02:21.076603
----------------------------------------------------------------------------------------------------------------------------------Start Calculations
2016-01-11 11:02:21.082678
-----------------------------------------------------------------ID Number
Name of Simulation
Seed Value for RNG
Number of Runs
Time Steps
4
Call_3_25_125_FF_1.0_1.0
100000
3
25
Paths
Moment Matching
Antithetic Paths
125000
False
False
Option Prices
Absolute Tolerance
60
0.0100
Relative Tolerance
Errors
Error Ratio
0.0100
2
0.0333
Aver Val Error
Aver Abs Val Error
0.0043
0.2666
Time in Seconds
Time in Minutes
Time per Option
6.9533
0.1159
0.1159
-----------------------------------------------------------------End Calculations
2016-01-11 11:02:28.035962
------------------------------------------------------------------
“Time in Minutes” and “Time per Option” values coincide since 60 options are valued per run.
In principle, the results from these three different parametrizations already illustrate
that valuation accuracy is, in general, higher the higher the computational effort (in this
case the number of paths). The number of errors drops from 9 for 75,000 paths to 2 for
125,000 paths.
Finally, the second major function contained in the Python module is
plot_error_ratio() and plots the error ratios against the computational effort,
i.e. (number of simulation paths) x (number of time steps).
150
LISTED VOLATILITY AND VARIANCE DERIVATIVES
def plot_error_ratio(filename=filename):
''' Show error ratio vs. paths * time_steps (i.e.granularity).
Parameters
==========
filename: string
name of file with data to be plotted
'''
h5 = pd. HDFStore(filename, mode='r')
sim_results = h5['sim_results']
x = np.array(sim_results['paths'] * sim_results['time_steps'], dtype='d')
x = x / max(x)
y = sim_results['error_ratio']
plt.plot(x, y, 'bo', label=' error ratio')
rg = np.polyfit(x, y, deg=1)
plt.plot(np.sort(x), np.polyval(rg, np.sort(x)), 'r', label='regression',
linewidth=2)
plt.xlabel('time steps * paths (normalized)')
plt.ylabel('errors / option valuations')
plt.legend()
plt.grid(True)
h5.close()
Let us generate an overview plot with the above function. The graphical output is shown as
Figure 6.5.
In [39]: from srd_simulation_results import plot_error_ratio
In [40]: plot_error_ratio(filename)
The linear regression illustrates graphically the statement that the higher the computational
effort the higher in general the accuracy of the MCS valuations. The high variability of the
error ratios stems from the fact that it is not only the computational effort which influences
valuation accuracy. For example, the use of variance reduction techniques also has a significant
influence.
This being said, one might be interested in analyzing the impact of the variance reduction
techniques on valuation accuracy. To this end, pandas provides powerful grouping methods
for the DataFrame object. First, we open the database in which the simulation results are
stored and read them into an in memory object:
Valuing Volatility Derivatives
FIGURE 6.5
Error ratios of Monte Carlo valuation vs. computational effort.
In [41]: import pandas as pd
In [42]: h5 = pd.HDFStore(filename, 'r')
In [43]: data = h5['sim_results']
In [44]: h5.close()
In [45]: data.info()
Int64Index: 96 entries, 0 to 95
Data columns (total 19 columns):
ab_val_err
96 non–null float64
abs_tol
anti_paths
av_val_err
96 non–null float64
96 non–null bool
96 non–null float64
end_date
error_ratio
96 non–null datetime64[ns]
96 non–null float64
errors
mo_match
opt_prices
96 non–null int64
96 non–null bool
96 non–null float64
paths
rel_tol
96 non–null int64
96 non–null float64
151
152
LISTED VOLATILITY AND VARIANCE DERIVATIVES
runs
seed
96 non–null int64
96 non–null int64
sim_name
start_date
time_min
96 non–null object
96 non–null datetime64[ns]
96 non–null float64
time_opt
time_sec
time_steps
96 non–null float64
96 non–null float64
96 non–null int64
dtypes: bool(2), datetime64[ns](2), float64(9), int64(5), object(1)
memory usage: 13.7+ KB
The simulation results are stored in 96 rows of the pandas DataFrame object. The method
groupby allows analyses of such data sets in “all directions and dimensions.” For example,
one gets the average error ratio depending on the use of moment matching with the following
code:
In [46]: data.groupby(['mo_match']).mean()[['error_ratio']]
Out[46]:
error_ratio
mo_match
False
0.193750
True
0.169097
In the simulations, moment matching obviously reduces the error ratio by slightly more than
2 percentage points from above 19% to less than 17%.
The same analysis can be done with more data columns, both for the grouping and the
output.
In [47]: data.groupby(['mo_match', 'anti_paths']).mean()[['error_ratio', 'errors']]
Out[47]:
error_ratio
errors
False
0.206944
12.416667
True
0.180556
10.833333
False
0.159028
9.541667
True
0.179167
10.750000
mo_match anti_paths
False
True
Here, the results indicate that moment matching alone should be used. Adding antithetic paths
increases the error ratio ceteris paribus.
In similar fashion, the data can be grouped by time steps and paths – by analogy with the
analysis above. Figure 6.6 presents the results.
Valuing Volatility Derivatives
FIGURE 6.6
153
Error ratios of Monte Carlo valuation vs. time per option.
In [48]: df = data.groupby(['time_steps', 'paths']).mean()[['error_ratio',
'time_opt']]
In [49]: df.plot(x='time_opt', y='error_ratio', style='bo');
In [50]: plt.xlabel('time per option valuation');
In [51]: plt.ylabel('error ratio');
As expected, the general rule that accuracy increases with increasing computational effort is
again supported by this specific analysis. Of course, other analyses can be done with the data
generated and stored on disk.
6.7 MODEL CALIBRATION
Previous sections implement a theoretical valuation approach in that a parametric model is
given and parameter values are simply assumed. However, any realistic valuation of volatility
options has to incorporate at least some information from the market itself. This is generally
done by taking the quotes of liquidly traded vanilla options as input for a so-called model
154
LISTED VOLATILITY AND VARIANCE DERIVATIVES
calibration. During the calibration of a financial model, parameters are determined for that
model which best replicate the observed option quotes. These parameters are in turn used to
value other, maybe more exotic, derivatives by means of numerical methods such as Monte
Carlo simulation. Such a procedure is often termed a market-based valuation approach.
This section illustrates how to calibrate the financial model of Grünbichler and Longstaff
(1996) to option quotes for European volatility calls traded at Eurex. In the following, the
semi-analytical formula as introduced in section 6.4, The Option Pricing Formula is used to
accomplish the model calibration.
6.7.1 The Option Quotes
The remainder of this section will work with option quotes of VSTOXX call options traded at
Eurex which are all of European type. The data is as of March 31, 2014 and contains quotes
for the options with maturities for each month from April 2014 to November 2014. Let us have
a look at the data.
In [52]: import pandas as pd
In [53]: path = './source/data/'
In [54]: h5 = pd.HDFStore(path + 'vstoxx_option_quotes.h5', 'r')
In [55]: option_quotes = h5['option_quotes']
In [56]: option_quotes.info()
Int64Index: 98 entries, 46219 to 46365
Data columns (total 8 columns):
DATE
EXP_YEAR
EXP_MONTH
98 non–null datetime64[ns]
98 non–null int64
98 non–null int64
TYPE
STRIKE
PRICE
98 non–null object
98 non–null float64
98 non–null float64
MATURITY
98 non–null datetime64[ns]
TTM
98 non–null float64
dtypes: datetime64[ns](2), float64(3), int64(2), object(1)
memory usage: 6.9+ KB
In [57]: option_quotes.iloc[25: 35]
Out[57]:
DATE EXP_YEAR EXP_MONTH TYPE
MATURITY
TTM
46244 2014-03-31
46245 2014-03-31
46246 2014-03-31
2014
2014
2014
5
5
5
C
C
C
STRIKE
26.0
27.0
28.0
PRICE
0.60 2014-05-16
0.50 2014-05-16
0.45 2014-05-16
0.126
0.126
0.126
46247 2014-03-31
2014
5
C
29.0
0.40 2014-05-16
0.126
155
Valuing Volatility Derivatives
46248 2014-03-31
2014
5
C
30.0
0.35 2014-05-16
0.126
46249 2014-03-31
46250 2014-03-31
46251 2014-03-31
2014
2014
2014
5
5
5
C
C
C
32.5
35.0
37.5
0.25 2014-05-16
0.20 2014-05-16
0.15 2014-05-16
0.126
0.126
0.126
46252 2014-03-31
46253 2014-03-31
2014
2014
5
5
C
C
40.0
42.5
0.10 2014-05-16
0.10 2014-05-16
0.126
0.126
As you can see, the data set comprises a total of 98 option quotes. Note that option quotes for
Eurex volatility options have tick sizes of 5 cents, i.e. 0.05 EUR. A detailed overview of the
terms of these options can be found in chapter 8, Terms of the VSTOXX and its Derivatives. In
what follows, we assume that each option is written on a single unit of the underlying, i.e. 1
point of the VSTOXX index translates into 1 EUR.
6.7.2 The Calibration Procedure
In simple terms, the problem of calibration is to find parameters for an option model such
that observed market quotes of liquidly traded plain vanilla options are replicated as closely
as possible. To this end, one defines an error function that is to be minimized. Such a function
could be the Mean Squared Absolute Error (MSAE). The task is then to solve the problem
min
𝜅,𝜃,𝜎
N
)2
1 ∑( ∗
Cn − CnGL96 (𝜅, 𝜃, 𝜎)
N n=1
with the Cn∗ being the market or input prices and the CnGL96 being the model or output prices
for the options n = 1, … , N.
For some constellations, it might be more appropriate to minimize the Mean Squared
Relative Error (MSRE):
N
1 ∑
min
𝜅,𝜃,𝜎 N
n=1
(
Cn∗ − CnGL96 (𝜅, 𝜃, 𝜎)
)2
Cn∗
The Python script (see sub-section 6.9.4, srd_model_calibration.py) that implements a procedure to calibrate the Grünbichler-Longstaff option pricing model is explained in the following
step-by-step procedure.
First, some imports and parameter specifications.
import numpy as np
import pandas as pd
from srd_functions import call_price
import scipy.optimize as sco
import matplotlib.pyplot as plt
path = './source/data/'
# Fixed Parameters
156
LISTED VOLATILITY AND VARIANCE DERIVATIVES
v0 = 17.6639 # VSTOXX index on 31. March 2014
r = 0.01 # risk-less short rate
zeta = 0.
# volatility risk premium factor
The function read_select_quotes() reads and selects option quotes for the calibration.
def read_select_quotes(path=path, tol=0.2):
''' Selects and read options quotes.
Parameters
==========
path: string
path to file with option quotes
Returns
=======
option_data: pandas DataFrame object
option data
'''
h5 = pd.HDFStore(path + 'vstoxx_march_2014.h5', 'r')
# read option data from file and close it
option_data = h5['vstoxx_options']
h5.close()
# select relevant date for call option quotes
option_data = option_data[(option_data.DATE == '2014-3-31')
& (option_data.TYPE == 'C')]
# calculate time-to-maturity in year fractions
option_data['TTM'] = (option_data.MATURITY - option_data.DATE).apply (
lambda x: x / np.timedelta64(1, 'D') / 365.)
# only those options close enough to the ATM level
option_data = option_data[(option_data.STRIKE > (1 - tol) * v0)
& (option_data.STRIKE < (1 + tol) * v0)]
return option_data
However, the core functions of the calibration script are the following. On the one hand,
valuation_function().
def valuation_function(p0):
''' Valuation function for set of strike prices
Parameters
==========
p0: list
set of model parameters
Valuing Volatility Derivatives
157
Returns
=======
call_prices: NumPy ndarray object
array of call prices
'''
kappa, theta, sigma = p0
call_prices = []
for strike in strikes:
call_prices.append(call_price(v0, kappa, theta,
sigma, zeta, ttm, r, strike))
call_prices = np.array(call_prices)
return call_prices
On the other hand, error_function().
def error_function(p0):
''' Error function for model calibration.
Parameters
==========
p0: tuple
set of model parameters
Returns
=======
MSE: float
mean squared (relative/absolute) error
'''
global i
call_prices = valuation_function(p0)
kappa, theta, sigma = p0
pen = 0.
if 2 * kappa * theta < sigma ** 2:
pen = 1000.0
if kappa < 0 or theta < 0 or sigma < 0:
pen = 1000.0
if relative is True:
MSE = (np.sum(((call_prices - call_quotes) / call_quotes) ** 2)
/ len(call_quotes) + pen)
else:
MSE = np.sum((call_prices - call_quotes) ** 2) / len(call_quotes) + pen
if i == 0:
print ("{:>6s}{:>6s}{:>6s}".format('kappa', 'theta', 'sigma')
+ "{:>12s}".format('MSE'))
# print intermediate results: every 100th iteration
if i % 100 == 0:
158
LISTED VOLATILITY AND VARIANCE DERIVATIVES
print "{:6.3f}{:6.3f}{:6.3f}".format(*p0) + "{:>12.5f}". format(MSE)
i += 1
return MSE
Function valuation_function() implements a valuation for the array of strikes under
consideration and the parameter values provided. Function error_function() calculates
either the MSAE or the MSRE – given the array of strikes and the parameter vector p0. Some
penalties are also added depending on the single parameter values or their constellation. For
details and a number of similar examples, refer to the book Hilpisch (2015, ch. 11).
Then standard functions from the scipy.optimize library (see Hilpisch (2014),
ch. 9) are used to implement both a global (i.e. via the brute() function) and a
local optimization (i.e. via the fmin() function). This is all wrapped into the function
model_calibration().
def model_calibration(option_data, rel=False, mat='2014-07-18'):
''' Function for global and local model calibration.
Parameters
==========
option_data: pandas DataFrame object
option quotes to be used
relative: boolean
relative or absolute MSE
maturity: string
maturity of option quotes to calibrate to
Returns
=======
opt: tuple
optimal parameter values
'''
global relative # if True: MSRE is used, if False: MSAE
global strikes
global call_quotes
global ttm
global i
relative = rel
# only option quotes for a single maturity
option_quotes = option_data[option_data.MATURITY == mat]
# time-to-maturity from the data set
ttm = option_quotes.iloc[0, -1]
# transform strike column and price column in ndarray object
strikes = option_quotes['STRIKE'].values
159
Valuing Volatility Derivatives
call_quotes = option_quotes['PRICE'].values
# global optimization
i = 0 # counter for calibration iterations
p0 = sco.brute(error_function, ((5.0, 20.1, 1.0), (10., 30.1, 1.25),
(1.0, 9.1, 2.0)), finish=None)
# local optimization
i = 0
opt = sco.fmin(error_function, p0, xtol=0.0000001, ftol=0.0000001,
maxiter=1000, maxfun=1500)
return opt
Finally, the results are visualized by plotting the model values against the market values
and providing the absolute differences separately. This is accomplished by the function
plot_calibration_results().
def plot_calibration_results(opt):
''' Function to plot market quotes vs. model prices.
Parameters
==========
opt: list
optimal parameters from calibration
'''
callalues = valuation_function(opt)
diffs = callalues - call_quotes
plt.figure()
plt.subplot(211)
plt.plot(strikes, call_quotes, label='market quotes')
plt.plot(strikes, callalues, 'ro', label='model prices')
plt.ylabel('option values')
plt.grid(True)
plt.legend()
plt.axis([min(strikes) - 0.5, max(strikes) + 0.5,
0.0, max(call_quotes) * 1.1])
plt.subplot(212)
wi = 0.3
plt.bar(strikes - wi / 2, diffs, width=wi)
plt.grid(True)
plt.xlabel('strike price')
plt.ylabel('difference')
plt.axis([min(strikes) - 0.5, max(strikes) + 0.5,
min(diffs) * 1.1, max(diffs) * 1.1])
plt.tight_layout()
160
LISTED VOLATILITY AND VARIANCE DERIVATIVES
6.7.3 The Calibration Results
We start with a calibration run for the option quotes with shorter maturity and using the MSAE
as a yardstick.
In [58]: from srd_model_calibration import *
In [59]: option_data = read_select_quotes()
In [60]: %time opt = model_calibration(option_data, rel=False, mat=
'2014-05-16')
kappa theta sigma
MSE
5.000 10.000
6.000 12.500
7.000 16.250
1.000
9.000
9.000
7.78606
0.23173
0.40860
8.000 20.000
9.000 23.750
10.000 27.500
9.000
9.000
9.000
2.54160
9.70521
26.90758
12.000 10.000
13.000 13.750
14.000 17.500
9.000
9.000
9.000
3.75104
1.50070
0.06953
15.000 21.250
16.000 25.000
17.000 28.750
9.000
9.000
9.000
3.39340
18.12298
51.74883
19.000 11.250
20.000 15.000
9.000
9.000
5.14886
1.96050
kappa theta
15.000 18.750
16.381 19.112
sigma
7.000
6.613
MSE
0.01356
0.00834
27.671 18.969
33.461 18.950
33.457 18.950
8.513
9.343
9.342
0.00831
0.00831
0.00831
Optimization terminated successfully.
Current function value: 0.008311
Iterations: 247
Function evaluations: 450
CPU times: user 5.53 s, sys: 0 ns, total: 5.53 s
Wall time: 5.54 s
Figure 6.7 shows the calibration results.
In [61]: plot_calibration_results(opt)
161
Valuing Volatility Derivatives
FIGURE 6.7
Calibration of option pricing model to option quotes for May 2014 maturity with
MSAE.
Now the calibration to the option quotes with a longer time-to-maturity and using the MSRE
as a yardstick.
In [62]: %time opt = model_calibration(option_data, rel=True, mat='2014-07-18')
theta
sigma
MSE
5.000 10.000
kappa
1.000
1.00000
6.000 12.500
9.000
0.12866
7.000 16.250
9.000
0.01167
8.000 20.000
9.000
0.18546
9.000 23.750
9.000
1.20473
10.000 27.500
9.000
3.86826
12.000 10.000
9.000
0.73475
13.000 13.750
9.000
0.41208
14.000 17.500
9.000
0.05791
15.000 21.250
9.000
0.15684
16.000 25.000
9.000
1.57671
17.000 28.750
9.000
5.26237
19.000 11.250
9.000
0.81013
20.000 15.000
9.000
0.45062
kappa
theta
sigma
MSE
5.000 20.000
5.000
0.00051
5.848 19.927
5.295
0.00050
162
LISTED VOLATILITY AND VARIANCE DERIVATIVES
FIGURE 6.8
Calibration of option pricing model to option quotes for July 2014 maturity with
MSRE.
11.500 19.608
7.260
0.00049
12.731 19.585
7.638
0.00049
12.732 19.585
7.638
0.00049
Optimization terminated successfully.
Current function value: 0.000495
Iterations: 252
Function evaluations: 450
CPU times: user 5.6 s, sys: 0 ns, total: 5.6 s
Wall time: 5.61 s
The results of this calibration run are shown in Figure 6.8.
In [63]: plot_calibration_results(opt)
Neither of the two fits is satisfactory for such small sets of option prices. The differences are
often higher than the tick size of 5 cents. Therefore, this model obviously does not paint too
realistic a picture of the real world. This may be due to the underlying, i.e. the VSTOXX
volatility index, exhibiting more and/or other properties than the ones captured in the model
Valuing Volatility Derivatives
163
of Grünbichler and Longstaff (1996). For example, a well documented stylized fact is that time
series of volatility indexes, like the VSTOXX, exhibit (positive) jumps of significant size and
with significant probability. Something that is, for example, not captured in the pricing model
used in this chapter.
6.8 CONCLUSIONS
Volatility derivatives, like futures and options on a volatility index, and their arbitrage-free
valuation are the focus of this chapter. The model of Grünbichler and Longstaff (1996) is
introduced since it serves pretty well as a benchmark case – due to its simplicity – for volatility modeling and volatility derivatives valuation. The model uses a square-root diffusion or
CIR process to describe the evolution of volatility (indexes) over time. Futures and European
call option pricing formulas are presented. Based on an exact discretization scheme, Monte
Carlo simulation is described as an alternative numerical pricing method. In addition, Python
code to implement automated Monte Carlo simulation studies is provided and used for such
a study. Finally, the GL96 model is calibrated to market quotes for European call options on
the VSTOXX volatility index. The results illustrate that the model might not be rich enough
to account for market realities.
6.9 PYTHON SCRIPTS
6.9.1 srd functions.py
#
# Module with functions for
# Grünbichler and Longstaff (1996) model
#
# (c) Dr. Yves J. Hilpisch
# Listed Volatility and Variance Derivatives
#
import math
import numpy as np
import scipy.stats as scs
def futures_price(v0, kappa, theta, zeta, T):
''' Futures pricing formula in GL96 model.
Parameters
==========
v0: float (positive)
current volatility level
kappa: float (positive)
mean-reversion factor
theta: float (positive)
long-run mean of volatility
164
LISTED VOLATILITY AND VARIANCE DERIVATIVES
zeta: float (positive)
volatility risk premium
T: float (positive)
time-to-maturity
Returns
=======
future: float
price of a future
'''
alpha = kappa * theta
beta = kappa + zeta
future = (alpha / beta * (1 - math.exp(-beta * T))
+ math.exp(-beta * T) * v0)
return future
def cx(K, gamma, nu, lamb):
''' Complementary distribution function of non-central chi-squared density.
Parameters
==========
K: float (positive)
strike price
gamma: float (positive)
as defined in the GL96 model
nu: float (positive)
degrees of freedom
lamb: float (positive)
non-centrality parameter
Returns
=======
complementary distribution of nc cs density
'''
return 1 - scs.ncx2.cdf(gamma * K, nu, lamb)
def call_price(v0, kappa, theta, sigma, zeta, T, r, K):
''' Call option pricing formula in GL96 Model
Parameters
==========
v0: float (positive)
current volatility level
kappa: float (positive)
mean-reversion factor
theta: float (positive)
long-run mean of volatility
sigma: float (positive)
volatility of volatility
165
Valuing Volatility Derivatives
zeta: float (positive)
volatility risk premium
T: float (positive)
time-to-maturity
r: float (positive)
risk-free short rate
K: float(positive)
strike price of the option
Returns
=======
call: float
present value of European call option
'''
D = math.exp(-r * T)
# discount factor
alpha = kappa * theta
beta = kappa + zeta
gamma = 4 * beta / (sigma ** 2 * (1 - math.exp(-beta * T)))
nu = 4 * alpha / sigma ** 2
lamb = gamma * math.exp(-beta * T) * v0
# the pricing formula
call = (D * math.exp(-beta * T) * v0 * cx(K, gamma, nu + 4, lamb)
+ D * (alpha / beta) * (1 - math.exp(-beta * T))
* cx(K, gamma, nu + 2, lamb)
- D * K * cx(K, gamma, nu, lamb))
return call
def generate_paths(x0, kappa, theta, sigma, T, M, I):
''' Simulation of square-root diffusion with exact discretization
Parameters
==========
x0: float (positive)
starting value
kappa: float (positive)
mean-reversion factor
theta: float (positive)
long-run mean
sigma: float (positive)
volatility (of volatility)
T: float (positive)
time-to-maturity
M: int
number of time intervals
I: int
number of simulation paths
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LISTED VOLATILITY AND VARIANCE DERIVATIVES
Returns
=======
x: NumPy ndarray object
simulated paths
'''
dt = float(T) / M
x = np.zeros((M + 1, I), dtype=np.float)
x[0, :] = x0
# matrix filled with standard normal distributed rv
ran = np.random.standard_normal((M + 1, I))
d = 4 * kappa * theta / sigma ** 2
# constant factor in the integrated process of x
c = (sigma ** 2 * (1 - math.exp(-kappa * dt))) / (4 * kappa)
if d > 1:
for t in range(1, M + 1):
# non-centrality parameter
l = x[t - 1, :] * math.exp(-kappa * dt) / c
# matrix with chi-squared distributed rv
chi = np.random.chisquare(d - 1, I)
x[t, :] = c * ((ran[t] + np.sqrt(l)) ** 2 + chi)
else:
for t in range(1, M + 1):
l = x[t - 1, :] * math.exp(-kappa * dt) / c
N = np.random.poisson(l / 2, I)
chi = np.random.chisquare(d + 2 * N, I)
x[t, :] = c * chi
return x
def call_estimator(v0, kappa, theta, sigma, T, r, K, M, I):
''' Estimation of European call option price in GL96 Model
via Monte Carlo simulation
Parameters
==========
v0: float (positive)
current volatility level
kappa: float (positive)
mean-reversion factor
theta: float (positive)
long-run mean of volatility
sigma: float (positive)
volatility of volatility
T: float (positive)
time-to-maturity
r: float (positive)
risk-free short rate
K: float (positive)
strike price of the option
Valuing Volatility Derivatives
M: int
number of time intervals
I: int
number of simulation paths
Returns
=======
callvalue: float
MCS estimator for European call option
'''
V = generate_paths(v0, kappa, theta, sigma, T, M, I)
callvalue = math.exp(-r * T) * np.sum(np.maximum(V[-1] - K, 0)) / I
return callvalue
6.9.2 srd simulation analysis.py
#
# Valuation of European volatility options
# by Monte Carlo simulation in
# Grünbichler and Longstaff (1996) model
# -- analysis of valuation results
#
# (c) Dr. Yves J. Hilpisch
# Listed Volatility and Variance Derivatives
#
import time
import math
import numpy as np
from datetime import datetime
from srd_functions import generate_paths, call_price
from srd_simulation_results import *
# Model Parameters
v0 = 20.0
# initial volatility
kappa = 3.0
theta = 20.0
sigma = 3.2
zeta = 0.0
r = 0.01
# speed of mean reversion
# long-term volatility
# standard deviation coefficient
# factor of the expected volatility risk premium
# risk-free short rate
# General Simulation Parameters
write = True
var_red = [(False, False), (False, True), (True, False), (True, True)]
# 1st = mo_match -- random number correction (std + mean + drift)
# 2nd = anti_paths -- antithetic paths for variance reduction
# number of time steps
steps_list = [25, 50, 75, 100]
167
168
LISTED VOLATILITY AND VARIANCE DERIVATIVES
# number of paths per valuation
paths_list = [2500, 50000, 75000, 100000, 125000, 150000]
SEED = 100000
runs = 3
# seed value
# number of simulation runs
PY1 = 0.010
# performance yardstick 1: abs. error in currency units
PY2 = 0.010
# performance yardstick 2: rel. error in decimals
maturity_list = [1.0 / 12 , 1.0 / 4, 1.0 / 2, 1.0]
strike_list = [15.0, 17.5, 20.0, 22.5, 25.0]
# maturity list
# strike list
def generate_paths(x0, kappa, theta, sigma, T, M, I):
''' Simulation of square-root diffusion with exact discretization
Parameters
==========
x0: float (positive)
starting value
kappa: float (positive)
mean-reversion factor
theta: float (positive)
long-run mean
sigma: float (positive)
volatility (of volatility)
T: float (positive)
time-to-maturity
M: int
number of time intervals
I: int
number of simulation paths
Returns
=======
x: NumPy ndarray object
simulated paths
'''
dt = float(T) / M
x = np.zeros((M + 1, I), dtype=np.float)
x[0, :] = x0
# matrix filled with standard normally distributed rv
ran = randoms(M, I)
d = 4 * kappa * theta / sigma ** 2
# constant factor in the integrated process of x
c = (sigma ** 2 * (1 - math.exp(-kappa * dt))) / (4 * kappa)
if d > 1:
for t in range(1, M + 1):
# non-centrality parameter
l = x[t - 1, :] * math.exp(-kappa * dt) / c
# matrix with chi-squared distributed rv
chi = np.random.chisquare(d - 1, I)
x[t, :] = c * ((ran[t] + np.sqrt(l)) ** 2 + chi)
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Valuing Volatility Derivatives
else:
for t in range(1, M + 1):
l = x[t - 1, :] * math.exp(-kappa * dt) / c
N = np.random.poisson(l / 2, I)
chi = np.random.chisquare(d + 2 * N, I)
x[t, :] = c * chi
return x
def randoms(M, I):
''' Function to generate pseudo-random numbers with variance reduction.
Parameters
==========
M: int
number of discrete time intervals
I: int
number of simulated paths
Returns
=======
rand: Numpy ndarray object
object with pseudo-random numbers
'''
if anti_paths is True:
rand_ = np.random.standard_normal((M + 1, I / 2))
rand = np.concatenate((rand_, -rand_), 1)
else:
rand = np.random.standard_normal((M + 1, I))
if mo_match is True:
rand = rand / np.std(rand)
rand = rand - np.mean(rand)
return rand
t0 = time.time()
sim_results = pd.DataFrame()
for vr in var_red:
# variance reduction techniques
mo_match, anti_paths = vr
for M in steps_list:
# number of time steps
for I in paths_list:
# number of paths
t1 = time.time()
d1 = datetime.now()
abs_errors = []
rel_errors = []
l = 0.0
errors = 0
# name of the simulation setup
name = ('Call_' + str(runs) + '_'
+ str(M) + '_' + str(I / 1000)
170
LISTED VOLATILITY AND VARIANCE DERIVATIVES
+ '_' + str(mo_match)[0] + str(anti_paths)[0] +
'_' + str(PY1 * 100) + '-' + str(PY2 * 100))
np.random.seed(SEED)
# RNG seed value
for run in range(runs):
# simulation runs
print "\nSimulation Run %d of %d" % (run + 1, runs)
print "-----------------------------------------------------------"
print ("Elapsed Time in Minutes %8.2f"
% ((time.time() - t0) / 60))
print "-----------------------------------------------------------"
z = 0
for T in maturity_list:
dt = T / M
# time-to-maturity
# time interval in year fractions
V = generate_paths(v0, kappa, theta, sigma, T, M, I)
# volatility process paths
print "\n Results for Time-to-Maturity %6.3f" % T
print "-----------------------------------------------------------"
for K in strike_list:
# Strikes
h = np.maximum(V[-1] - K, 0)
# inner value matrix
# MCS estimator
call_estimate = math.exp(-r * T) * np.sum(h) / I * 100
# BSM analytical value
callalue = call_price(v0, kappa, theta, sigma,
zeta, T, r, K) * 100
# errors
diff = call_estimate - callalue
rdiff = diff / callalue
abs_errors.append(diff)
rel_errors.append(rdiff * 100)
# output
br = "
----------------------------------------------"
print "\n Results for Strike %4.2f\n" % K
print ("
European Op. Value MCS
%8.4f" %
call_estimate)
print ("
European Op. Value Closed %8.4f" %
callalue)
print "
Valuation Error (abs)
%8.4f" % diff
print "
Valuation Error (rel)
%8.4f" % rdiff
if abs(diff) < PY1 or abs(diff) / callalue < PY2:
print "
Accuracy ok!\n" + br
CORR = True
else:
print "
Accuracy NOT ok!\n" + br
CORR = False
errors = errors + 1
print "
%d Errors, %d Values, %.1f Min. "
% (errors, len(abs_errors),
float((time.time() - t1) / 60))
print (" %d Time Intervals, %d Paths"
% (M, I))
\
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Valuing Volatility Derivatives
z = z + 1
l = l + 1
t2 = time.time()
d2 = datetime.now()
if write is True:
# append simulation results
sim_results = write_results(sim_results, name, SEED,
runs, M, I, mo_match, anti_paths,
l, PY1, PY2, errors,
float(errors) / l, np.array(abs_errors),
np.array(rel_errors), t2 - t1, (t2 - t1) / 60, d1, d2)
if write is True:
# write/append DataFrame to HDFStore object
write_to_database(sim_results)
6.9.3 srd simulation results.py
#
# Valuation of European volatility options
# by Monte Carlo simulation in
# Grünbichler and Longstaff (1996) model
# -- Creating a database for simulation results
# with pandas and PyTables
#
# (c) Dr. Yves J. Hilpisch
# Listed Volatility and Variance Derivatives
#
import numpy as np
import pandas as pd
import datetime as dt
import matplotlib.pyplot as plt
# filname for HDFStore to save results
filename = "../data/simulation_results.h5"
def write_results(sim_results, name, SEED, runs, steps, paths, mo_match,
anti_paths, l, PY1, PY2, errors, error_ratio,
abs_errors, rel_errors, t1, t2, d1, d2):
''' Appends simulation results to pandas DataFrame df and returns it.
Parameters
==========
see srd_simulation_analysis.py
Returns
=======
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LISTED VOLATILITY AND VARIANCE DERIVATIVES
df: pandas DataFrame object
updated results object
'''
results = {
'sim_name': name,
'seed': SEED,
'runs': runs,
'time_steps': steps,
'paths': paths,
'mo_match': mo_match,
'anti_paths': anti_paths,
'opt_prices': l,
'abs_tol': PY1,
'rel_tol': PY2,
'errors': errors,
'error_ratio': error_ratio,
'aval_err': sum(abs_errors) / l,
'abal_err': sum(abs(rel_errors)) / l,
'time_sec': t1,
'time_min': t2,
'time_opt': t1 / l,
'start_date': d1,
'end_date': d2
}
df = pd.concat([sim_results,
pd.DataFrame([results])],
ignore_index=True)
return df
def write_to_database(sim_results, filename=filename):
''' Write pandas DataFrame sim_results to HDFStore object.
Parameters
==========
sim_results: pandas DataFrame object
object with simulation results
filename: string
name of the file for storage
'''
h5 = pd. HDFStore(filename, 'a')
h5.append('sim_results', sim_results, min_itemsize={'values': 30},
ignore_index=True)
h5.close()
def print_results(filename=filename, idl=0, idh=50):
''' Prints valuation results in detailed form.
Parameters
==========
173
Valuing Volatility Derivatives
filename: string
HDFStore with pandas.DataFrame with results
idl: int
start index value
idh: int
stop index value
'''
h5 = pd. HDFStore(filename, 'r')
sim_results = h5['sim_results']
br = "-----------------------------------------------------------"
for i in range(idl, min(len(sim_results), idh + 1)):
row = sim_results.iloc[i]
print br
print "Start Calculations
%32s" % row['start_date'] + "\n" + br
print "ID Number
%32d" % i
print "Name of Simulation
%32s" % row['sim_name']
print "Seed Value for RNG
%32d" % row['seed']
print "Number of Runs
%32d" % row['runs']
print "Time Steps
%32d" % row['time_steps']
print "Paths
%32d" % row['paths']
print "Moment Matching
%32s" % row['mo_match']
print "Antithetic Paths
%32s" % row['anti_paths'] + "\n"
print "Option Prices
%32d" % row['opt_prices']
print "Absolute Tolerance
%32.4f" % row['abs_tol']
print "Relative Tolerance
%32.4f" % row['rel_tol']
print "Errors
%32d" % row['errors']
print "Error Ratio
%32.4f" % row['error_ratio'] + "\n"
print "Aver Val Error
%32.4f" % row['aval_err']
print "Aver Abs Val Error
%32.4f" % row['abal_err']
print "Time in Seconds
%32.4f" % row['time_sec']
print "Time in Minutes
%32.4f" % row['time_min']
print "Time per Option
%32.4f" % row['time_opt'] + "\n" + br
print "End Calculations
%32s" % row['end_date']
+ "\n" + br + "\n"
\
print "Total number of rows in table %d" % len(sim_results)
h5.close()
def plot_error_ratio(filename=filename):
''' Show error ratio vs. paths * time_steps (i.e. granularity).
Parameters
==========
filename: string
name of file with data to be plotted
'''
h5 = pd. HDFStore(filename, mode='r')
sim_results = h5['sim_results']
x = np.array(sim_results['paths'] * sim_results['time_steps'], dtype='d')
x = x / max(x)
174
LISTED VOLATILITY AND VARIANCE DERIVATIVES
y = sim_results['error_ratio']
plt.plot(x, y, 'bo', label='error ratio')
rg = np.polyfit(x, y, deg=1)
plt.plot(np.sort (x), np.polyval(rg, np.sort(x)), 'r', label='regression',
linewidth=2)
plt.xlabel('time steps * paths (normalized)')
plt.ylabel('errors / option valuations')
plt.legend()
plt.grid(True)
h5.close()
6.9.4 srd model calibration.py
#
# Calibration of Grünbichler and Longstaff (1996)
# square-root diffusion model to
# VSTOXX call options traded at Eurex
# Data as of 31. March 2014
# All data from www.eurexchange.com
#
# (c) Dr. Yves J. Hilpisch
# Listed Volatility and Variance Derivatives
#
import numpy as np
import pandas as pd
from srd_functions import call_price
import scipy.optimize as sco
import matplotlib.pyplot as plt
path = './source/data/'
# Fixed Parameters
v0 = 17.6639
r = 0.01
# VSTOXX index on 31. March 2014
# risk-less short rate
zeta = 0.
# volatility risk premium factor
def read_select_quotes(path=path, tol=0.2):
''' Selects and read options quotes.
Parameters
==========
path: string
path to file with option quotes
Returns
=======
option_data: pandas DataFrame object
175
Valuing Volatility Derivatives
option data
'''
h5 = pd.HDFStore(path + 'vstoxx_march_2014.h5', 'r')
# read option data from file and close it
option_data = h5['vstoxx_options']
h5.close()
# select relevant date for call option quotes
option_data = option_data[(option_data.DATE == '2014-3-31')
& (option_data.TYPE == 'C')]
# calculate time-to-maturity in year fractions
option_data['TTM'] = (option_data.MATURITY - option_data.DATE).apply(
lambda x: x / np.timedelta64(1, 'D') / 365.)
# only those options close enough to the ATM level
option_data = option_data[(option_data.STRIKE > (1 - tol) * v0)
& (option_data.STRIKE < (1 + tol) * v0)]
return option_data
def valuation_function(p0):
''' Valuation function for set of strike prices
Parameters
==========
p0: list
set of model parameters
Returns
=======
call_prices: NumPy ndarray object
array of call prices
'''
kappa, theta, sigma = p0
call_prices = []
for strike in strikes:
call_prices.append(call_price(v0, kappa, theta,
sigma, zeta, ttm, r, strike))
call_prices = np.array(call_prices)
return call_prices
def error_function(p0):
''' Error function for model calibration.
Parameters
==========
p0: tuple
set of model parameters
Returns
=======
176
LISTED VOLATILITY AND VARIANCE DERIVATIVES
MSE: float
mean squared (relative/absolute) error
'''
global i
call_prices = valuation_function(p0)
kappa, theta, sigma = p0
pen = 0.
if 2 * kappa * theta < sigma ** 2:
pen = 1000.0
if kappa < 0 or theta < 0 or sigma < 0:
pen = 1000.0
if relative is True:
MSE = (np.sum(((call_prices - call_quotes) / call_quotes) ** 2)
/ len(call_quotes) + pen)
else:
MSE = np.sum((call_prices - call_quotes) ** 2) / len(call_quotes) + pen
if i == 0:
print ("{:>6s}{:>6s}{:>6s}".format('kappa', 'theta', 'sigma')
+ "{:>12s}".format('MSE'))
# print intermediate results: every 100th iteration
if i % 100 == 0:
print "{:6.3f} {:6.3f} {:6.3f}".format(*p0) + "{:>12.5f}".format(MSE)
i += 1
return MSE
def model_calibration(option_data, rel=False, mat=' 2014-07-18'):
''' Function for global and local model calibration.
Parameters
==========
option_data: pandas DataFrame object
option quotes to be used
relative: boolean
relative or absolute MSE
maturity: string
maturity of option quotes to calibrate to
Returns
=======
opt: tuple
optimal parameter values
'''
global relative
# if True: MSRE is used, if False: MSAE
global strikes
global call_quotes
global ttm
177
Valuing Volatility Derivatives
global i
relative = rel
# only option quotes for a single maturity
option_quotes = option_data[option_data.MATURITY == mat]
# time-to-maturity from the data set
ttm = option_quotes.iloc[0, -1]
# transform strike column and price column in ndarray object
strikes = option_quotes['STRIKE'].values
call_quotes = option_quotes['PRICE'].values
# global optimization
i = 0
# counter for calibration iterations
p0 = sco.brute(error_function, ((5.0, 20.1, 1.0), (10., 30.1, 1.25),
(1.0, 9.1, 2.0)), finish=None)
# local optimization
i = 0
opt = sco.fmin(error_function, p0, xtol=0.0000001, ftol=0.0000001,
maxiter=1000, maxfun=1500)
return opt
def plot_calibration_results(opt):
''' Function to plot market quotes vs. model prices.
Parameters
==========
opt: list
optimal parameters from calibration
'''
callalues = valuation_function(opt)
diffs = callalues - call_quotes
plt.figure()
plt.subplot(211)
plt.plot(strikes, call_quotes, label='market quotes')
plt.plot(strikes, callalues, 'ro', labe='model prices')
plt.ylabel('option values')
plt.grid(True)
plt.legend()
plt.axis([min(strikes) - 0.5, max(strikes) + 0.5,
0.0, max(call_quotes) * 1.1])
plt.subplot(212)
wi = 0.3
plt.bar(strikes - wi / 2, diffs, width=wi)
plt.grid(True)
plt.xlabel('strike price')
178
LISTED VOLATILITY AND VARIANCE DERIVATIVES
plt.ylabel('difference')
plt.axis([min(strikes) - 0.5, max(strikes) + 0.5,
min(diffs) * 1.1, max(diffs) * 1.1])
plt.tight_layout()
if _ _name_ _ == '_ _main_ _':
option_data = read_select_quotes()
opt = model_calibration(option_data=option_data)
Listed Volatility and Variance Derivatives: A
Python-based Guide
By Dr. Yves J. Hilpisch
© 2017 Yves Hilpisch
CHAPTER
7
Advanced Modeling of the
VSTOXX Index
7.1 INTRODUCTION
This chapter is somewhat different in style compared to the other chapters about the VSTOXX
volatility index and related derivatives. It introduces another parsimonious model, which is
called square-root jump diffusion (SRJD) to model the VSTOXX volatility index. This model,
which is essentially an extension of the Grünbichler and Longstaff (1996) model as analyzed
in the previous chapter, is capable of reproducing prices of European options written on the
VSTOXX reasonably well.
Two major enhancements characterize the SRJD model:
term structure: it allows us to capture the term structure as observed in the prices of
futures on the VSTOXX index
jump component: including a jump component allows better replication of option prices
in the short term
Adding these two components makes a market-consistent calibration of the model to a comprehensive set of European options on the VSTOXX index possible. For similar analyses and
modeling approaches including jumps for the volatility index refer, for example, to Psychoyios
(2005), Sepp (2008) or Psychoyios et al. (2010). For empirical evidence with regard to jumps
in volatility see Todorov and Tauchen (2011).
7.2 MARKET QUOTES FOR CALL OPTIONS
Before we introduce the model, let us set the stage by revisiting market quotes for European
call options written on the VSTOXX volatility index. First, we import the data:
In [1]: import pandas as pd
In [2]: path = './source/data/'
In [3]: h5 = pd. HDFStore(path + 'vstoxx_data_31032014.h5', 'r')
179
180
LISTED VOLATILITY AND VARIANCE DERIVATIVES
We have both options data and futures data stored in this file with quotes from March 31, 2014.
In [4]: h5
Out[4]:
File path: ./source/data/vstoxx_data_31032014.h5
/futures_data
frame
(shape->[8,6])
/options_data
frame
(shape->[395,8])
Option market quotes are what we are concerned with for the moment.
In [5]: option_quotes = h5['options_data']
In [6]: option_quotes.info()
Int64Index: 395 entries, 46170 to 46564
Data columns (total 8 columns):
DATE
395 non-null datetime64[ns]
EXP_YEAR
395 non-null int64
EXP_MONTH
TYPE
STRIKE
395 non-null int64
395 non-null object
395 non–null float64
PRICE
MATURITY
395 non–null float64
395 non-null datetime64[ns]
TTM
395 non–null float64
dtypes: datetime64[ns](2), float64(3), int64(2), object(1)
memory usage: 27.8+ KB
In [7]: option_quotes.head()
Out[7]:
DATE
46170 2014-03-31
46171 2014-03-31
EXP_YEAR
2014
2014
46172 2014-03-31
46173 2014-03-31
46174 2014-03-31
2014
2014
2014
EXP_MONTH TYPE
4
C
4
C
4
4
4
C
C
C
STRIKE
1.0
2.0
PRICE
MATURITY
TTM
16.85 2014-04-18 0.049
15.85 2014-04-18 0.049
3.0
4.0
5.0
14.85 2014-04-18 0.049
13.85 2014-04-18 0.049
12.85 2014-04-18 0.049
At any given point in time, there are options on the VSTOXX available for eight maturities.
In [8]: mats = sorted(set(option_quotes['MATURITY']))
In [9]: mats
Out[9]:
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Advanced Modeling of the VSTOXX Index
[Timestamp('2014-04-18 00:00:00'),
Timestamp('2014-05-16 00:00:00'),
Timestamp('2014-06-20 00:00:00'),
Timestamp('2014-07-18 00:00:00'),
Timestamp('2014-08-15 00:00:00'),
Timestamp('2014-09-19 00:00:00'),
Timestamp('2014-10-17 00:00:00'),
Timestamp('2014-11-21 00:00:00')]
The spot level of the VSTOXX index on March 31, 2014 was 17.6639.
In [10]: v0 = 17.6639
In what follows, we only want to plot call option market quotes which are not too far in-themoney nor out-of-the-money.
In [11]: tol = 0.4
In [12]: to_plot = option_quotes[(option_quotes['STRIKE'] > (1 - tol) * v0)
& (option_quotes['STRIKE'] < (1 + tol) * v0)]
....:
....:
Figure 7.1 shows the VSTOXX European call option quotes which fulfill the requirements.
The goal of this chapter is to replicate “all these option quotes” as closely as possible.
In [13]: import matplotlib.pyplot as plt
In [14]: import seaborn as sns; sns.set()
In [15]: import matplotlib
In [16]: matplotlib.rcParams['font.family'] = 'serif'
In [17]: markers = ['.', 'o', 'ˆ', 'v', 'x', 'D', 'd', '>', '<']
In [18]: plt.figure(figsize=(10, 6));
In [19]: for i, mat in enumerate(mats):
....:
strikes = to_plot[(to_plot['MATURITY'] == mat)]['STRIKE']
....:
....:
....:
prices = to_plot[(to_plot['MATURITY'] == mat)]['PRICE']
plt.plot(strikes, prices, 'b%s' % markers[i],
label=str(mat)[:10])
182
FIGURE 7.1
LISTED VOLATILITY AND VARIANCE DERIVATIVES
VSTOXX European call option quotes on March 31, 2014.
In [20]: plt.legend();
In [21]: plt.xlabel('strike');
In [22]: plt.ylabel('option quote');
7.3 THE SRJD MODEL
A filtered probability space {Ω, , 𝔽 , P} representing uncertainty in the model economy is
given with final date T where 0 < T < ∞. Ω denotes the continuous state space, a 𝜎−algebra,
𝔽 a filtration – i.e. a family of non-decreasing 𝜎−algebras 𝔽 ≡ {t∈[0,T] } with 0 ≡ {∅, Ω} and
T ≡ – and P the real or objective probability measure.
In the SRJD model, which is an affine jump diffusion (see Duffie et al. (2000)), the riskneutral dynamics of the VSTOXX volatility index are given by the following stochastic differential equation (SDE):
√
dvt = 𝜅(𝜃 − vt )dt + 𝜎 vt dZt + Jt vt dNt − rJ dt
The meaning of the variables and parameters is:
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Advanced Modeling of the VSTOXX Index
vt volatility index level at date t
𝜅 speed of adjustment of vt to…
… 𝜃, the long-term mean of the index
𝜎 volatility coefficient of the index level
Zt standard Brownian motion
Jt jump at date t with distribution…
… log(1 + Jt ) ≈ N(log(1 + 𝜇) − 𝛿2 , 𝛿 2 )
N cumulative distribution function of a standard normal random variable
Nt Poisson process with intensity 𝜆
2
rJ ≡ 𝜆 ⋅ (e𝜇+𝛿 ∕2 − 1) drift correction for jump.
2
The stochastic process for v is adapted to the filtration 𝔽 . Moreover, Z and N are not correlated.
The time t value of a zero-coupon bond paying one unit of currency at T, 0 ≤ t < T, is Bt (T) =
e−r(T−t) with r ≥ 0 the constant risk-less short rate.
By the Fundamental Theorem of Asset Pricing, the time t value of an attainable, T measurable contingent claim VT ≡ hT (XT ) ≥ 0 (satisfying suitable integrability conditions) is
given by arbitrage as
(
)
Vt = EQ
t Bt (T)VT
with V0 = EQ
(B0 (T)VT ) as the important special case for valuation purposes. Q is a P0
equivalent martingale measure. The contingent claim could be a European call option maturing
at T with payoff VT = hT (vT ) ≡ max[vT − K, 0]. It could also be a European put with payoff
VT = hT (vT ) ≡ max[K − vT , 0]. In both cases, K is the fixed strike price of the option.
To simulate the financial model, i.e. to generate numerical values for vt , it has to be discretized. To this end, divide the given time interval [0, T] in equidistant sub-intervals Δt such
that now t ∈ {0, Δt, 2Δt, … , T}, i.e. there are M + 1 points in time with M ≡ T∕Δt. With
s = t − Δt, a discretization of the continuous time market model is given by
√ √
)Δt
+
𝜎
ṽ +
Δtz1t
ṽ t = ṽ s + 𝜅(𝜃 − ṽ +
s
s
(
)
2 2
+ e𝜇J +𝛿 zt − 1 ṽ +
s yt − rJ Δt
+
vt = ṽ t
for t ∈ {Δt, … , T} with x+ ≡ max[x, 0] and the znt being standard normally distributed and yt
Poisson distributed. z1t , z2t and yt are uncorrelated. This discretization scheme is an Euler discretization and is generally called full trunction scheme. See Lord et al. (2008) for an analysis
of this and other biased discretization schemes for the square-root diffusion process.
7.4 TERM STRUCTURE CALIBRATION
The first step in the calibration of the SRJD model is with regard to the futures term structure.
184
LISTED VOLATILITY AND VARIANCE DERIVATIVES
7.4.1 Futures Term Structure
It is difficult for parsimonious short rate models like that of Cox et al. (1985) to account for
different term structures of the interest rate. A possible solution is the introduction of timedependent parameters which, however, enlarges the number of parameters significantly, sacrificing at the same time the convenience of a limited number of economic parameters. Another
solution is a deterministic shift approach according to Brigo and Mercurio (2001) which preserves the basic structure of the model with all its advantages and which nevertheless allows
us to better account for different term structures of the short rate.
In this section, we transfer the deterministic shift approach for the short rate models of
Brigo and Mercurio (2001) to the SRJD model. Since the square-root diffusion and the jumps
are not correlated, we can first apply the approach to the diffusion part and use this enhanced
component later on in conjunction with the jump component.
Before we present the theory, a look at the VSTOXX futures data first.
In [23]: futures_quotes = h5['futures_data']
In [24]: futures_quotes.info()
Int64Index: 8 entries, 496 to 503
Data columns (total 6 columns):
DATE
EXP_YEAR
EXP_MONTH
8 non-null datetime64[ns]
8 non-null int64
8 non-null int64
PRICE
MATURITY
TTM
8 non-null float64
8 non-null datetime64[ns]
8 non-null float64
dtypes: datetime64[ns](2), float64(2), int64(2)
memory usage: 448.0 bytes
Figure 7.2 presents the futures quotes for all eight maturities.
In [25]: ax = futures_quotes.plot(x='MATURITY', y='PRICE',
....:
figsize=(10, 6), legend=False)
....:
In [26]: futures_quotes.plot(x='MATURITY', y='PRICE', style='ro', ax=ax);
Consider for now the square-root diffusion volatility model of Grünbichler and Longstaff
(1996) as presented in the previous chapter, which is formally the same as the short rate model
of Cox et al. (1985) and which is a special case of the SRJD model:
√
dvt = 𝜅(𝜃 − vt )dt + 𝜎 vt dZt
185
Advanced Modeling of the VSTOXX Index
FIGURE 7.2
VSTOXX futures quotes on March 31, 2014.
We want to calibrate this model to the observed volatility term structure, given by the above
presented set of prices for futures on the VSTOXX index with different maturities. We have
to minimize for all considered times t and a parameter set 𝛼 = (𝜅, 𝜃, 𝜎, v0 ) simultaneously the
single differences
Δf (0, t) ≡ f (0, t) − f GL96 (0, t; 𝛼)
where f (0, t) is the time 0 market (instantaneous) forward volatility for time t and the quantity
f GL96 (0, t; 𝛼) is the model (instantaneous) forward volatility for time t given parameter set 𝛼.
Assume that there is a continuously differentiable volatility term structure function F(0, t)
available (i.e. there are infinitely many volatility futures prices). The forward volatility then is
f (0, t) =
𝜕F(0, t)
𝜕t
On the other hand, the model implied forward volatility is given as (see Brigo and Mercurio
(2001))
f GL96 (0, t; 𝛼) =
𝜅𝜃(e𝛾t − 1)
2𝛾 + (𝜅 + 𝛾)(e𝛾t − 1)
4𝛾 2 e𝛾t
+ v0
(2𝛾 + (𝜅 + 𝛾)(e𝛾t − 1))2
186
LISTED VOLATILITY AND VARIANCE DERIVATIVES
with
𝛾≡
√
𝜅 2 + 2𝜎 2
The Python script srjd_fwd_calibration.py contains the Python code to calibrate the forward volatilities to the VSTOXX futures prices (see sub-section 7.8.1,
srjd_fwd_calibration.py for the complete script). The beginning of the script is about library
imports, importing the data sets and making some selections.
import math
import numpy as np
import pandas as pd
import scipy.optimize as sco
v0 = 17.6639 # initial VSTOXX index level
i = 0 # counter for calibration runs
# reading the VSTOXX futures quotes
path = 'source/data/'
h5 = pd.HDFStore(path + 'vstoxx_data_31032014.h5', 'r')
futures_quotes = h5['futures_data']
h5.close()
# selecting needed data columns and adding spot value
forwards = list(futures_quotes['PRICE'].values)
forwards.insert(0, v0)
forwards = np.array(forwards)
ttms = list(futures_quotes['TTM'].values)
ttms.insert(0, 0)
ttms = np.array(ttms)
The function srd_forwards() implements the forward formula from above for a given
parameter set.
def srd_forwards(p0):
''' Function for forward volatilities in GL96 Model.
Parameters
==========
p0: list
set of model parameters, where
kappa: float
mean-reversion factor
187
Advanced Modeling of the VSTOXX Index
theta: float
long-run mean
sigma: float
volatility factor
Returns
=======
forwards: NumPy ndarray object
forward volatilities
'''
t = ttms
kappa, theta, sigma = p0
g = math.sqrt(kappa ** 2 + 2 * sigma ** 2)
sum1 = ((kappa * theta * (np.exp(g * t) - 1)) /
(2 * g + (kappa + g) * (np.exp(g * t) - 1)))
sum2 = v0 * ((4 * g ** 2 * np.exp(g * t)) /
(2 * g + (kappa + g) * (np.exp(g * t) - 1)) ** 2)
forwards = sum1 + sum2
return forwards
To operationalize the calibration, we use the mean squared error (MSE) as our yardstick
min
𝛼
N
)2
1 ∑(
fn − fnGL96 (𝛼)
N n=1
which is to be minimized. Here, we assume that we have N observations for the forward volatility. In Python this takes on the form of function srd_fwd_error().
def srd_fwd_error(p0):
''' Error function for GL96 forward volatilities calibration.
Parameters
==========
p0: tuple
parameter vector
Returns
=======
MSE: float
mean-squared error for p0
'''
global i
kappa, theta, sigma = p0
srd_fwds = srd_forwards(p0)
MSE = np.sum((forwards – srd_fwds) ** 2) / len(forwards)
if 2 * kappa * theta < sigma ** 2:
MSE = MSE + 100
# penalty
188
LISTED VOLATILITY AND VARIANCE DERIVATIVES
elif sigma < 0:
MSE = MSE + 100
# print intermediate results: every 50th iteration
if i % 50 == 0:
print "{:6.3f} {:6.3f} {:6.3f}".format(* p0) + "{:>12.5f}".format(MSE)
i += 1
return MSE
Executing the script yields optimal parameters for the Grünbichler and Longstaff (1996) model
given the VSTOXX futures prices.
In [27]: %run source/scripts/srjd_fwd_calibration.py
1.000 17.500
0.258 40.585
1.575 46.996
1.000
0.057
0.392
35.99817
5.61895
0.22958
2.525 46.207
3.447 44.752
3.538 44.458
0.710
0.972
0.903
0.06348
0.04264
0.04261
3.947 43.495
4.113 43.102
4.187 42.969
0.610
0.275
0.019
0.04243
0.04219
0.04213
4.169 42.984 0.002
0.04213
4.170 42.984 0.000
0.04213
Optimization terminated successfully.
Current function value: 0.042129
Iterations: 292
Function evaluations: 526
In [28]: opt.round(3)
Out[28]: array([ 4.17 ,
42.984,
0.
])
These optimal values can be used to calculate the model forward volatilities using function
srd_forwards().
In [29]: from srjd_fwd_calibration import *
In [30]: srd_fwds = srd_forwards(opt)
In [31]: srd_fwds
Out[31]:
array([ 17.6639
, 18.37130183, 19.22831373, 19.97504105,
20.3916346, 20.69048824, 20.95488999, 21.10239774, 21.23092346])
Advanced Modeling of the VSTOXX Index
FIGURE 7.3
189
VSTOXX futures market quotes vs. model prices.
The numerical differences to the market futures prices are:
In [32]: srd_fwds - forwards
Out[32]:
array([ 0.
, 0.52130183, –0.32168627, 0.02504105, –0.0083654,
–0.00951176, 0.00488999, 0.05239774, –0.01907654])
Figure 7.3 compares the model futures prices (forward volatilities) with the VSTOXX futures
market quotes. For longer maturities the fit is quite good.
In [33]: plt.figure(figsize=(10, 6));
In [34]: plt.plot(ttms, forwards, 'b', label='market prices');
In [35]: plt.plot(ttms, srd_fwds, 'ro', label='model prices');
In [36]: plt.legend(loc=0);
In [37]: plt.xlabel('time-to-maturity (year fractions)');
In [38]: plt.ylabel('forward volatilities');
190
LISTED VOLATILITY AND VARIANCE DERIVATIVES
Finally, we save the results from the term structure calibration for later use during the simulation of the model.
In [39]: import pickle
In [40]: f = open('varphi', 'w')
# open file on disk
# write ttms object and differences (varphi values) as dictionary
In [41]: pickle.dump({'ttms': ttms, 'varphi': srd_fwds – forwards}, f)
In [42]: f.close()
# close file
7.4.2 Shifted Volatility Process
Assume that we are given a continuously differentiable futures price function (i.e. through
splines interpolation of discrete futures prices for different maturities). We consider now the
deterministically shifted volatility process (see Brigo and Mercurio (2001))
v̂ t ≡ vt + 𝜑(t, 𝛼 ∗ )
with 𝜑(t, 𝛼 ∗ ) ≡ f (0, t) − f GL96 (0, t; 𝛼 ∗ ), the difference at time t between the market implied
forward volatility and the model implied forward volatility after calibration, i.e.for the optimal
parameter set 𝛼 ∗ . 𝜑(t, 𝛼 ∗ ) corresponds to the differences (bars) in Figure 7.4.
FIGURE 7.4
Deterministic shift values to account for VSTOXX futures term structure.
191
Advanced Modeling of the VSTOXX Index
In [43]: plt.figure(figsize=(10, 6));
In [44]: plt.bar(ttms, srd_fwds – forwards,
....:
....:
width=0.05, label='$\\varphi(t,\\alphaˆ*)$');
In [45]: plt.legend(loc=0);
In [46]: plt.xlabel('time-to-maturity (year fractions)');
In [47]: plt.ylabel('deterministic shift');
The SRJD model discretization can now be adjusted as follows:
√ √
ṽ t = ṽ s + 𝜅(𝜃 − ṽ +
)Δt
+
𝜎
ṽ +
Δtz1t
s
s
(
)
2 2
+ e𝜇J +𝛿 zt − 1 ṽ +
s yt − rJ Δt
+
∗
v̂ t = ṽ t + 𝜑(t, 𝛼 )
(7.1)
(7.2)
This is consistent since the diffusion and jump parts are not correlated and since the jump
part is added in such a way that the first moment of the stochastic volatility process does not
change.
7.5 OPTION VALUATION BY MONTE CARLO SIMULATION
This section implements Monte Carlo simulation procedures for the SRJD model.
7.5.1 Monte Carlo Valuation
In what follows, the model option values are computed by Monte Carlo simulation (MCS).
Given the discrete version of the financial model, the value of a European call option on the
volatility index is estimated by MCS as follows:
Algorithm 1: Simulation algorithm
1
2
3
4
for i = 1, 2, … , I do
for t = Δt, … , T do
draw pseudo-random numbers z1t (i), z2t (i), yt (i)
apply these to equations (7.1) and (7.2) to calculate vt (i)
calculate VT (i) = hT (i) ≡ max[vT (i) − K, 0]
sum up all payoffs at T, take the average and discount back to t = 0:
V0 (K, T) ≈ e−rT ⋅
1∑
max[vT (i) − K, 0]
I I
V0 is the MCS estimator for the European call option value.
192
LISTED VOLATILITY AND VARIANCE DERIVATIVES
7.5.2 Technical Implementation
Because we are using a numerical method like MCS for all valuation and calibration tasks,
the parametrization and implementation of the MCS algorithm play an important role. Some
major features of our implementation are:
discretization: the algorithm uses the Euler discretization scheme which is an approximation only but which might bring performance benefits
random numbers: for every single option valuation the seed can be held constant such
that every option is valued with the same set of (pseudo-)random numbers
variance reduction: both antithetic variates and moment matching (for the first two
moments of the pseudo-random numbers) are used as generic variance reduction techniques
deterministic shift: the deterministic shift values 𝜑 are determined only once through a
separate calibration and are held constant afterwards (even if model parameters change).
𝜑 only has to be deterministic and integrable on closed intervals (see Brigo and Mercurio
(2001)), which is of course the case. For the approach to be valid, it is not important how we
originally came up with the 𝜑.
The Python code for simulating the SRJD model is found in the script
srjd_simulation.py (see sub-section 7.8.2, srjd_simulation.py for the complete
script). The beginning of the script shows several imports, the definition of example parameters and also the cubic splines interpolation to be used for the estimation of the deterministic
shift parameters.
import math
import pickle
import numpy as np
import scipy.interpolate as scint
v0 = 17.6639
# initial VSTOXX index level
# parameters of square-root diffusion
kappa = 2.0 # speed of mean reversion
theta = 15.0 # long-term volatility
sigma = 1.0 # standard deviation coefficient
# parameters of log-normal jump
lamb = 0.4 # intensity (jumps per year)
mu = 0.4 # average jump size
delta = 0.1 # volatility of jump size
# general parameters
r = 0.01 # risk-free interest rate
K = 17.5 # strike
T = 0.5
M = 150
# time horizon
# time steps
Advanced Modeling of the VSTOXX Index
193
I = 10000 # number of MCS paths
anti_paths = True # antithetic variates
mo_match = True
# moment matching
# deterministic shift parameters
varphi = pickle.load(open('varphi'))
tck = scint.splrep(varphi['ttms'], varphi['varphi'], k=1)
# linear splines interpolation of
# term structure calibration differences
The Python function random_number_gen() generates arrays of standard normally distributed pseudo-random numbers using both antithetic variates and moment matching as
generic variance reduction techniques.
def random_number_gen(M, I, fixed_seed=False):
''' Generate standard normally distributed pseudo-random numbers
Parameters
==========
M: int
number of time intervals
I: int
number of paths
Returns
=======
ran: NumPy ndarrayo object
random number array
'''
if fixed_seed is True:
np.random.seed(10000)
if anti_paths is True:
ran = np.random.standard_normal((M + 1, I / 2))
ran = np.concatenate((ran, -ran), axis=1)
else:
ran = np.standard_normal((M + 1, I))
if mo_match is True:
ran = ran / np.std(ran)
ran -= np.mean(ran)
return ran
The major function of this script is srjd_simulation() which implements the Monte
Carlo simulation for the SRJD model based on an Euler discretization scheme. The scheme
used here is usually called a full truncation scheme.
194
LISTED VOLATILITY AND VARIANCE DERIVATIVES
def srjd_simulation(x0, kappa, theta, sigma,
lamb, mu, delta, T, M, I, fixed_seed=False):
''' Function to simulate square-root jump Difusion.
Parameters
==========
x0: float
initial value
kappa: float
mean-reversion factor
theta: float
long-run mean
sigma: float
volatility factor
lamb: float
jump intensity
mu: float
expected jump size
delta: float
standard deviation of jump
T: float
time horizon/maturity
M: int
time steps
I: int
number of simulation paths
Returns
=======
x: NumPy ndarray object
array with simulated SRJD paths
'''
dt = float(T) / M # time interval
shift = scint.splev(np.arange(M + 1) * dt, tck, der=0)
# deterministic shift values
xh = np.zeros((M + 1, I), dtype=np.float)
x = np.zeros((M + 1, I), dtype=np.float)
xh[0, :] = x0
x[0, :] = x0
# drift contribution of jump p.a.
rj = lamb * (math.exp(mu + 0.5 * delta ** 2) - 1)
# 1st matrix with standard normal rv
ran1 = random_number_gen(M + 1, I, fixed_seed)
# 2nd matrix with standard normal rv
ran2 = random_number_gen(M + 1, I, fixed_seed)
# matrix with Poisson distributed rv
Advanced Modeling of the VSTOXX Index
195
ran3 = np.random.poisson(lamb * dt, (M + 1, I))
for t in range(1, M + 1):
xh[t, :] = (xh[t - 1, :] +
kappa * (theta - np.maximum(0, xh[t - 1, :])) * dt
+ np.sqrt(np.maximum(0, xh[t - 1, :])) * sigma
* ran1[t] * np.sqrt(dt)
+ (np.exp(mu + delta * ran2[t]) - 1) * ran3[t]
* np.maximum(0, xh[t - 1, :]) - rj * dt)
x[t, :] = np.maximum(0, xh[t, :]) + shift[t]
return x
Finally, the function srjd_call_valuation() estimates the value of a European call
option given the simulated volatility paths from srjd_simulation().
def srjd_call_valuation(v0, kappa, theta, sigma,
lamb, mu, delta, T, r, K, M=M, I=I,
fixed_seed=False):
''' Function to value European volatility call option in SRDJ model.
Parameters see function srjd_simulation.
Returns
=======
call_value: float
estimator for European call present value for strike K
'''
v = srjd_simulation(v0, kappa, theta, sigma,
lamb, mu, delta, T, M, I, fixed_seed)
call_value = np.exp(- r * T) * sum(np.maximum(v[-1] - K, 0)) / I
return call_value
Executing the script yields a MCS estimator for the European call option with the parameters
as assumed in the script of about 1 currency unit.
In [48]: %run source/scripts/srjd_simulation.py
Value of European call by MCS:
0.9959
7.6 MODEL CALIBRATION
This section now calibrates the SRJD model to market quotes for European call options on
VSTOXX futures. It considers calibrations to a single maturity as well as to multiple maturities.
196
LISTED VOLATILITY AND VARIANCE DERIVATIVES
7.6.1 The Python Code
The calibration of the SRJD model is similar to the procedure for the Grünbichler and Lonstaff
(1996) square-root diffusion model as presented in the previous chapter. The major difference
now is that we have to take into account more parameter values for the optimization. The
Python code is contained in script srjd_model_calibration.py (see sub-section 7.8.3,
srjd_model_calibration.py for the complete script). As usual a few imports and parameter
definitions first.
import numpy as np
import pandas as pd
import scipy.optimize as sco
import matplotlib.pyplot as plt
from srd_model_calibration import path, read_select_quotes
from srjd_simulation import srjd_call_valuation
# fixed parameters
r = 0.01 # risk-less short rate
v0 = 17.6639 # VSTOXX index at 31.03.2014
M = 15
I = 100
# number of time intervals
# number of simulated paths
In what follows, we want to calibrate the model simultaneously to multiple maturities for
the VSTOXX European call options. The valuation function srjd_valuation_function()
therefore now calculates the differences between model and market values directly and returns
an array with all differences (relative or absolute).
def srjd_valuation_function(p0):
''' Valuation ('difference') function for all options
of a given DataFrame object.
Parameters
==========
p0: list
set of model parameters
Returns
=======
diffs: NumPy ndarray object
array with valuation differences
'''
global relative, option_data
kappa, theta, sigma, lamb, mu, delta = p0
diffs = []
for i, option in option_data.iterrows():
197
Advanced Modeling of the VSTOXX Index
value = srjd_call_valuation(v0, kappa, theta, sigma,
lamb, mu, delta,
option['TTM'], r, option['STRIKE'],
M=M, I=I, fixed_seed=True)
if relative is True:
diffs.append((value - option['PRICE']) / option['PRICE'])
else:
diffs.append(value - option['PRICE'])
diffs = np.array(diffs)
return diffs
The error function srjd_error_function() for the SRJD model has to be enhanced
compared to the SRD case to account for the additional parameters of the model.
def srjd_error_function(p0):
''' Error function for model calibration.
Parameters
==========
p0: tuple
set of model parameters
Returns
=======
MSE: float
mean squared (relative/absolute) error
'''
global i, min_MSE, option_data
OD = len(option_data)
diffs = srjd_valuation_function(p0)
kappa, theta, sigma, lamb, mu, delta = p0
# penalties
pen = 0.
if 2 * kappa * theta < sigma ** 2:
pen = 1000.0
if kappa < 0 or theta < 0 or sigma < 0 or lamb < 0 or delta < 0:
pen = 1000.0
MSE = np.sum(diffs ** 2) / OD + pen
min_MSE = min(min_MSE, MSE)
# mean squared error
# running minimum value
if i == 0:
print
\'n' + ('{:>5s}'.format('its')
+ '{:>7s} {:>6s} {:>6s} {:>6s} {:>6s} {:>6s}'.format(
198
LISTED VOLATILITY AND VARIANCE DERIVATIVES
'kappa', 'theta', 'sigma', 'lamb', 'mu', 'delta')
+ '{:>12s}'.format('MSE') + '{:>12s}'.format('min_MSE'))
# print intermediate results: every 100th iteration
if i % 100 == 0:
print ('{:>5d}'.format(i)
+ '{:7.3f} {:6.3f} {:6.3f} {:6.3f} {:6.3f} {:6.3f}'.format(*p0)
+ '{:>12.5f}'.format(MSE) + '{:>12.5f}'.format(min_MSE))
i += 1
return MSE
The same holds true for the calibration function srjd_model_calibration() itself.
This function allows us to select certain maturities for the calibration.
def srjd_model_calibration(data, p0=None, rel=False, mats=None):
''' Function for global and local model calibration.
Parameters
==========
option_data: pandas DataFrame object
option quotes to be used
relative: bool
relative or absolute MSE
mats: list
list of maturities of option quotes to calibrate to
Returns
=======
opt: tuple
optimal parameter values
'''
global i, min_MSE, option_data
global relative # if True: MSRE is used, if False: MSAE
min_MSE = 5000. # dummy value
relative = rel # relative or absolute
option_data = data
if mats is not None:
# select the option data for the given maturities
option_data = option_data[option_data['MATURITY'].isin(mats)]
# global optimization
if p0 is None:
i = 0 # counter for calibration iterations
p0 = sco.brute(srjd_error_function, (
(1.0, 9.1, 4.0), # kappa
199
Advanced Modeling of the VSTOXX Index
(10., 20.1, 10.0), # theta
(1.0, 3.1, 2.0), # sigma
(0.0, 0.81, 0.4), # lambda
(-0.2, 0.41, 0.3), # mu
(0.0, 0.31, 0.15)), # delta
finish=None)
# local optimization
i = 0
opt = sco.fmin(srjd_error_function, p0,
xtol=0.0000001, ftol=0.0000001,
maxiter=550, maxfun=700)
return opt
7.6.2 Short Maturity
The addition of a jump component shall allow a better fit to short-term call option market
quotes. Therefore, consider the following calibration to the shortest option maturity available:
In [49]: from srjd_model_calibration import *
# read option data, allow for 30% moneyness tolerance
In [50]: option_data = read_select_quotes(tol=0.3)
In [51]: %%time
....: opt_1 = srjd_model_calibration(option_data, p0=None,
....:
rel=False, mats=['2014-4-18'])
....:
its
kappa
theta
sigma
mu
delta
MSE
min_MSE
0
100
200
1.000 10.000
1.000 20.000
5.000 20.000
1.000
3.000
3.000
0.000 -0.200
0.800 -0.200
0.400 -0.200
lamb
0.000
0.000
0.150
0.15785
0.10867
0.27793
0.15785
0.01976
0.01976
300
9.000 20.000
3.000
0.000 -0.200
0.300
0.70669
0.01976
its
0
100
kappa theta
1.000 10.000
1.016 9.216
sigma
3.000
2.996
lamb
mu
0.400 -0.200
0.476 -0.124
delta
0.000
0.001
MSE
0.01976
0.01956
min_MSE
0.01976
0.01955
200
300
400
1.020
0.976
0.977
9.213
8.895
8.869
3.006
3.000
3.002
0.494 -0.128
0.392 -0.128
0.394 -0.128
0.001
0.001
0.001
0.01955
0.02214
0.01955
0.01955
0.01955
0.01955
500
0.976
8.866
3.002
0.394 -0.128
0.001
0.01955
0.01955
200
LISTED VOLATILITY AND VARIANCE DERIVATIVES
600
0.976
8.867
3.002
0.394 -0.128
0.001
0.01955
0.01955
Warning: Maximum number of function evaluations has been exceeded.
CPU times: user 14.7 s, sys: 7.97 ms, total: 14.7 s
Wall time: 14.7 s
The optimal parameter values are:
In [52]: opt_1
Out[52]:
array([
9.75770825e-01,
3.93539089e-01,
8.86539094e+00,
-1.28094441e-01,
3.00188833e+00,
1.26180667e-03])
Using these optimal parameter values, add the model prices to the DataFrame object containing the option data.
In [53]: values = []
In [54]: kappa, theta, sigma, lamb, mu, delta = opt_1
In [55]: for i, option in option_data.iterrows():
....:
value = srjd_call_valuation(v0, kappa, theta, sigma,
....:
lamb, mu, delta,
....:
....:
....:
option['TTM'], r, option['STRIKE'],
M=M, I=I, fixed_seed=True)
values.append(value)
....:
In [56]: option_data['MODEL'] = values
Figure 7.5 shows the calibration results graphically. Indeed, the fit seems to be quite good,
reflecting a MSAE of about 0.0015 only.
# selecting the data for the shortest maturity
In [57]: os = option_data[option_data.MATURITY == '2014-4-18']
# selecting corresponding strike prices
In [58]: strikes = os.STRIKE.values
# comparing the model prices with the market quotes
In [59]: fig, ax = plt.subplots(2, 1, sharex=True, figsize=(10, 6));
In [60]: ax[0].plot(strikes, os.PRICE.values, label='market quotes');
In [61]: ax[0].plot(strikes, os.MODEL.values, 'ro', label='model prices');
201
Advanced Modeling of the VSTOXX Index
FIGURE 7.5
Calibration of SRJD model to European call options on the VSTOXX for shortest
option maturity (April 2014).
In [62]: ax[1].bar(strikes - 0.15, os.MODEL.values - os.PRICE.values,
....:
width=0.3);
....:
In [63]: ax[1].set_xlim(12.5, 23);
7.6.3 Two Maturities
Let us proceed with the simultaneous calibration of the model to the May and July maturities.
The MSE is also pretty low in this case (i.e. below 0.01).
# read option data, allow for 17.5% moneyness tolerance
In [64]: option_data = read_select_quotes(tol=0.175)
In [65]: %%time
....: opt_2 = srjd_model_calibration(option_data, rel=False,
....:
....:
mats=['2014-5-16', '2014-7-18'])
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LISTED VOLATILITY AND VARIANCE DERIVATIVES
its
0
kappa theta
1.000 10.000
sigma
1.000
lamb
mu
0.000 -0.200
delta
0.000
MSE
8.71137
min_MSE
8.71137
100
200
300
1.000 20.000
5.000 20.000
9.000 20.000
3.000
3.000
3.000
0.800 -0.200
0.400 -0.200
0.000 -0.200
0.000
0.150
0.300
0.85126
0.59218
0.45081
0.10351
0.09015
0.01539
its
0
kappa theta
5.000 20.000
sigma
3.000
lamb
0.800
mu
0.400
delta
0.000
MSE
0.01539
min_MSE
0.01539
100
200
300
4.471 19.655
4.010 19.231
4.023 18.987
3.256
2.730
2.321
0.762
0.761
0.721
0.424
0.575
0.668
0.000
0.002
0.003
0.00892
0.00792
0.01182
0.00892
0.00792
0.00688
400
500
600
3.803 18.830
3.823 18.799
3.998 18.681
2.115
2.106
1.916
0.722
0.723
0.728
0.701
0.706
0.749
0.003
0.003
0.003
0.00660
0.00644
0.00598
0.00660
0.00644
0.00598
Warning: Maximum number of function evaluations has been exceeded.
CPU times: user 17.1 s, sys: 10 us, total: 17.1 s
Wall time: 17.1 s
The optimal parameter values are:
In [66]: opt_2
Out[66]:
array([
3.99997568e+00,
7.28298815e-01,
1.86836808e+01,
7.48260996e-01,
1.91679210e+00,
3.26197421e-03])
In what follows, we use the Python function plot_calibration_results() to generate the plots for the different valuation runs. This function allows for different numbers of
sub-plots, i.e. when the number of option maturities is changed. It is mainly a generalization
of the plotting code used above.
def plot_calibration_results(option_data, opt, mats):
''' Function to plot market quotes vs. model prices.
Parameters
==========
option_data: pandas DataFrame object
option data to plot
opt: list
optimal results from calibration
mats: list
maturities to be plotted
'''
kappa, theta, sigma, lamb, mu, delta = opt
# adding model values for optimal parameter set
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Advanced Modeling of the VSTOXX Index
# to the DataFrame object
values = []
for i, option
in option_data.iterrows():
value = srjd_call_valuation(v0, kappa, theta, sigma,
lamb, mu, delta,
option['TTM'], r, option['STRIKE'],
M=M, I=I, fixed_seed=True)
values.append(value)
option_data['MODEL'] = values
# plotting the market and model values
height = min(len(mats) * 3, 12)
fig, axarr = plt.subplots(len(mats), 2, sharex=True, figsize=(10, height))
for z, mat in enumerate(mats):
if z == 0:
axarr[z, 0].set_title('values')
axarr[z, 1].set_title('differences')
os = option_data[option_data.MATURITY == mat]
strikes = os. STRIKE.values
axarr[z, 0].set_ylabel('%s' % str(mat)[: 10])
axarr[z, 0].plot(strikes, os.PRICE.values, label='market quotes')
axarr[z, 0].plot(strikes, os.MODEL.values, 'ro', label='model prices')
axarr[z, 0].legend(loc=0)
wi = 0.3
axarr[z, 1].bar(strikes - wi / 2, os.MODEL.values - os.PRICE.values,
width=wi)
if mat == mats[- 1]:
axarr[z, 0].set_xlabel('strike')
axarr[z, 1].set_xlabel('strike')
Figure 7.6 shows the results of the calibration graphically. It is obvious that the SRJD model
is able to account for multiple maturities at the same time which is mainly due to the term
structure component introduced by the deterministic shift approach.
In [67]: plot_calibration_results(option_data, opt_2, ['2014-5-16',
'2014-7-18'])
7.6.4 Four Maturities
In a next step, we consider four maturities – before we try to calibrate the model to all eight
maturities.
204
LISTED VOLATILITY AND VARIANCE DERIVATIVES
FIGURE 7.6
Calibration of SRJD model to European call options on the VSTOXX for May and
July 2014 maturities.
In [68]: mats = sorted(set(option_data['MATURITY']))
In [69]: mats
Out[69]:
[Timestamp('2014-04-18 00:00:00'),
Timestamp('2014-05-16 00:00:00'),
Timestamp('2014-06-20 00:00:00'),
Timestamp('2014-07-18 00:00:00'),
Timestamp('2014-08-15 00:00:00'),
Timestamp('2014-09-19 00:00:00'),
Timestamp('2014-10-17 00:00:00'),
Timestamp('2014-11-21 00:00:00')]
The four maturities for this particular calibration run are:
In [70]: mats[::2]
Out[70]:
[Timestamp('2014-04-18 00:00:00'),
Timestamp('2014-06-20 00:00:00'),
Timestamp('2014-08-15 00:00:00'),
Timestamp('2014-10-17 00:00:00')]
205
Advanced Modeling of the VSTOXX Index
For this calibration run, we use the optimal parameters from the previous calibration to two
maturities. Obviously, the more options to calibrate the model to, the longer the procedure
takes.
In [71]: %%time
....: opt_4 = srjd_model_calibration(option_data, p0=opt_2,
....:
rel=False, mats=mats[::2])
....:
its
0
kappa theta
4.000 18.684
sigma
1.917
lamb
0.728
mu
0.748
delta
0.003
MSE
0.52345
min_MSE
0.52345
100
200
4.202 18.938
3.650 21.465
1.980
3.293
0.727 0.623
0.744 -0.075
0.003
0.002
0.27887
0.18617
0.27681
0.18617
300
400
3.195 22.481
3.107 22.191
2.867
2.990
0.743 -0.223
0.745 -0.141
0.000
0.000
0.14915
0.14609
0.14915
0.14608
500
3.081 22.162
3.028
0.743 -0.138
0.000
0.14572
0.14570
600 3.057 22.100 3.041 0.738 -0.124 0.000
0.14521
0.14521
700 3.061 22.110 3.034 0.738 -0.126 0.000
0.14517
0.14517
Warning: Maximum number of function evaluations has been exceeded.
CPU times: user 22.9 s, sys: 16 ms, total: 22.9 s
Wall time: 22.9 s
In [72]: opt_4
Out[72]:
array([ 3.06082557e+00,
7.38054016e-01,
2.21100415e+01,
3.03422643e+00,
-1.25823900e-01,
1.19890649e-06])
Even calibrating the model to four maturities yields quite a good fit over these maturities as
Figure 7.7 illustrates.
In [73]: plot_calibration_results(option_data, opt_4, mats[::2])
7.6.5 All Maturities
Finally, let us attack the hardest calibration problem – the one involving all eight option
maturities.
In [74]: %%time
....: opt_8_MSAE = srjd_model_calibration(option_data,
....:
....:
rel=False, mats=mats)
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LISTED VOLATILITY AND VARIANCE DERIVATIVES
FIGURE 7.7
Calibration of SRJD model to European call options on the VSTOXX for four
maturities.
its
kappa
theta
sigma
mu
delta
MSE
min_MSE
0
100
1.000 10.000
1.000 20.000
1.000
3.000
0.000 -0.200
0.800 -0.200
lamb
0.000
0.000
12.79035
1.13623
12.79035
0.13849
200
300
5.000 20.000
9.000 20.000
3.000
3.000
0.400 -0.200
0.000 -0.200
0.150
0.300
1.08765
1.27175
0.13849
0.13849
207
Advanced Modeling of the VSTOXX Index
its
0
kappa theta
1.000 20.000
sigma
3.000
lamb
0.400
mu
0.400
delta
0.150
MSE
0.13849
min_MSE
0.13849
100
200
300
1.007 20.567
1.018 21.609
1.033 22.713
2.921
3.266
3.222
0.396 0.326
0.404 0.032
0.407 -0.123
0.155
0.163
0.155
0.12039
0.10182
0.09869
0.11935
0.10096
0.09863
400
500
1.542 23.755
1.860 23.848
3.235
2.967
0.409 -0.985
0.410 -1.297
0.067
0.002
0.08329
0.07686
0.08120
0.07686
600 1.868 23.863 2.956 0.410 -1.303 -0.000 1000.07683
0.07683
Warning: Maximum number of function evaluations has been exceeded.
CPU times: user 1min 7s, sys: 16 ms, total: 1min 7s
Wall time: 1min 7s
In [75]: opt_8_MSAE
Out[75]:
array([ 1.86781293e+00,
4.10158642e-01,
2.38625952e+01,
2.95539873e+00,
-1.30281337e+00,
8.30841775e-08])
Figure 7.8 shows that the fit is still reasonable for eight maturities and that many options.
In [76]: plot_calibration_results(option_data, opt_8_MSAE, mats)
To check whether there is a (larger) difference when we calibrate the model using relative
differences (i.e. the MSRE) as a yardstick, consider the following calibration run:
In [77]: %%time
....: opt_8_MSRE = srjd_model_calibration(option_data, p0=opt_8_MSAE,
....:
rel=True, mats=mats)
....:
its
kappa
theta
sigma
mu
delta
MSE
min_MSE
0
100
200
1.868 23.863
1.599 24.888
1.015 26.443
2.955
2.619
2.737
0.410 -1.303
0.392 -1.381
0.388 -1.676
lamb
0.000
0.000
0.000
0.02551
0.02099
0.02017
0.02551
0.02099
0.02017
300
400
0.915 27.147
0.920 27.175
2.723
2.719
0.388 -1.947
0.388 -2.142
0.000
0.000
0.02046
0.02010
0.02011
0.02010
500
600
700
0.927 27.082
0.925 27.083
0.925 27.086
2.724
2.722
2.722
0.389 -2.373
0.389 -2.409
0.389 -2.409
0.000
0.000
0.000
0.02009
0.02009
0.02009
0.02009
0.02009
0.02009
Warning: Maximum number of function evaluations has been exceeded.
CPU times: user 46.9 s, sys: 16 ms, total: 46.9 s
Wall time: 46.9 s
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LISTED VOLATILITY AND VARIANCE DERIVATIVES
FIGURE 7.8
Calibration of SRJD model to European call options on the VSTOXX for all eight
maturities (MSAE used).
Figure 7.9 presents the results. They are not too dissimilar to the ones obtained using the MSAE
as a yardstick. The major difference is the weighting of the options in that now those options
with lower market quotes (higher strikes) get more weight.
In [78]: plot_calibration_results(option_data, opt_8_MSRE, mats)
Advanced Modeling of the VSTOXX Index
209
FIGURE 7.9
Calibration of SRJD model to European call options on the VSTOXX for all eight
maturities (MSRE used).
7.7 CONCLUSIONS
This chapter introduces a more sophisticated model, the so-called square-root jump diffusion model (SRJD), for the evolution of the VSTOXX volatility index over time. It enhances
the Grünbichler and Longstaff (1996) square-root diffusion model by two components: a log
normally distributed jump component and a deterministic shift component. While the first
210
LISTED VOLATILITY AND VARIANCE DERIVATIVES
allows us to better calibrate the model to short term option quotes, the latter makes it possible to take the volatility term structure – as embodied by the eight futures on the VSTOXX
index – into account. All in all, the model yields good calibration results even in cases where
all eight option maturities are accounted for.
7.8 PYTHON SCRIPTS
7.8.1 srjd fwd calibration.py
#
# Script for term structure calibration of
# Square-Root Jump Diffusion (SRJD) model
#
# (c) Dr. Yves J. Hilpisch
# Listed Volatility and Variance Derivatives
#
import math
import numpy as np
import pandas as pd
import scipy.optimize as sco
v0 = 17.6639
i = 0
# initial VSTOXX index level
# counter for calibration runs
# reading the VSTOXX futures quotes
path = 'source/data/'
h5 = pd.HDFStore(path + 'vstoxx_data_31032014.h5', 'r')
futures_quotes = h5['futures_data']
h5.close()
# selecting needed data columns and adding spot value
forwards = list(futures_quotes['PRICE'].values)
forwards.insert(0, v0)
forwards = np.array(forwards)
ttms = list(futures_quotes['TTM'].values)
ttms.insert(0, 0)
ttms = np.array(ttms)
def srd_forwards(p0):
''' Function for forward volatilities in GL96 Model.
Parameters
==========
p0: list
Advanced Modeling of the VSTOXX Index
set of model parameters, where
kappa: float
mean-reversion factor
theta: float
long-run mean
sigma: float
volatility factor
Returns
=======
forwards: NumPy ndarray object
forward volatilities
'''
t = ttms
kappa, theta, sigma = p0
g = math.sqrt(kappa ** 2 + 2 * sigma ** 2)
sum1 = ((kappa * theta * (np.exp(g * t) - 1)) /
(2 * g + (kappa + g) * (np.exp(g * t) - 1)))
sum2 = v0 * ((4 * g ** 2 * np.exp(g * t)) /
(2 * g + (kappa + g) * (np.exp(g * t) - 1)) ** 2)
forwards = sum1 + sum2
return forwards
def srd_fwd_error(p0):
''' Error function for GL96 forward volatilities calibration.
Parameters
==========
p0: tuple
parameter vector
Returns
=======
MSE: float
mean-squared error for p0
'''
global i
kappa, theta, sigma = p0
srd_fwds = srd_forwards(p0)
MSE = np.sum((forwards - srd_fwds) ** 2) / len(forwards)
if 2 * kappa * theta < sigma ** 2:
MSE = MSE + 100
# penalty
elif sigma < 0:
MSE = MSE + 100
# print intermediate results: every 50th iteration
211
212
LISTED VOLATILITY AND VARIANCE DERIVATIVES
if i % 50 == 0:
print "{:6.3f} {:6.3f} {:6.3f}".format(*p0) + "{:>12.5f}".format(MSE)
i += 1
return MSE
if _ _name_ _
is '_ _main_ _':
p0 = 1.0, 17.5, 1.0
opt = sco.fmin(srd_fwd_error, p0,
xtol=0.00001, ftol=0.00001,
maxiter=1500, maxfun=2000)
7.8.2 srjd simulation.py
#
# Module with simulation functions for
# Square-Root Jump Diffusion (SRJD) model
#
# (c) Dr. Yves J. Hilpisch
# Listed Volatility and Variance Derivatives
#
import math
import pickle
import numpy as np
import scipy.interpolate as scint
v0 = 17.6639
# initial VSTOXX index level
# parameters of square-root diffusion
kappa = 2.0 # speed of mean reversion
theta = 15.0 # long-term volatility
sigma = 1.0 # standard deviation coefficient
# parameters of log-normal jump
lamb = 0.4 # intensity (jumps per year)
mu = 0.4 # average jump size
delta = 0.1
# volatility of jump size
# general parameters
r = 0.01
K = 17.5
# risk-free interest rate
# strike
T = 0.5 # time horizon
M = 150 # time steps
I = 10000 # number of MCS paths
anti_paths = True # antithetic variates
mo_match = True # moment matching
Advanced Modeling of the VSTOXX Index
# deterministic shift parameters
varphi = pickle.load(open('varphi'))
tck = scint.splrep(varphi['ttms'], varphi['varphi'], k=1)
# linear splines interpolation of
# term structure calibration differences
def random_number_gen(M, I, fixed_seed=False):
''' Generate standard normally distributed pseudo-random numbers
Parameters
==========
M: int
number of time intervals
I: int
number of paths
Returns
=======
ran: NumPy ndarrayo object
random number array
'''
if fixed_seed is True:
np.random.seed(10000)
if anti_paths is True:
ran = np.random.standard_normal((M + 1, I / 2))
ran = np.concatenate((ran, -ran), axis=1)
else:
ran = np.standard_normal((M + 1, I))
if mo_match is True:
ran = ran / np.std(ran)
ran -= np.mean(ran)
return ran
def srjd_simulation(x0, kappa, theta, sigma,
lamb, mu, delta, T, M, I, fixed_seed=False):
''' Function to simulate square-root jump Difusion.
Parameters
==========
x0: float
initial value
kappa: float
mean-reversion factor
213
214
LISTED VOLATILITY AND VARIANCE DERIVATIVES
theta: float
long-run mean
sigma: float
volatility factor
lamb: float
jump intensity
mu: float
expected jump size
delta: float
standard deviation of jump
T: float
time horizon/maturity
M: int
time steps
I: int
number of simulation paths
Returns
=======
x: NumPy ndarray object
array with simulated SRJD paths
'''
dt = float(T) / M # time interval
shift = scint.splev(np.arange (M + 1) * dt, tck, der=0)
# deterministic shift values
xh = np.zeros((M + 1, I), dtype=np.float)
x = np.zeros((M + 1, I), dtype=np.float)
xh[0, :] = x0
x[0, :] = x0
# drift contribution of jump p.a.
rj = lamb * (math.exp(mu + 0.5 * delta ** 2) - 1)
# 1st matrix with standard normal rv
ran1 = random_number_gen(M + 1, I, fixed_seed)
# 2nd matrix with standard normal rv
ran2 = random_number_gen(M + 1, I, fixed_seed)
# matrix with Poisson distributed rv
ran3 = np.random.poisson(lamb * dt, (M + 1, I))
for t in range(1, M + 1):
xh[t, :] = (xh[t - 1, :] +
kappa * (theta - np.maximum(0, xh[t - 1, :])) * dt
+ np.sqrt(np.maximum(0, xh[t - 1, :])) * sigma
* ran1[t] * np.sqrt(dt)
+ (np.exp(mu + delta * ran2[t]) - 1) * ran3[t]
* np.maximum(0, xh[t - 1, :]) - rj * dt)
x[t, :] = np.maximum(0, xh[t, :]) + shift[t]
return x
Advanced Modeling of the VSTOXX Index
def srjd_call_valuation(v0, kappa, theta, sigma,
lamb, mu, delta, T, r, K, M=M, I=I,
fixed_seed=False):
''' Function to value European volatility call option in SRDJ model.
Parameters see function srjd_simulation.
Returns
=======
call_value: float
estimator for European call present value for strike K
'''
v = srjd_simulation(v0, kappa, theta, sigma,
lamb, mu, delta, T, M, I, fixed_seed)
call_value = np.exp(-r * T) * sum(np.maximum(v[-1] - K, 0)) / I
return call_value
if _ _name_ _ is '_ _main_ _':
call_value = srjd_call_valuation(v0, kappa, theta, sigma,
lamb, mu, delta, T, r, K, M, I)
print "Value of European call by MCS: %10.4f" % call_value
7.8.3 srjd model calibration.py
#
# Calibration of square-root jump diffusion (SRJD) model
# to VSTOXX European call options traded at Eurex
# Data as of 31. March 2014
# All data from www.eurexchange.com
#
# (c) Dr. Yves J. Hilpisch
# Listed Volatility and Variance Derivatives
#
import numpy as np
import pandas as pd
import scipy.optimize as sco
import matplotlib.pyplot as plt
from srd_model_calibration import path, read_select_quotes
from srjd_simulation import srjd_call_valuation
# fixed parameters
r = 0.01
# risk-less short rate
v0 = 17.6639
M = 15
I = 100
# VSTOXX index at 31.03.2014
# number of time intervals
# number of simulated paths
215
216
LISTED VOLATILITY AND VARIANCE DERIVATIVES
def srjd_valuation_function(p0):
''' Valuation ('difference') function for all options
of a given DataFrame object.
Parameters
==========
p0: list
set of model parameters
Returns
=======
diffs: NumPy ndarray object
array with valuation differences
'''
global relative, option_data
kappa, theta, sigma, lamb, mu, delta = p0
diffs = []
for i, option in option_data.iterrows():
value = srjd_call_valuation(v0, kappa, theta, sigma,
lamb, mu, delta,
option['TTM'], r, option['STRIKE'],
M=M, I=I, fixed_seed=True)
if relative is True:
diffs.append((value - option['PRICE']) / option['PRICE'])
else:
diffs.append(value - option['PRICE'])
diffs = np.array(diffs)
return diffs
def srjd_error_function(p0):
''' Error function for model calibration.
Parameters
==========
p0: tuple
set of model parameters
Returns
=======
MSE: float
mean squared (relative/absolute) error
'''
global i, min_MSE, option_data
OD = len(option_data)
diffs = srjd_valuation_function(p0)
kappa, theta, sigma, lamb, mu, delta = p0
217
Advanced Modeling of the VSTOXX Index
# penalties
pen = 0.
if 2 * kappa * theta < sigma ** 2:
pen = 1000.0
if kappa < 0 or theta < 0 or sigma < 0 or lamb < 0 or delta < 0:
pen = 1000.0
MSE = np.sum(diffs ** 2) / OD + pen
min_MSE = min(min_MSE, MSE)
# mean squared error
# running minimum value
if i == 0:
print '\n' + ('{:>5s}'.format('its')
+ '{:>7s} {:>6s} {:>6s} {:>6s} {:>6s} {:>6s}'.format(
'kappa', 'theta', 'sigma', 'lamb', 'mu', 'delta')
+ '{:>12s}'.format('MSE') + '{:>12s}'.format('min_MSE'))
# print intermediate results: every 100th iteration
if i % 100 == 0:
print ('{:>5d}'.format(i)
+ '{:7.3f} {:6.3f} {:6.3f} {:6.3f} {:6.3f} {:6.3f}'.format(*p0)
+ '{:>12.5f}'.format(MSE) + '{:>12.5f}'.format(min_MSE))
i += 1
return MSE
def srjd_model_calibration(data, p0= None, rel=False, mats=None):
''' Function for global and local model calibration.
Parameters
==========
option_data: pandas DataFrame object
option quotes to be used
relative: bool
relative or absolute MSE
mats: list
list of maturities of option quotes to calibrate to
Returns
=======
opt: tuple
optimal parameter values
'''
global i, min_MSE, option_data
global relative
min_MSE = 5000.
relative = rel
# if True: MSRE is used, if False: MSAE
# dummy value
# relative or absolute
option_data = data
218
LISTED VOLATILITY AND VARIANCE DERIVATIVES
if mats is
not None:
# select the option data for the given maturities
option_data = option_data[option_data['MATURITY'].isin(mats)]
# global optimization
if p0
is None:
i = 0
# counter for calibration iterations
p0 = sco.brute(srjd_error_function, (
(1.0, 9.1, 4.0),
# kappa
(10., 20.1, 10.0),
(1.0, 3.1, 2.0),
# theta
# sigma
(0.0, 0.81, 0.4),
(-0.2, 0.41, 0.3),
# lambda
# mu
(0.0, 0.31, 0.15)),
# delta
finish=None)
# local optimization
i = 0
opt = sco.fmin(srjd_error_function, p0,
xtol=0.0000001, ftol=0.0000001,
maxiter=550, maxfun=700)
return opt
def plot_calibration_results(option_data, opt, mats):
''' Function to plot market quotes vs. model prices.
Parameters
==========
option_data: pandas DataFrame object
option data to plot
opt: list
optimal results from calibration
mats: list
maturities to be plotted
'''
kappa, theta, sigma, lamb, mu, delta = opt
# adding model values for optimal parameter set
# to the DataFrame object
values = []
for i, option in option_data.iterrows():
value = srjd_call_valuation(v0, kappa, theta, sigma,
lamb, mu, delta,
option['TTM'], r, option['STRIKE'],
M=M, I=I, fixed_seed=True)
values.append(value)
option_data['MODEL'] = values
Advanced Modeling of the VSTOXX Index
219
# plotting the market and model values
height = min(len(mats) * 3, 12)
fig, axarr = plt.subplots(len(mats), 2, sharex=True,
figsize=(10, height))
for z, mat in enumerate(mats):
if z == 0:
axarr[z, 0].set_title('values')
axarr[z, 1].set_title('differences')
os = option_data[option_data. MATURITY == mat]
strikes = os. STRIKE.values
axarr[z, 0].set_ylabel('%s' % str(mat)[:10])
axarr[z, 0].plot(strikes, os.PRICE.values, label='market quotes')
axarr[z, 0].plot(strikes, os.MODEL.values, 'ro', label='model prices')
axarr[z, 0].legend(loc=0)
wi = 0.3
axarr[z, 1].bar(strikes - wi / 2, os. MODEL.values - os.PRICE.values,
width=wi)
if mat == mats[-1]:
axarr[z, 0].set_xlabel('strike')
axarr[z, 1].set_xlabel('strike')
if _ _name_ _ == '_ _main_ _':
option_data = read_select_quotes('./source/data/', tol=0.1)
option_data['VALUE'] = 0.0
opt = srjd_model_calibration()
Listed Volatility and Variance Derivatives: A
Python-based Guide
By Dr. Yves J. Hilpisch
© 2017 Yves Hilpisch
CHAPTER
8
Terms of the VSTOXX and
its Derivatives
his brief chapter is about the terms of the VSTOXX volatility index as well as futures and
options traded on it. The chapter starts, however, with a brief review of some facts about
the EURO STOXX 50 index whose options build the basis for the VSTOXX.
T
8.1 THE EURO STOXX 50 INDEX
The EURO STOXX 50 index is a European blue chip index introduced in February 1998 by
STOXX Limited, a company of Deutsche Boerse Group. It comprises the stocks of the 50
largest companies by market capitalization in the euro zone given their free float. The weighting of the single components is also according to the market capitalization with a 10% cap for
any given stock. The composition and single memberships are reviewed on an annual basis
in September. The index level itself is calculated during trading days between 9 am and 6 pm
CET. Although it was only introduced in 1998, it has been recalculated on a daily basis dating
back to December 31, 1986.
Table 8.1 shows the composition of the index by sector.
Table 8.2 shows the composition of the index by country. Note that countries like the
United Kingdom or Switzerland are not represented due to the euro zone criterion.
While the VSTOXX is based on an equity index with 50 stocks only, the VIX index is on
a broader index, namely the S&P 500 which comprises 500 companies. This and the fact that
the EURO STOXX 50 index is heavy on the financial sector gives a generally higher level for
the VSTOXX index compared to the VIX index. In other words, 30 day implied volatility for
the EURO STOXX 50 index is on average higher than for the S&P 500 index.
8.2 THE VSTOXX INDEX
The calculation of the VSTOXX index is explained in detail in Chapter 5, VSTOXX Index. Its
calculation is based on out-of-the-money put and call options on the EURO STOXX 50 index.
221
222
LISTED VOLATILITY AND VARIANCE DERIVATIVES
TABLE 8.1
Industry sector weighting in the
EURO STOXX 50 index.
Sector
Weight
Banks
Industrial Goods & Services
Chemicals
Insurance
Health Care
Oil & Gas
Personal & Household Goods
Automobiles & Parts
Telecommunications
Food & Beverage
16.9%
9.9%
9.5%
8.0%
6.8%
6.8%
6.5%
6.2%
6.1%
5.9%
Data source: STOXX Limited, data as of 30
November 2015.
The methodology for its calculation makes use of the model-free replication approach for
variance swaps (see Chapter 3, Model-Free Replication of Variance). It yields an estimate for
the 30 day implied volatility by interpolating in general the implied volatilities from the two
nearby maturity months.
In addition to the main VSTOXX 30-day index with ticker symbol V2TX, eight subindexes are calculated with maturities of 1, 2, 3, 6, 9, 12, 18 and 24 months reflecting the
option series available for the EURO STOXX 50 index (see also Chapter 5, VSTOXX Index for
more details). There are other main VSTOXX indexes available with time ranges from 60 to
360 days (in intervals of 30 days).
Figure 8.1 shows the starting screen for the VSTOXX index at the Thomson Reuters Eikon
terminal. As you can see in the figure, the index mainly varied between 15% and 40% over the
period shown.
TABLE 8.2 Country weighting in the EURO
STOXX 50 index.
Country
Weight
France
Germany
Spain
Italy
Netherlands
Belgium
Finland
36.6%
32.1%
10.9%
7.7%
7.6%
4.1%
1.1%
Data source: STOXX Limited, data as of 30
November 2015
Terms of the VSTOXX and its Derivatives
223
FIGURE 8.1 Thomson Reuters Eikon starting screen for the VSTOXX volatility index (vendor
code: .V2TX).
8.3 VSTOXX FUTURES CONTRACTS
VSTOXX (mini) futures contracts, in the form they are traded today, were introduced in 2009.
Their product ID is FVS. Their main contract terms are described in the following list:
contract value: 100 EUR per index point of the underlying
settlement: cash settlement, on the first day after final settlement day
price quotation: the price quotation is in (volatility) points to two decimal places
minimum price change: the minimum price change is 0.05 (volatility) points, i.e. 5 EUR
contract months: the eight nearest successive calendar months
last trading day: last trading day is the final settlement day
final settlement day: 30 calendar days before the expiration day of the underlying options,
i.e. 30 calendar days before the third Friday of the expiration month of the underlying
options
daily settlement price: determinded for the current maturity month during the closing
auction of the respective futures contract; for the other maturity months it is determined
by the average bid/ask spread of the combination order book
final settlement price: determined by Eurex Exchange on the last trading day based on
the values of the underlying VSTOXX index between 11:30 am and 12:00 pm CET
trading hours: 9 am to 10 pm CET (order book and Eurex Trade Entry Services)
block trade size: 1,000 contracts
vendor codes: FVSA Index (Bloomberg) and 0#FVS (Thomson Reuters).
Figure 8.2 shows a screen from the Thomson Reuters Eikon terminal with data about the
VSTOXX futures contracts.
224
LISTED VOLATILITY AND VARIANCE DERIVATIVES
FIGURE 8.2
Thomson Reuters Eikon screen with VSTOXX futures contract information
(vendor code: 0#FVS).
8.4 VSTOXX OPTIONS CONTRACTS
The VSTOXX options contracts, with product ID OVS, were introduced in 2010. Their main
contract terms are described in the following list. They are mainly the same as for the futures
contracts:
contract value: 100 EUR per index point of the underlying
settlement: cash settlement, on the first day after final settlement day
price quotation: the price quotation is in (volatility) points with two decimal places
minimum price change: the minimum price change is 0.05 (volatility) points, i.e. 5 EUR
contract months: the eight nearest successive calendar months
last trading day: last trading day is the final settlement day
final settlement day: 30 calendar days before the expiration day of the underlying options,
i.e. 30 calendar days before the third Friday of the expiration month of the underlying
options
daily settlement price: determined by Eurex Exchange on the basis of the Black (1976)
model
final settlement price: determined by Eurex Exchange on the last trading day based on
the values of the underlying VSTOXX index between 11:30 am and 12:00 pm CET
exercise: European style; exercise only on the final settlement day of the option series
until 9 pm CET
exercise prices: minimum exercise price difference of 1 volatility point
number of exercise prices: at least 11 different exercise prices for both calls and puts
such that 5 are in-the-money, 1 at-the-money, 5 out-of-the-money
trading hours: 9 am to 10 pm CET (order book and Eurex Trade Entry Services)
block trade size: 1,000 contracts
vendor codes: V2X Index OMON (Bloomberg) and <0#FVS+> (Thomson Reuters).
Terms of the VSTOXX and its Derivatives
225
FIGURE 8.3 Thomson Reuters Eikon screen with VSTOXX options contract information (vendor
code: FVS270A6.EX).
Figure 8.3 shows a screen from the Thomson Reuters Eikon terminal with VSTOXX call and
put options data for an exercise price of 27.
8.5 CONCLUSIONS
The previous three chapters cover the VSTOXX volatility index and derivatives written on
it from a conceptual and valuation point of view. This chapter adds information about the
concrete terms of the contracts to the mix. While perhaps not as important from an academic
point of view, they are of paramount importance for practitioners.
PART
Three
Listed Variance Derivatives
Listed Volatility and Variance Derivatives: A
Python-based Guide
By Dr. Yves J. Hilpisch
© 2017 Yves Hilpisch
CHAPTER
9
Realized Variance and Variance Swaps
9.1 INTRODUCTION
This chapter discusses basic notions and concepts needed in the context of variance swaps and
futures. It covers among others the following topics:
realized variance: the basic measure on which variance swaps and variance futures are
defined
variance swap: the definition of a variance swap and some numerical examples
mark-to-market: the mark-to-market valuation approach for a variance swap
variance swap on EURO STOXX 50: simple re-calculation of a variance swap given
historical data
variance vs. volatility: major differences between the two measures.
9.2 REALIZED VARIANCE
Historical or realized variance 𝜎 2 generally is defined as
𝜎2 ≡
N
252 ∑ 2
R
⋅
N n=1 n
where, for a time series Sn , n = 0, 1, ..., N, the log returns are given by
Rn ≡ log
Sn
Sn−1
Here, it is assumed that there are 252 trading days per year and that the average daily return is
zero. The simple application of these definitions yields values as decimals. Scaling by a factor
of 1002 = 10,000 gives values in percent.
𝜎 2 ≡ 10000 ⋅
N
252 ∑ 2
R
⋅
N n=1 n
229
230
LISTED VOLATILITY AND VARIANCE DERIVATIVES
To simplify notation, we use the notation 𝜎 2 instead of 𝜎̂ 2 for the realized variance from
here on.
The concept of realized variance is easily illustrated by the use of historical data for the
EURO STOXX 50 stock index. To this end, we read data from the index provider’s website http://www.stoxx.com with Python and the pandas library. For details on using the pandas library for interactive financial analytics see chapter 2, Introduction to Python or refer to
Hilpisch (2014). As usual, we start with some Python library imports.
In [1]: import math
In [2]: import numpy as np
In [3]: import pandas as pd
First, we need the complete URL of the data set.
# text/csv file containing daily closing levels of EURO STOXX 50 index
In [4]: path = 'https://www.stoxx.com/document/Indices/Current/HistoricalData/'
In [5]: es_url = path + 'hbrbcpe.txt'
Second, we read the data with the pandas library from that source. The row structure of that
file changes at the end of 2001 such that we need to define a helper column which is to be
deleted after importing the data (see also chapter 4, Data Analysis and Strategies).
# column names for the data set
In [6]: cols = ['Date', 'SX5P', 'SX5E', 'SXXP', 'SXXE',
...:
...:
'SXXF', 'SXXA', 'DK5F', 'DKXF', 'DEL']
In [7]: try: # reading the data with pandas
...:
es = pd.read_csv(es_url, # filename
...:
...:
...:
header=None, # ignore column names
index_col=0, # index column (dates)
parse_dates=True, # parse these dates
...:
...:
...:
dayfirst=True, # format of dates
skiprows=4, # ignore these rows
sep=';', # data separator
...:
names=cols)
# use these column names
...:
del es['DEL']
...: except: # read stored data if there is no Internet connection
...:
...:
es = pd.HDFStore('./source/data/SX5E.h5', 'r')['SX5E']
In [8]: es = es[es.index < '2015-12-31']
231
Realized Variance and Variance Swaps
Let us inspect the final five data rows.
In [9]: es.tail()
Out[9]:
SX5P
SX5E
SXXP
SXXE
SXXF
SXXA
DK5F
DKXF
2015-12-23
3109.23
3286.68
366.39
346.14
433.58
375.53
9927.33
614.12
2015-12-24
3108.11
3284.47
366.28
346.05
433.43
375.39
9931.72
614.38
2015-12-28
3093.61
3256.49
364.49
343.54
431.26
374.32
9873.94
611.58
2015-12-29
3139.28
3314.28
369.68
349.29
438.43
378.86
10023.66
620.66
2015-12-30
3118.07
3287.98
367.70
347.02
435.82
377.20
9956.22
617.48
Date
Third, we select the EURO STOXX 50 index data from the data set just downloaded and
imported, i.e. the data sub-set for the symbol SX5E. Using this sub-set, we generate a new
pandas DataFrame object to store the data. The historical time series of daily closing levels
of the EURO STOXX 50 can then easily be inspected by a call of the plot method. Figure 9.1
shows the graphical output.
In [10]: import seaborn as sns; sns.set()
In [11]: import matplotlib
In [12]: matplotlib.rcParams['font.family'] = 'serif'
In [13]: data = pd.DataFrame({'SX5E': es['SX5E']})
In [14]: data.plot(figsize=(10, 6));
FIGURE 9.1
Historical index levels of EURO STOXX 50 index.
232
LISTED VOLATILITY AND VARIANCE DERIVATIVES
Fourth, the log returns are calculated (in vectorized fashion, i.e. simultaneously over the whole
time series) and stored as a new column in the pandas DataFrame object.
In [15]: data['R_n'] = np.log(data['SX5E'] / data['SX5E'].shift(1))
Let us inspect the last five data rows of this new DataFrame object.
In [16]: data.tail()
Out[16]:
SX5E
R_n
Date
2015-12-23
3286.68
0.022262
2015-12-24
2015-12-28
3284.47
3256.49
-0.000673
-0.008555
2015-12-29
2015-12-30
3314.28
3287.98
0.017590
-0.007967
In the fifth step, we calculate the realized variance, again in vectorized fashion. With
the following code we calculate the realized variance for every single date of the time
series.
# np.cumsum calculates the element-wise cumulative sum of an array/time series
# np.arange(N) gives an array of the form [0, 1, ..., N-1]
In [17]: data['sigma**2'] = 10000 * 252 * (np.cumsum(data['R_n'] ** 2)
/ np.arange(len(data)))
....:
....:
The third column of the DataFrame object now contains the realized variance.
In [18]: data.tail()
Out[18]:
SX5E
R_n
sigma**2
3286.68
3284.47
0.022262
-0.000673
446.280005
446.220423
2015-12-28
2015-12-29
3256.49
3314.28
-0.008555
0.017590
446.185389
446.230025
2015-12-30
3287.98
-0.007967
446.191722
Date
2015-12-23
2015-12-24
In the sixth and final step, one can now compare the index level time series with the realized
variance over time graphically – see Figure 9.2.
233
Realized Variance and Variance Swaps
FIGURE 9.2
Historical index levels of EURO STOXX 50 index and realized variance (long
term).
In [19]: data[['SX5E', 'sigma**2']].plot(subplots=True,
....:
figsize=(10, 8),
....:
color='blue',
....:
....:
grid=True);
Now let us implement the same approach for a shorter, recent period of time, i.e. the second
half of the year 2015. The realized variance has to be re-calculated since there is now a new
starting date.
# select time series data with date later/earlier than given dates
In [20]: short = data[['SX5E', 'R_n']][(data.index > '2015-7-1')
....:
& (data.index <= '2015-12-31')]
....:
# calculate the realized variance in percent values
In [21]: short['sigma**2'] = 10000 * 252 * (np.cumsum(short['R_n'] ** 2)
....:
/ np.arange(len(short)))
....:
234
LISTED VOLATILITY AND VARIANCE DERIVATIVES
The first five rows of the new DataFrame object are as follows:
In [22]: short.head()
Out[22]:
SX5E
R_n
sigma**2
Date
2015-07-02
3463.25 -0.009492
inf
2015-07-03
3441.76 -0.006224
324.686954
2015-07-06
3365.20 -0.022496
799.967421
2015-07-07
3294.19 -0.021327
915.381074
2015-07-08
3327.50
750.306250
0.010061
A graphical comparison of the EUROS STOXX 50 time series data with its realized variance
for the shorter time frame is displayed in Figure 9.3.
In [23]: short[['SX5E', 'sigma**2']].plot(subplots=True,
....:
figsize=(10, 8),
....:
color='blue',
....:
grid=True);
....:
FIGURE 9.3
term).
Historical index levels of EURO STOXX 50 index and realized variance (short
235
Realized Variance and Variance Swaps
9.3 VARIANCE SWAPS
Nowadays, variance swaps are popular financial instruments for volatility/variance trading and
hedging purposes. See, for instance, the paper of Bossu et al. (2005) for an overview of the
features and characteristics of variance swaps.
9.3.1 Definition of a Variance Swap
A variance swap is a financial instrument that allows investors to trade future realized variance
against current implied volatility (the “strike”). The characteristics and payoff of a variance
swap are more like those of a forward contract than those of a typical swap on interest rates,
currencies, equities, etc.
The payoff hT of a variance swap maturing at some future date T is
2
hT = 𝜎0,T
− 𝜎K2
with 𝜎K2 being the variance strike and 𝜎K the volatility strike.
At inception, i.e. at t = 0, the volatility strike is set such that the value of the variance
swap is zero. This implies that the volatility strike is set equal to the implied volatility 𝜎i (0, T)
for the maturity T.
9.3.2 Numerical Example
Consider a Black-Scholes-Merton (1973) world with a geometric Brownian motion driving
uncertainty for the index level of relevance (see Black and Scholes (1973) and Merton (1973)).
The risk-neutral stochastic differential equation (SDE) in this model (without dividends) is
given by:
dSt = rSt dt + 𝜎St dZt
St is the index level at time t, r the constant risk-less short rate, 𝜎 the instantaneous volatility and
Zt a standard Brownian motion. For a comprehensive treatment of this and other continuous
time financial models refer, for example, to Björk (2009).
Instantaneous volatility (and variance) in this model world is constant which makes
implied volatility also constant, say 𝜎t = 𝜎i = 𝜎 = 0.2. Given, for example, a Monte Carlo simulation of this model, realized variance might deviate from 𝜎 2 = 0.22 = 0.04. Let us implement
such a Monte Carlo simulation for the model. An Euler discretization scheme for the above
SDE is given for t ≥ Δt by
((
)
)
√
𝜎2
r−
Δt + 𝜎 Δtzt
St = St−Δt exp
2
with Δt being the fixed time interval used for the discretization and zt a standard normally
distributed random variable.
A Python implementation might look as follows (refer to Hilpisch (2014) for details on
Monte Carlo simulation with Python):
236
LISTED VOLATILITY AND VARIANCE DERIVATIVES
def generate_path(S0, r, sigma, T, M):
''' Function to simulate a geometric Brownian motion.
Parameters
==========
S0: float
initial index level
r: float
constant risk-less short rate
sigma: float
instantaneous volatility
T: float
date of maturity (in year fractions)
M: int
number of time intervals
Returns
=======
path: pandas DataFrame object
simulated path
'''
# length of time interval
dt = float(T) / M
# random numbers
np.random.seed(100000)
rn = np.random.standard_normal(M + 1)
rn[0] = 0 # to keep the initial value
# simulation of path
path = S0 * np.exp(np.cumsum((r - 0.5 * sigma ** 2) * dt
+ sigma * math.sqrt(dt) * rn))
# setting initial value
path = pd.DataFrame(path, columns=['index'])
return path
Using this function and providing numerical parameters returns a pandas DataFrame with a
single simulated path for the model. A sample path is shown in Figure 9.4.
In [24]: S0 = 100
# initial index level
In [25]: r = 0.005
In [26]: sigma = 0.2
In [27]: T = 1.0
# risk-less short rate
# instantaneous volatility
# maturity date
237
Realized Variance and Variance Swaps
In [28]: M = 50
# number of time intervals
In [29]: data = generate_path(S0, r, sigma, T, M)
In [30]: data.plot(figsize=(10, 5));
Given such a simulated path, one can calculate realized variance over time in the same fashion
as above for the EURO STOXX 50 index.
In [31]: data['R_t'] = np.log(data['index'] / data['index'].shift(1))
# scaling now by M / T since returns are not necessarily daily returns
In [32]: data['sigma**2'] = 10000 * M / T * (np.cumsum(data['R_t'] ** 2)
/ np.arange(len(data)))
....:
....:
In [33]: data.tail()
Out[33]:
index
R_t
sigma**2
46
47
86.914274 –0.020492
86.263467 –0.007516
462.916655
453.668339
48
49
50
87.101314 0.009666
87.020414 –0.000929
83.157399 –0.045408
445.190118
436.113416
448.009603
Figure 9.5 shows the results graphically.
FIGURE 9.4
Sample path based on geometric Brownian motion.
238
LISTED VOLATILITY AND VARIANCE DERIVATIVES
FIGURE 9.5
Geometric Brownian motion sample path with realized variance.
In [34]: data[['index', 'sigma**2']].plot(subplots=True,
....:
figsize=(10, 8),
color='blue',
grid=True);
....:
....:
....:
In this case, the payoff hT of the variance swap at maturity is:
In [35]: data['sigma**2'].iloc[–1] – 20 ** 2
Out[35]: 48.009603247833468
In general, variance swaps have a notional that differs from a value of 1, i.e. the above value
would have to be multiplied by a notional not equal to 1. It is market practice to define the
variance swap notional in volatility terms:
Notional =
VegaNotional
2 ⋅ Strike
239
Realized Variance and Variance Swaps
This can be done consistently due to the following relationship for a derivative instrument f
depending on some underlying S with volatility 𝜎 (and satisfying further technical assumptions):
𝜕f
𝜕f
⋅ 2𝜎
=
𝜕𝜎
𝜕(𝜎 2 )
𝜕f
𝜕f
⇔
= 𝜕𝜎
2𝜎
𝜕(𝜎 2 )
Say we want a vega notional of 100,000 currency units, i.e.we want a payoff of 100,000 currency units per volatility point difference (e.g.when realized volatility is 1 percentage point
above the volatility strike). The variance notional then is
Notional =
100000
= 2500
2 ⋅ 20
In [36]: Notional = 100000. / (2 * 20)
In [37]: Notional
Out[37]: 2500.0
Given this value for the variance notional, the payoff of the variance swap in the above numerical example would be:
In [38]: Notional * (data['sigma**2'].iloc[–1] – 20 ** 2)
Out[38]: 120024.00811958367
9.3.3 Mark-to-Market
What about the value of a variance swap over time? A major advantage of working with variance (instead of volatility) is that variance is additive over time (when a mean of about zero
is assumed). This gives rise to the following present value of a variance swap at time t, for a
constant short rate r:
(
Vt = Notional ⋅ e
−r(T−t)
⋅
2 + (T − t) ⋅ 𝜎 2 (t, T)
t ⋅ 𝜎0,t
i
T
)
− 𝜎K2
The major component of the mark-to-market value of the variance swap is the time weighted
2 up until time t and implied variance 𝜎 2 (t, T) for the remaining
average of realized variance 𝜎0,t
i
life time from t onwards.
240
LISTED VOLATILITY AND VARIANCE DERIVATIVES
In the model economy, 𝜎i2 (t, T) = 𝜎K2 = 𝜎 2 = 400. Therefore:
(
Vt = Notional ⋅ e
−r(T−t)
⋅
2 + (T − t) ⋅ 400
t ⋅ 𝜎0,t
T
)
− 400
For t = 0, this obviously gives Vt = 0 as desired.
This is readily implemented in Python given that we already have realized variance in a
pandas DataFrame object. We calculate it again in vectorized fashion for t = 0, Δt, 2Δt..., T.
In [39]: dt = T / M
In [40]: t = np.arange(M + 1) * dt
In [41]: t
Out[41]:
array([0.
,
0.02,
0.04,
0.06,
0.08,
0.1 ,
0.12,
0.14,
0.16,
0.18,
0.2 ,
0.22,
0.24,
0.26,
0.28,
0.3 ,
0.32,
0.34,
0.36,
0.38,
0.4 ,
0.42,
0.44,
0.46,
0.48,
0.5 ,
0.52,
0.54,
0.56,
0.58,
0.6 ,
0.62,
0.64,
0.66,
0.68,
0.7 ,
0.72,
0.74,
0.76,
0.78,
0.8 ,
0.82,
0.84,
0.86,
0.88,
0.9 ,
0.92,
0.94,
0.96,
0.98,
1.
])
In [42]: sigma_K = 20
In [43]: data['V_t'] = Notional * np.exp(–r * (T – t)) * ((t * data['sigma**2']
+ (T – t) * sigma_K ** 2) / T – sigma_K ** 2)
....:
....:
In [44]: data.tail()
Out[44]:
R_t
sigma**2
V_t
46
86.914274 –0.020492
index
462.916655
144650.434548
47
86.263467 –0.007516
453.668339
126082.766551
48
87.101314
0.009666
445.190118
108434.594165
49
87.020414 –0.000929
436.113416
88469.022898
50
83.157399 –0.045408
448.009603
120024.008120
Graphically, we get the result as presented in Figure 9.6.
In [45]: data[['index', 'sigma**2', 'V_t']].plot(subplots=True,
....:
....:
figsize=(10, 8),
color='blue',
....:
....:
grid=True);
241
Realized Variance and Variance Swaps
FIGURE 9.6
Geometric Brownian motion sample path with realized variance and variance swap
mark-to-market values.
9.3.4 Vega Sensitivity
What is the sensitivity of the mark-to-market value of a variance swap with regard to implied
volatility? Recall that the value itself is given by
(
Vt = Notional ⋅ e
−r(T−t)
⋅
2 + (T − t) ⋅ 𝜎 2 (t, T)
t ⋅ 𝜎0,t
i
T
)
− 𝜎K2
Differentiation with respect to 𝜎i gives a vega of
Vegat =
𝜕Vt
T −t
= Notional ⋅ e−r(T−t) ⋅
⋅ 2𝜎i (t, T)
𝜕𝜎i
T
At inception of the variance swap, we have a vega of
𝜕V0
𝜕𝜎i
VegaNotional −r(T−t) T − t
=
⋅
⋅e
⋅ 2𝜎i (0, T)
2 ⋅ 𝜎i (0, T)
T
Vega0 =
= e−rT ⋅ VegaNotional
242
LISTED VOLATILITY AND VARIANCE DERIVATIVES
In this case, vega equals the discounted vega notional. For general t, we get
𝜕Vt
𝜕𝜎i
VegaNotional −r(T−t) T − t
=
⋅
⋅e
⋅ 2𝜎i (t, T)
2 ⋅ 𝜎i (t, T)
T
T −t
= e−r(T−t) ⋅ VegaNotional ⋅
T
Vegat =
This illustrates that vega sensitivity diminishes over time and that it is proportional to the
time-to-maturity.
9.3.5 Variance Swap on the EURO STOXX 50
We are now ready to do a historical re-calculation of a variance swap on the EURO STOXX
50. We will re-calculate a variance swap during June 2015. To this end we also use VSTOXX
sub-index data for the shortest maturity available which provides us with a time series for the
correct implied volatilities.
EURO STOXX 50 data is already available.
In [46]: es.info()
DatetimeIndex: 7475 entries, 1986-12-31 to 2015-12-30
Data columns (total 8 columns):
SX5P
7475 non–null float64
SX5E
SXXP
7475 non–null float64
7475 non–null float64
SXXE
SXXF
SXXA
7475 non–null float64
7475 non–null float64
7475 non–null float64
DK5F
7475 non–null float64
DKXF
7475 non–null float64
dtypes: float64(8)
memory usage: 525.6 KB
The VSTOXX data can be read from the same source (see chapter 4, Data Analysis and Strategies).
In [47]: vs_url = path + 'h_vstoxx.txt'
In [48]: try: # reading the data with pandas
....:
vs = pd.read_csv(vs_url, # filename
....:
index_col=0, # index column (dates)
....:
parse_dates=True,
# parse date information
243
Realized Variance and Variance Swaps
dayfirst=True, # day before month
header=2, # header/column names
....:
....:
....:
sep=',') # separator character
....: except: # read stored data if there is no Internet connection
....:
vs = pd. HDFStore('./source/data/V2TX.h5', 'r')
....:
In [49]: vs.info()
DatetimeIndex: 4357 entries, 1999-01-04 to 2016-02-12
Data columns (total 9 columns):
V2TX
V6I1
V6I2
4357 non–null float64
3906 non–null float64
4357 non–null float64
V6I3
V6I4
V6I5
4296 non–null float64
4357 non–null float64
4357 non–null float64
V6I6
V6I7
4340 non–null float64
4357 non–null float64
V6I8
4343 non–null float64
dtypes: float64(9)
memory usage: 340.4 KB
The data column V6I2 contains the index values (= implied volatility) for the nearest option
series maturity available (i.e within a maximum of one month). For example, on June 1, 2015,
the index values represent implied volatilities for the maturity on the third Friday in June 2015,
i.e. June 19. Maturity T then is:
# 15 trading days
In [50]: T = 15.
The variance swap we want to re-calculate should start on June 1, 2015 and will have a maturity
until June 19. It will have a vega notional of 100,000 EUR.
First, let us select and collect the data needed from the available data sets.
In [51]: data = pd.DataFrame(es['SX5E'][(es.index > '2015-5-31')
....:
....:
& (es.index < '2015-6-20')])
In [52]: data['V6I1'] = vs['V6I1'][(vs.index > '2015-5-31')
....:
& (vs.index < '2015-6-20')]
....:
244
LISTED VOLATILITY AND VARIANCE DERIVATIVES
The new data set looks as follows. Note that the VSTOXX sub-index is only available up until
two days before the maturity date.
In [53]: data
Out[53]:
SX5E
V6I1
2015-06-01
2015-06-02
2015-06-03
3575.04
3561.89
3583.82
25.8710
25.9232
25.7958
2015-06-04
2015-06-05
2015-06-08
3556.38
3510.01
3468.31
26.2418
27.4496
27.2996
2015-06-09
2015-06-10
2015-06-11
3456.79
3526.48
3551.91
26.8020
25.8610
26.3897
2015-06-12
2015-06-15
3502.77
3438.07
29.7725
34.5593
2015-06-16
2015-06-17
2015-06-18
3454.09
3428.76
3450.45
36.2222
34.7235
NaN
2015-06-19
3455.80
NaN
Date
We forward fill the NA values since for the vectorized calculations to follow we want to use
these data points but they will have a negligible or zero influence anyway.
In [54]: data = data.fillna(method='ffill')
We save the data set for later re-use.
In [55]: h5 = pd.HDFStore('./source/data/SX5E_V6I1.h5')
In [56]: h5['SX5E_V6I1'] = data
In [57]: h5.close()
The implied volatility on June 1 was 25.871%. This gives rise to a variance swap
strike of 𝜎K2 = 25.8712 = 669.31. For a vega notional of 100,000, the variance notional
therefore is
Notional =
100000
= 1932.67
2 ⋅ 25.871
Realized Variance and Variance Swaps
245
In [58]: data['V6I1'][0]
Out[58]: 25.870999999999999
In [59]: sigma_K = data['V6I1'][0]
In [60]: Notional = 100000 / (2. * sigma_K)
In [61]: Notional
Out[61]: 1932.665919369178
Three time series now have to be calculated:
log returns of the EURO STOXX 50 index
realized variance of the EURO STOXX 50 index
mark-to-market values of the variance swap.
First, the log returns.
In [62]: data['R_t'] = np.log(data['SX5E'] / data['SX5E'].shift (1))
Second, the realized variance which we scale by a factor of 10,000 to end up with percentage
values and not decimal values.
In [63]: data['sigma**2'] = 10000 * 252 * (np.cumsum(data['R_t'] ** 2)
....:
/ np.arange(len(data)))
....:
Third, the mark-to-market values. We start with the array of elapsed days.
In [64]: t = np.arange(1, 16)
In [65]: t
Out[65]: array([ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15])
We assume a fixed short rate of 0.1%.
In [66]: r = 0.001
In [67]: data['V_t'] = np.exp(–r * (T – t) / 365.) * ((t * data['sigma**2']
....:
+ (T – t) * data['V6I1'] ** 2) / T – sigma_K ** 2)
....:
The initial value of the variance swap is zero.
246
LISTED VOLATILITY AND VARIANCE DERIVATIVES
In [68]: data['V_t'].loc['2015-06-01'] = 0.0
The complete results data set is given below.
In [69]: data
Out[69]:
SX5E
V6I1
R_t
sigma**2
V_t
Date
2015-06-01
2015-06-02
2015-06-03
3575.04
3561.89
3583.82
25.8710
NaN
25.9232 -0.003685
25.7958 0.006138
NaN
0.000000
34.220799 -82.332277
64.580457 -124.049833
2015-06-04
2015-06-05
3556.38
3510.01
26.2418 -0.007686
27.4496 -0.013124
92.677552 -139.593571
178.023718 -107.644092
2015-06-08
2015-06-09
2015-06-10
3468.31
3456.79
3526.48
27.2996 -0.011951
26.8020 -0.003327
25.8610 0.019960
214.408816 -136.380856
183.323049 -200.634978
300.555651 -196.905901
2015-06-11
2015-06-12
2015-06-15
3551.91
3502.77
3438.07
26.3897 0.007185
29.7725 -0.013931
34.5593 -0.018644
279.249096 -223.189008
302.564935 -172.129074
359.901600 -86.887791
2015-06-16
2015-06-17
2015-06-18
3454.09
3428.76
3450.45
36.2222 0.004649
34.7235 -0.007360
34.7235 0.006306
332.134168 -141.190592
315.833038 -234.822528
299.246550 -309.629583
2015-06-19
3455.80
34.7235
278.303868 -391.004773
0.001549
The payoff of the variance swap at maturity given the variance notional then is:
In [70]: Notional * data['V_t'][-1]
Out[70]: -755681.59959400445
Finally, a plot of the major results is presented as Figure 9.7.
In [71]: data[['SX5E', 'sigma**2', 'V_t']].plot(subplots=True,
....:
color='blue',
....:
....:
figsize=(10, 8));
We save the data for use in the next chapter.
In [72]: h5 = pd.HDFStore('./source/data/var_data.h5', 'a')
In [73]: h5['var_swap'] = data
In [74]: h5.close()
Realized Variance and Variance Swaps
FIGURE 9.7
247
EURO STOXX 50 historical index levels with realized variance and futures prices.
9.4 VARIANCE VS. VOLATILITY
Both variance and volatility are tradable asset classes. The following sub-sections discuss some
differences between the two measures of variability and the two asset classes, respectively. See
also Bennett and Gil (2012) on this topic.
9.4.1 Squared Variations
Squared variations are in many application scenarios the better measure for variability compared to simple variations. By squaring variations, one makes sure that variations do not cancel
each other out. Since volatility is generally defined as the square root of variance, both measures avoid the cancelling of positive and negative variations.
9.4.2 Additivity in Time
Although both volatility and variance avoid the cancelling out of variations, there is a major
difference between both when it comes to additivity. While variance is additive (linear) in time,
volatility is convex (nonlinear) in time.
248
LISTED VOLATILITY AND VARIANCE DERIVATIVES
Assume we have N return observations and assume 0 < M < N. We then have:
N
252 ∑ 2
R
⋅
𝜎 ≡
N n=1 n
(M
)
N
∑
∑
252
2
2
=
Rn +
Rn
⋅
N
n=1
n=M+1
2
=
M
N
252 ∑ 2 252 ∑ 2
Rn +
R
⋅
⋅
N n=1
N n=M+1 n
≡ 𝜎12 + 𝜎22
Here, 𝜎12 is the variance for the first part and 𝜎22 for the second part of the return observations.
in order to retain additivity.
Note that one needs to keep the weighting factor constant at 252
N
This aspect can be illustrated by a simple numerical example. Consider first a function to
calculate realized variance that we can re-use.
# function to calculate the realized variance
In [75]: rv = lambda ret_dat: 10000 * 252. / N * np.sum(ret_dat ** 2)
Second, a simple example data set...
In [76]: data = np.array([0.01, 0.02, 0.03, 0.04, 0.05])
… of length N = 5.
In [77]: N = len(data)
In [78]: N
Out[78]: 5
Then, we can easily see additivity.
In [79]: rv(data[:2]) + rv(data[2:])
Out[79]: 2772.0000000000005
In [80]: rv(data)
Out[80]: 2772.0000000000005
249
Realized Variance and Variance Swaps
Next, we use the EURO STOXX 50 index data from before. Let us have a look at the year
2013 and the two halves of the year.
In [81]: data = pd.DataFrame(es['SX5E'][(es.index > '31-12-2012')
....:
....:
& (es.index < '01-01-2014')])
# we need log returns
In [82]: data['R_t'] = np.log(data['SX5E'] / data['SX5E'].shift(1))
We have 256 index level observations and 255 return observations.
In [83]: N = len(data) – 1
In [84]: N
Out[84]: 255
In [85]: var_1st = rv(data['R_t'][data.index < '2013-07-01'])
In [86]: var_1st
Out[86]: 159.1490165508348
In [87]: var_2nd = rv(data['R_t'][data.index > '2013-06-30'])
In [88]: var_2nd
Out[88]: 98.3424775210876
Again, additivity is given for the realized variance.
In [89]: var_1st + var_2nd
Out[89]: 257.4914940719224
In [90]: var_full = rv(data['R_t'])
In [91]: var_full
Out[91]: 257.4914940719223
Obviously, this is different when considering realized volatility instead of variance.
In [92]: vol_1st = math.sqrt(rv(data['R_t'][data.index < '2013-07-01']))
In [93]: vol_1st
Out[93]: 12.615427719694438
250
LISTED VOLATILITY AND VARIANCE DERIVATIVES
In [94]: vol_2nd = math.sqrt(rv(data['R_t'][data.index > '2013-06-30']))
In [95]: vol_2nd
Out[95]: 9.916777577473823
In [96]: vol_1st + vol_2nd
Out[96]: 22.53220529716826
In [97]: vol_full = math.sqrt(rv(data['R_t']))
In [98]: vol_full
Out[98]: 16.04654149877544
This is something to be expected due to the sub-additivity
root function.
√
a+b≤
√
√
a + b of the square-
9.4.3 Static Hedges
Realized variance can be statically replicated (hedged) by positions in out-of-the money put
and call options. This is a well-known result which is presented in detail in chapter 3, ModelFree Replication of Variance. It is the basic idea and approach underlying volatility indexes
like the VSTOXX and the VIX. This also makes it possible to statically replicate and hedge
variance swaps by the use of options – something not true for volatility swaps, for example.
9.4.4 Broad Measure of Risk
Implied volatility generally is only defined for a certain maturity and a certain strike. When the
spot moves, at-the-money implied volatility changes as well. By contrast, (implied) variance
is a measure taking into account all strikes for a given maturity. This can be seen from the fact
that the traded variance level of a variance swap is applicable independently of the spot of the
underlying (index).
9.5 CONCLUSIONS
This chapter introduces variance swaps both theoretically as well as based on concrete numerical examples. Central notions are realized variance, variance/volatility strike and variance
notional. Mark-to-market valuations of such instruments are easily accomplished due to their
very nature. As a consequence, sensitivities of variance swaps, for example, with regard to vega
are also easily derived. The major numerical example is based on EURO STOXX 50 index and
log return data. A hypothetical variance swap with inception on June 1, 2015 and maturity on
June 19, 2015 is valued by the mark-to-market approach using VSTOXX sub-index data with
the very same maturity as a proxy for the implied volatility during the life time of the variance
swap.
Listed Volatility and Variance Derivatives: A
Python-based Guide
By Dr. Yves J. Hilpisch
© 2017 Yves Hilpisch
CHAPTER
10
Variance Futures at Eurex
10.1 INTRODUCTION
Based on the previous chapter, this chapter introduces the Eurex variance futures contracts and
their major characteristics. It covers mainly the following topics:
introduction to and motivation for Eurex variance futures
variance futures concepts needed to understand and trade in Eurex variance futures
example calculation for a variance future illustrating the concepts by numerical results
comparison between variance swaps and futures based on the numerical example.
By introducing variance futures on the EURO STOXX 50 stock index, Eurex standardizes one
of the most popular types of volatility/variance derivatives, namely variance swaps. Such a
standardization brings a number of benefits to the markets:
unified terms: standardization leads to unified, well-documented terms
transparency: centralized trading at Eurex increases transparency for all market participants
liquidity: standardization and centralization increase liquidity (market depth) in the variance futures
fungibility: variance futures are fully fungible and can be traded in and out at any point
over their life time
increased market integrity: clearing by Eurex ensures, among other things, low counterparty risk and high risk management standards.
Variance futures replicate the payoff of Over-the-Counter (OTC) variance swaps and are traded
based on OTC conventions in vega notional and at volatility strikes. Daily margin payments
based on settlement prices add up to the final settlement payoff of the OTC variance swap. The
start of trading for the variance futures of Eurex was September 22, 2014.
Net present value (NPV) effects are accounted for by two different pricing components:
discount factor as for the mark-to-market for variance swaps
accumulated return on modified variation margin (ARMVM) for the daily margin
payments.
251
252
LISTED VOLATILITY AND VARIANCE DERIVATIVES
In order to perfectly mimic the cash flow of a variance swap with the same terms, variance
futures trading makes certain conversions necessary. This is mainly due to the variance swaps
being based on end-of-day returns of the respective index (which cannot be exactly known
during the day). Therefore, there are two conversions:
intraday: intraday trades are booked with the desired quantity and at preliminary futures
prices
end-of-day: at the end of the day, intraday trades are cancelled and re-booked with the
same quantity at the futures settlement prices (taking into account the realized variance
for the rest of the day, i.e. since the time the trade was initially booked).
All trades are booked and conversions are made by the Eurex T7 trading system. While variance futures are quoted in vega notional and volatility strikes, bookings and clearings are made
only in futures and futures prices.
10.2 VARIANCE FUTURES CONCEPTS
Standardized trading and settlement in variance futures is based on a number of financial concepts. This section introduces all necessary concepts and provides formal definitions.
10.2.1 Realized Variance
Realized variance 𝜎 2 for the Eurex variance futures is defined as
N
252 ∑ 2
𝜎 ≡ 10000 ⋅
R
⋅
N n=1 n
2
where, for a time series Sn , n = 0, 1, … , N, of daily EURO STOXX 50 closing values, the log
returns are given by
Rn ≡ log
Sn
Sn−1
Here, it is assumed that there are 252 trading days per year and that the average daily return is
zero. Note the scaling factor of 10,000 to get to the market convention of quoting variance in
percent and not in decimal values.
10.2.2 Net Present Value Concepts
There are two concepts that take into account that the daily margin cash flows of trading futures
leads, time-wise, to a different cash flow profile than the single payoff of a variance future at
maturity.
253
Variance Futures at Eurex
Discount Factor The first is the discount factor DFt which is defined for a maturity date
T and a current date t by
DFt = e−
r(T−t)
365
Here, r is the relevant interest rate to be applied to the remaining time-to-maturity (T − t),
calculated in days. Eurex uses Euribor rates to calculate the discount factor.
Let us have a look at the development of the Euribor rates in 2014 from January to the
beginning of September. Euribor rates are available on the website www.euribor-rates.eu or
in a slightly more convenient form as Excel spreadsheet files for download on the website
www.emmi-benchmarks.eu/euribor-org/euribor-rates.html. From this last website, we have
downloaded Euribor rates data for 2014 which we use in the following. For details on the
Python usage that follows, see chapter 2, Introduction to Python or refer to Hilpisch (2014).
pandas is once again the library we are working with.
In [1]: import pandas as pd
With pandas, you can read data from a spreadsheet file as follows:
# read data from CSV file
In [2]: eb = pd.read_csv('./source/data/hist_EURIBOR_2015.csv',
...:
index_col=0,
# index column
# filename
parse_dates=True,
# parsing date information
dayfirst=True)
# European date convention
...:
...:
...:
In [3]: eb. info()
DatetimeIndex: 135 entries, 2015-01-02 to 2015-07-14
Data columns (total 8 columns):
1w
2w
1m
135 non-null float64
135 non-null float64
135 non-null float64
2m
3m
6m
135 non-null float64
135 non-null float64
135 non-null float64
9m
135 non-null float64
12m
135 non-null float64
dtypes: float64(8)
memory usage: 9.5 KB
254
FIGURE 10.1
LISTED VOLATILITY AND VARIANCE DERIVATIVES
Historical Euribor rates.
Figure 10.1 shows the evolution of the eight different rates over time (values in percent).
In [4]: import seaborn as sns; sns. set()
In [5]: import matplotlib
In [6]: matplotlib.rcParams['font.family'] = 'serif'
In [7]: eb.plot(figsize=(10, 6));
Short term interest rates are pretty low, leading to a rather low impact of the discount factor in
such an environment. Let us calculate the discount factor for the longest maturity with the last
available rate for the 12 month horizon.
In [8]: eb.tail()
Out[8]:
1w
2w
1m
2m
3m
2015-07-08 -0.131 -0.118 -0.071 -0.038 -0.018
2015-07-09 -0.131 -0.118 -0.071 -0.039 -0.018
6m
0.049
0.049
9m
0.101
0.100
12m
0.164
0.163
2015-07-10 -0.133 -0.118 -0.071 -0.039 -0.018
2015-07-13 -0.131 -0.118 -0.071 -0.038 -0.019
2015-07-14 -0.131 -0.118 -0.071 -0.039 -0.019
0.049
0.049
0.049
0.101
0.102
0.101
0.164
0.166
0.168
For a 12 month period the discount effect is less than half a percent.
255
Variance Futures at Eurex
In [9]: import math
In [10]: math.exp(-0.00168 * 365. / 365)
Out[10]: 0.9983214104100598
ARMVM The accumulated return on modified variation margin (ARMVM) takes into account
that variance futures trading leads to daily margin payments that have an impact on the present
value of the variance future. For given dates t and t − 1, ARMVM is given by
)
rΔt
( S
) ( rΔt
− C e 365 − 1
ARMV Mt = ARMV Mt−1 ⋅ e 365 + Ft−1
Here, r is the relevant interest rate to be used, Δt is the difference between dates t and t − 1 in
S is the variance futures settlement price at t − 1 and C is a constant fixed at a level
days, Ft−1
of 3,000. If t = 1, i.e. on the first settlement day (say Monday), ARMVM1 ≡ 0 and F1S ≡ C.
The relevant rate for the ARMVM calculation is the Eonia rate. On the website www.emmi-benchmarks.eu/euribor-eonia-org/about-eonia.html you find the following
explanation:
“Eonia® (Euro OverNight Index Average) is the effective overnight reference
rate for the euro. It is computed as a weighted average of all overnight
unsecured lending transactions in the interbank market, undertaken in the
European Union and European Free Trade Association (EFTA) countries.”
This website also provides historical Eonia data for download. Let us have a look at the historical development of the Eonia rates for a couple of weeks in 2015.
# read data from Excel spreadsheet file
In [11]: eo = pd.read_csv('./source/data/hist_EONIA_2015.csv',
....:
....:
....:
# filename
index_col=0, # index column
parse_dates=True, # parsing date information
dayfirst=True) # European date convention
....:
In [12]: eo. info()
DatetimeIndex: 60 entries, 2015-04-21 to 2015-07-14
Data columns (total 1 columns):
EONIA
60 non-null float64
dtypes: float64(1)
memory usage: 960.0 bytes
Again, we can easily visualize the data set (see Figure 10.2).
256
FIGURE 10.2
LISTED VOLATILITY AND VARIANCE DERIVATIVES
Historical Eonia rates.
In [13]: eo.plot(figsize=(10, 5))
Out[13]:
The last available values in the data set for Eonia are:
In [14]: eo.tail()
Out[14]:
EONIA
2015-07-08 -0.118
2015-07-09 -0.121
2015-07-10 -0.120
2015-07-13 -0.118
2015-07-14 -0.112
Let us work with a value (in decimals) of reo = −0.001.
In [15]: r_eo = -0.001
Now consider the second settlement day t = 2 (say Tuesday), assume that a rate of reo applies
and remember that F1S = 3000. We then have
ARMV M2 = 0 ⋅ e
reo ⋅1
365
(
)
reo ⋅1
+ (3000 − 3000) e 365 − 1 = 0
257
Variance Futures at Eurex
Consider now the third settlement day t = 3 (say Wednesday). Assume that the futures settlement price has risen the day before to F2S = 3100. Now
ARMV M3 = 0 ⋅ e
reo ⋅1
365
(
+ (3100 − 3000) e
reo ⋅1
365
)
−1
The effect is rather small.
In [16]: ARMVM3 = 0 * math.exp(r_eo / 365) + (3100 - 3000)
* (math.exp(r_eo / 365) - 1)
In [17]: ARMVM3
Out[17]: -0.0002739722274380796
One day later, on the fourth settlement day t = 4, and assuming that the variance futures settlement price has fallen the day before to F3 = 3050, we get
ARMV M4 = ARMV M3 ⋅ e
reo ⋅1
365
(
+ (3050 − 3000) e
reo ⋅1
365
)
−1
In [18]: ARMVM4 = (ARMVM3 * math.exp(r_eo / 365)
....:
+ (3050 - 3000) * (math.exp(r_eo / 365) - 1))
....:
In [19]: ARMVM4
Out[19]: -0.0004109575905493053
10.2.3 Traded Variance Strike
2
The traded variance strike 𝜎t2 at date t is the time weighted average of the realized variance 𝜎0,t
weighted by the number of elapsed days t and the traded implied variance 𝜎i2 (t, T) weighted
by the remaining life time of the variance future in days:
𝜎t2 =
2 + (T − t) ⋅ 𝜎 2 (t, T)
t ⋅ 𝜎0,t
i
T
10.2.4 Traded Futures Price
The traded futures price Ft given the other quantities from before is then defined as
(
)
Ft = DFt ⋅ 𝜎t2 − 𝜎i2 (t, T) − ARMV Mt + 3000
The scaling constant C = 3000 is chosen to ensure that the futures price cannot become
negative.
258
LISTED VOLATILITY AND VARIANCE DERIVATIVES
10.2.5 Number of Futures
Variance futures are traded in vega notional and volatility strikes. However, settlement takes
place in futures (or variance notional) and variance strikes. The number of futures is calculated
according to the formula
futures =
VegaNotional
T
⋅
2 ⋅ 𝜎i (t, T)
T −t
with 𝜎i (t, T) as the traded implied volatility strike.
10.2.6 Par Variance Strike
At daily settlement, i.e. when the log return for the EURO STOXX 50 for that day is
known, the par variance strike 𝜎p2 is calculated using the then current settled implied
volatility 𝜎i .
𝜎p2
=
2 + (T − t) ⋅ 𝜎 2
t ⋅ 𝜎0,t
i
T
10.2.7 Futures Settlement Price
Finally, the daily futures settlement price FtS is calculated according to the following formula
where all components are as defined as before:
)
(
FtS = DFt ⋅ 𝜎p2 − 𝜎K2 − ARMV Mt + 3000
The futures settlement price is the mark-to-market value of the corresponding variance swap
minus the ARMVM plus the scaling factor of 3,000.
10.3 EXAMPLE CALCULATION FOR A VARIANCE FUTURE
In this section, we conduct an example calculation for the Eurex variance futures contract given
historical data as used in sub-section 9.3.5, Variance Swap on the EURO STOXX 50.
In [20]: import pandas as pd
In [21]: h5 = pd.HDFStore('./source/data/SX5E_V6I1.h5', 'r')
In [22]: data = h5['SX5E_V6I1']
In [23]: h5.close()
In [24]: data
259
Variance Futures at Eurex
Out[24]:
SX5E
V6I1
2015-06-01
2015-06-02
2015-06-03
3575.04
3561.89
3583.82
25.8710
25.9232
25.7958
2015-06-04
2015-06-05
2015-06-08
3556.38
3510.01
3468.31
26.2418
27.4496
27.2996
2015-06-09
2015-06-10
2015-06-11
3456.79
3526.48
3551.91
26.8020
25.8610
26.3897
2015-06-12
2015-06-15
3502.77
3438.07
29.7725
34.5593
2015-06-16
2015-06-17
2015-06-18
3454.09
3428.76
3450.45
36.2222
34.7235
34.7235
2015-06-19
3455.80
34.7235
Date
We add Euribor and Eonia data to the pandas DataFrame object. For simplicity, we use the
2 week Euribor values throughout.
In [25]: data = data.join(eb['2w'], how='left')
In [26]: data = data.join(eo, how='left')
In [27]: data
Out[27]:
SX5E
V6I1
2w
EONIA
Date
2015-06-01
3575.04
25.8710
-0.108
-0.106
2015-06-02
2015-06-03
2015-06-04
3561.89
3583.82
3556.38
25.9232
25.7958
26.2418
-0.109
-0.109
-0.109
-0.122
-0.143
-0.138
2015-06-05
2015-06-08
2015-06-09
3510.01
3468.31
3456.79
27.4496
27.2996
26.8020
-0.109
-0.110
-0.110
-0.115
-0.127
-0.126
2015-06-10
2015-06-11
2015-06-12
3526.48
3551.91
3502.77
25.8610
26.3897
29.7725
-0.111
-0.111
-0.113
-0.117
-0.120
-0.125
2015-06-15
2015-06-16
2015-06-17
3438.07
3454.09
3428.76
34.5593
36.2222
34.7235
-0.108
-0.109
-0.109
-0.119
-0.125
-0.110
2015-06-18
2015-06-19
3450.45
3455.80
34.7235
34.7235
-0.109
-0.108
-0.118
-0.120
260
LISTED VOLATILITY AND VARIANCE DERIVATIVES
Let us add the log returns to the data set as well as the realized variance.
In [28]: import numpy as np
In [29]: data['R_t'] = np.log(data['SX5E'] / data['SX5E'].shift(1))
In [30]: data['sigma**2'] = 10000 * 252 * (np.cumsum(data['R_t'] ** 2)
....:
/ np.arange(len(data)))
....:
In [31]: data
Out[31]:
SX5E
V6I1
2w
EONIA
R_t
sigma**2
2015-06-01
2015-06-02
2015-06-03
3575.04
3561.89
3583.82
25.8710
25.9232
25.7958
-0.108
-0.109
-0.109
-0.106
-0.122
-0.143
NaN
-0.003685
0.006138
NaN
34.220799
64.580457
2015-06-04
2015-06-05
2015-06-08
3556.38
3510.01
3468.31
26.2418
27.4496
27.2996
-0.109
-0.109
-0.110
-0.138
-0.115
-0.127
-0.007686
-0.013124
-0.011951
92.677552
178.023718
214.408816
2015-06-09
2015-06-10
2015-06-11
3456.79
3526.48
3551.91
26.8020
25.8610
26.3897
-0.110
-0.111
-0.111
-0.126
-0.117
-0.120
-0.003327
0.019960
0.007185
183.323049
300.555651
279.249096
2015-06-12
2015-06-15
3502.77
3438.07
29.7725
34.5593
-0.113
-0.108
-0.125
-0.119
-0.013931
-0.018644
302.564935
359.901600
2015-06-16
2015-06-17
2015-06-18
3454.09
3428.76
3450.45
36.2222
34.7235
34.7235
-0.109
-0.109
-0.109
-0.125
-0.110
-0.118
0.004649
-0.007360
0.006306
332.134168
315.833038
299.246550
2015-06-19
3455.80
34.7235
-0.108
-0.120
0.001549
278.303868
Date
Assume that the variance future comes to life on June 1, 2015 and that it matures on June 19,
2015. This is a maturity of T = 15 trading days. Let us generate in addition an array with all
(elapsed) trading days over time.
In [32]: T = 15.
In [33]: data['t'] = np. arange(1, 16)
In [34]: data['t']
Out[34]:
Date
2015-06-01
2015-06-02
2015-06-03
1
2
3
261
Variance Futures at Eurex
2015-06-04
4
2015-06-05
2015-06-08
2015-06-09
5
6
7
2015-06-10
2015-06-11
2015-06-12
8
9
10
2015-06-15
2015-06-16
2015-06-17
11
12
13
2015-06-18
2015-06-19
14
15
Name: t, dtype: int64
Assuming a constant Euribor rate of −0.1%, we can add a new column to the DataFrame
object with the discount factors in vectorized fashion (see chapter 11, Trading and Settlement
for further details on the discount factor calculation).
In [35]: r_eb = -0.001
In [36]: data['DF_t'] = np.exp(-r_eb * (T - data['t']) / 365.)
In [37]: data
Out[37]:
SX5E
V6I1
2w
EONIA
R_t
sigma**2
t
DF_t
2015-06-01 3575.04 25.8710 -0.108 -0.106
NaN
2015-06-02 3561.89 25.9232 -0.109 -0.122 -0.003685
NaN
34.220799
1
2
1.000038
1.000036
2015-06-03 3583.82 25.7958 -0.109 -0.143 0.006138 64.580457
2015-06-04 3556.38 26.2418 -0.109 -0.138 -0.007686 92.677552
2015-06-05 3510.01 27.4496 -0.109 -0.115 -0.013124 178.023718
3
4
5
1.000033
1.000030
1.000027
2015-06-08 3468.31 27.2996 -0.110 -0.127 -0.011951 214.408816
2015-06-09 3456.79 26.8020 -0.110 -0.126 -0.003327 183.323049
2015-06-10 3526.48 25.8610 -0.111 -0.117 0.019960 300.555651
6
7
8
1.000025
1.000022
1.000019
Date
2015-06-11 3551.91 26.3897 -0.111 -0.120
0.007185 279.249096
9
1.000016
2015-06-12 3502.77 29.7725 -0.113 -0.125 -0.013931 302.564935
2015-06-15 3438.07 34.5593 -0.108 -0.119 -0.018644 359.901600
10
11
1.000014
1.000011
2015-06-16 3454.09 36.2222 -0.109 -0.125 0.004649 332.134168
2015-06-17 3428.76 34.7235 -0.109 -0.110 -0.007360 315.833038
2015-06-18 3450.45 34.7235 -0.109 -0.118 0.006306 299.246550
12
13
14
1.000008
1.000005
1.000003
2015-06-19 3455.80 34.7235 -0.108 -0.120
15
1.000000
0.001549 278.303868
The standard volatility strike is 𝜎K = 25.871 and the standard variance strike is 𝜎K2 =
25.8712 = 669.31.
262
LISTED VOLATILITY AND VARIANCE DERIVATIVES
In [38]: sigma_K = data['V6I1'][0]
In [39]: sigma_K
Out[39]: 25.870999999999999
Assume a vega notional of 100,000. This translates into a variance notional of:
In [40]: Notional = 100000 / (2 * sigma_K)
In [41]: Notional
Out[41]: 1932.665919369178
The settlement price on the first trading day is standardized to 3,000. We generate a new column
in the pandas DataFrame object and initialize the first value (and all the others) accordingly.
In [42]: data['F_tS'] = 3000
In [43]: data
Out[43]:
SX5E
V6I1
2w
EONIA
R_t
sigma**2
t
Date
2015-06-01
2015-06-02
3575.04
3561.89
25.8710
25.9232
-0.108
-0.109
-0.106
-0.122
NaN
-0.003685
NaN
34.220799
1
2
2015-06-03
2015-06-04
2015-06-05
3583.82
3556.38
3510.01
25.7958
26.2418
27.4496
-0.109
-0.109
-0.109
-0.143
-0.138
-0.115
0.006138
-0.007686
-0.013124
64.580457
92.677552
178.023718
3
4
5
2015-06-08
2015-06-09
2015-06-10
3468.31
3456.79
3526.48
27.2996
26.8020
25.8610
-0.110
-0.110
-0.111
-0.127
-0.126
-0.117
-0.011951
-0.003327
0.019960
214.408816
183.323049
300.555651
6
7
8
2015-06-11
2015-06-12
2015-06-15
3551.91
3502.77
3438.07
26.3897
29.7725
34.5593
-0.111
-0.113
-0.108
-0.120
-0.125
-0.119
0.007185
-0.013931
-0.018644
279.249096
302.564935
359.901600
9
10
11
2015-06-16
2015-06-17
2015-06-18
3454.09
3428.76
3450.45
36.2222
34.7235
34.7235
-0.109
-0.109
-0.109
-0.125
-0.110
-0.118
0.004649
-0.007360
0.006306
332.134168
315.833038
299.246550
12
13
14
2015-06-19
3455.80
34.7235
-0.108
-0.120
0.001549
278.303868
15
DF_t
F_tS
Date
2015-06-01
1.000038
3000
2015-06-02
2015-06-03
2015-06-04
1.000036
1.000033
1.000030
3000
3000
3000
2015-06-05
1.000027
3000
\
263
Variance Futures at Eurex
2015-06-08
1.000025
3000
2015-06-09
2015-06-10
2015-06-11
1.000022
1.000019
1.000016
3000
3000
3000
2015-06-12
2015-06-15
2015-06-16
1.000014
1.000011
1.000008
3000
3000
3000
2015-06-17
2015-06-18
2015-06-19
1.000005
1.000003
1.000000
3000
3000
3000
The ARMVM on the first settlement day is zero. We again generate a new column and initialize
all values with zero.
In [44]: data['ARMVM_t'] = 0.0
The futures price on the second settlement day is given as:
In [45]: data['F_tS'][1] = data['DF_t'][1] * ((data['t'][1] * data['sigma**2'][1]
....:
+ (T - data['t'][1]) * data['V6I1'][1] ** 2) / T
....:
- sigma_K ** 2) + 3000
....:
In [46]: data['F_tS'][1]
Out[46]: 2917
Analogously, we can calculate the settlement prices for all other settlement days. Note that
we take the index level of the EURO STOXX 50 at close and not the average of its level
between 11:50 and 12:00 CET at the final settlement day (see chapter 11, Trading and
Settlement).
In [47]: for t in data.index[1:]:
....:
....:
....:
data['ARMVM_t'][t] = (data['ARMVM_t'].shift(1)[t]
* math.exp(data['EONIA'].shift(1)[t] / 252)
+ (data['F_tS'].shift(1)[t] - 3000)
....:
* (math.exp(data['EONIA'].shift(1)[t] / 252) - 1))
....:
data['F_tS'][t] = data['DF_t'][t] * ((data['t'][t] * data['sigma**2'][t]
....:
+ (T - data['t'][t]) * data['V6I1'][t] ** 2) / T
....:
-sigma_K ** 2)- data['ARMVM_t'][t] + 3000
....:
We end up with a complete data set, and in particular the simulated variance futures settlement
values.
264
LISTED VOLATILITY AND VARIANCE DERIVATIVES
In [48]: data
Out[48]:
SX5E
V6I1
2w
EONIA
Date
2015-06-01
3575.04
25.8710
-0.108
-0.106
NaN
NaN
1
2015-06-02
2015-06-03
2015-06-04
3561.89
3583.82
3556.38
25.9232
25.7958
26.2418
-0.109
-0.109
-0.109
-0.122
-0.143
-0.138
-0.003685
0.006138
-0.007686
34.220799
64.580457
92.677552
2
3
4
2015-06-05
2015-06-08
2015-06-09
3510.01
3468.31
3456.79
27.4496
27.2996
26.8020
-0.109
-0.110
-0.110
-0.115
-0.127
-0.126
-0.013124
-0.011951
-0.003327
178.023718
214.408816
183.323049
5
6
7
2015-06-10
2015-06-11
2015-06-12
3526.48
3551.91
3502.77
25.8610
26.3897
29.7725
-0.111
-0.111
-0.113
-0.117
-0.120
-0.125
0.019960
0.007185
-0.013931
300.555651
279.249096
302.564935
8
9
10
2015-06-15
2015-06-16
2015-06-17
3438.07
3454.09
3428.76
34.5593
36.2222
34.7235
-0.108
-0.109
-0.109
-0.119
-0.125
-0.110
-0.018644
0.004649
-0.007360
359.901600
332.134168
315.833038
11
12
13
2015-06-18
2015-06-19
3450.45
3455.80
34.7235
34.7235
-0.109
-0.108
-0.118
-0.120
0.006306
0.001549
299.246550
278.303868
14
15
DF_t F_tS
R_t
sigma**2
ARMVM_t
Date
2015-06-01
1.000038
3000
0.000000
2015-06-02
2015-06-03
1.000036
1.000033
2917
2875
0.000000
0.040173
2015-06-04
2015-06-05
2015-06-08
1.000030
1.000027
1.000025
2860
2892
2863
0.111062
0.187647
0.236836
2015-06-09
2015-06-10
2015-06-11
1.000022
1.000019
1.000016
2799
2802
2776
0.305743
0.406065
0.497784
2015-06-12
2015-06-15
2015-06-16
1.000014
1.000011
1.000008
2827
2912
2858
0.604188
0.689681
0.730901
2015-06-17
2015-06-18
1.000005
1.000003
2764
2689
0.800957
0.903601
2015-06-19
1.000000
2607
1.048771
Figure 10.3 shows the results graphically.
In [49]: data[['SX5E', 'sigma**2', 'F_tS']]. plot(subplots=True,
....:
color='blue',
....:
....:
figsize=(10, 9));
t
\
Variance Futures at Eurex
FIGURE 10.3
265
Calculated variance futures settlement values for the EURO STOXX 50.
We save the generated data set for re-use in the next section.
In [50]: h5 = pd.HDFStore('./source/data/var_data.h5', 'a')
In [51]: h5['var_future'] = data
In [52]: h5.close()
10.4 COMPARISON OF VARIANCE SWAP AND FUTURE
Eurex variance futures are a means to replicate the payoff of OTC-traded variance swaps with
a listed, standardized product. Let us compare the mark-to-market values of the variance swap
from sub-section 9.3.5, Variance Swap on the EURO STOXX 50 with the settlement values
calculated in the previous section.
266
LISTED VOLATILITY AND VARIANCE DERIVATIVES
To this end, we import the data for the variance swap first.
In [53]: h5 = pd.HDFStore('./source/data/var_data.h5', 'r')
In [54]: var_swap = h5['var_swap']
In [55]: h5. close()
Next, we combine and plot the two time series for the variance swap and the variance future,
respectively, against each other. Note that we subtract the constant C for the comparison.
In [56]: comp = pd.DataFrame({'F_tS': data['F_tS'] - 3000,
....:
'V_t': var_swap['V_t']}, index=data.index)
....:
In [57]: comp
Out[57]:
F_tS
V_t
0
-83
0.000000
-82.332277
Date
2015-06-01
2015-06-02
2015-06-03
-125 -124.049833
2015-06-04
2015-06-05
-140 -139.593571
-108 -107.644092
2015-06-08
2015-06-09
2015-06-10
-137 -136.380856
-201 -200.634978
-198 -196.905901
2015-06-11
2015-06-12
2015-06-15
-224 -223.189008
-173 -172.129074
-88 -86.887791
2015-06-16
2015-06-17
2015-06-18
-142 -141.190592
-236 -234.822528
-311 -309.629583
2015-06-19
-393 -391.004773
Figure 10.4 compares the two time series.
In [58]: comp.plot(style=['b', 'ro'], figsize=(9, 5));
Finally, Figure 10.5 presents the absolute differences which are quite small.
In [59]: (comp['F_tS'] - comp['V_t']).plot(style='rˆ', figsize=(9, 5));
Variance Futures at Eurex
FIGURE 10.4
Variance swap and futures prices for EURO STOXX 50 compared.
FIGURE 10.5
Differences between variance swap and futures prices.
267
268
LISTED VOLATILITY AND VARIANCE DERIVATIVES
10.5 CONCLUSIONS
This chapter deals with the basic notions and concepts that need to be understood to trade
Eurex variance futures based on the EURO STOXX 50 stock index. It introduces all concepts
in a formal fashion and illustrates their calculation based on a concrete numerical example
which is based on the same data as the variance swap example calculation in the previous
chapter. This makes it possible to numerically show that the construction of the Eurex variance futures contract does indeed replicate the payoff of typical OTC variance contracts while
adding intraday liquidity and tradability. The next chapter covers special aspects that relate to
the intraday trading and end-of-day settlement of these derivatives products.
Listed Volatility and Variance Derivatives: A
Python-based Guide
By Dr. Yves J. Hilpisch
© 2017 Yves Hilpisch
CHAPTER
11
Trading and Settlement
11.1 INTRODUCTION
This chapter covers practical aspects of trading Eurex variance futures. Among others, topics
are:
overview of major variance futures contract terms
intraday trading and trade conventions
trade matching at the end of the trading day
different traded volatilities with their impact on margining
after the trade matching until maturity
further details about interest rate calculation and market disruption events.
11.2 OVERVIEW OF VARIANCE FUTURES TERMS
The following list provides an overview of the major terms of the Eurex variance futures contract. All technical terms are explained in chapter 10 Variance Futures at Eurex.
underlying: realized variance of EURO STOXX 50 stock index
interest rates: Euribor for present value factor DFt and Eonia for ARMVMt
standard strike: determined on first trading day and equal to settled implied volatility for
the relevant maturity
1st settlement price: equal to 3,000 (i.e. for setting standard strike = settled implied
variance)
contract value: 1 EUR per variance futures point
minimum price change/tick value: 0.0001 variance futures points and 0.0001 EUR
settlement: in cash
terms: next three month ends and the next three quarter end months thereafter as well as
June and December thereafter
last trading day: one day before final settlement day
final settlement day/expiration: third Friday of maturity month
269
270
LISTED VOLATILITY AND VARIANCE DERIVATIVES
final settlement price: based on the average index level of EURO STOXX 50 between
11:50 am and 12:00 CET on final settlement day
trade matching: variance futures trade in notional vega at volatility; upon matching
notional vega and volatility are converted into variance futures and variance futures prices,
respectively
continuous trading: from 9:00 am until 5:30 pm CET on each trading day of Eurex
Eurex trade entry services: from 6:30 pm until 9:00 pm CET on each trading day of
Eurex
order maintenance: notional vega at volatility, minimum order size 1 vega, minimum
price change 0.05 volatility points.
11.3 INTRADAY TRADING
Variance futures are traded on-exchange in terms of notional vega at volatility. However,
intraday there are no live futures prices quoted; they have to be calculated using live volatility quotes and the conversion parameters provided by Eurex. The conversion parameters are
those used for the calculation of the settlement price from the previous day, except that the
time-to-maturity (T − t) is reduced by 1 day and the number of elapsed days t is increased
by 1 day.
Let us get back to the numerical example from section 10.3 Example Calculation for a
Variance Future. The example simulates the calculation of daily settlement prices for a variance futures contract with inception on June 1, 2015 and a maturity on June 19, 2015 (i.e.
15 trading days). Since we have saved the data, we can continue from where we ended.
In [1]: import numpy as np
In [2]: a = np.random.standard_normal((10))
In [3]: np.round(a)
Out[3]: array([1., -0., -1., 1., 1., -0., 0., 0., 1., 1.])
In [4]: import pandas as pd
In [5]: h5 = pd.HDFStore('./source/data/var_data.h5', 'r')
In [6]: data = h5['var_future']
In [7]: h5.close()
In [8]: data.head()
271
Trading and Settlement
Out[8]:
SX5E
V6I1
2w
EONIA
R_t
sigma**2
t
DF_t
\
Date
2015-06-01 3575.04 25.8710 -0.108 -0.106
NaN
NaN
1
1.000038
2015-06-02 3561.89 25.9232 -0.109 -0.122 -0.003685
34.220799
2
1.000036
2015-06-03 3583.82 25.7958 -0.109 -0.143
0.006138
64.580457
3
1.000033
2015-06-04 3556.38 26.2418 -0.109 -0.138 -0.007686
92.677552
4
1.000030
2015-06-05 3510.01 27.4496 -0.109 -0.115 -0.013124 178.023718
5
1.000027
F_tS ARMVM_t
Date
2015-06-01
3000
0.000000
2015-06-02
2917
0.000000
2015-06-03
2875
0.040173
2015-06-04
2860
0.111062
2015-06-05
2892
0.187647
We need to read out two central terms of the variance, namely the time-to-maturity in year
fractions and the standard volatility strike.
In [9]: T = data['t'][-1]
In [10]: T
Out[10]: 15
In [11]: sigma_K = data['V6I1'][0]
In [12]: sigma_K
Out[12]: 25.870999999999999
Similarly to the calculation in section 10.3, Example Calculation for a Variance Future, we
can now do the following calculation where we use the previous day’s input parameters but
today’s remaining time-to-maturity and today’s elapsed time (in days).
In [13]: data['F_ti'] = 3000.0
In [14]: for t in data.index[2:]:
....:
data.loc[t, 'F_ti'] = data['DF_t'].shift(1)[t] * (
....:
(data['t'][t] * data['sigma**2'].shift(1)[t]
....:
+(T - data['t'][t]) * data['V6I1'].shift(1)[t]
** 2) / T
....:
- sigma_K ** 2) - data['ARMVM_t'].shift(1)[t] + 3000
....:
The results are shown in the following table in the column F_ti for interim futures prices.
272
LISTED VOLATILITY AND VARIANCE DERIVATIVES
In [15]: data
Out[15]:
SX5E
V6I1
2w
EONIA
Date
2015-06-01
R_t
3575.04
25.8710
-0.108
-0.106
NaN
NaN
1
2015-06-02
2015-06-03
2015-06-04
3561.89
3583.82
3556.38
25.9232
25.7958
26.2418
-0.109
-0.109
-0.109
-0.122
-0.143
-0.138
-0.003685
0.006138
-0.007686
34.220799
64.580457
92.677552
2
3
4
2015-06-05
2015-06-08
2015-06-09
3510.01
3468.31
3456.79
27.4496
27.2996
26.8020
-0.109
-0.110
-0.110
-0.115
-0.127
-0.126
-0.013124
-0.011951
-0.003327
178.023718
214.408816
183.323049
5
6
7
2015-06-10
2015-06-11
2015-06-12
3526.48
3551.91
3502.77
25.8610
26.3897
29.7725
-0.111
-0.111
-0.113
-0.117
-0.120
-0.125
0.019960
0.007185
-0.013931
300.555651
279.249096
302.564935
8
9
10
2015-06-15
2015-06-16
2015-06-17
3438.07
3454.09
3428.76
34.5593
36.2222
34.7235
-0.108
-0.109
-0.109
-0.119
-0.125
-0.110
-0.018644
0.004649
-0.007360
359.901600
332.134168
315.833038
11
12
13
2015-06-18
2015-06-19
3450.45
3455.80
34.7235
34.7235
-0.109
-0.108
-0.118
-0.120
0.006306
0.001549
299.246550
278.303868
14
15
DF_t
F_tS
ARMVM_t
F_ti
Date
2015-06-01
1.000038
3000
0.000000
3000.000000
2015-06-02
2015-06-03
1.000036
1.000033
2917
2875
0.000000
0.040173
3000.000000
2875.140910
2015-06-04
2015-06-05
2015-06-08
1.000030
1.000027
1.000025
2860
2892
2863
0.111062
0.187647
0.236836
2835.844331
2820.555454
2853.797523
2015-06-09
2015-06-10
2015-06-11
1.000022
1.000019
1.000016
2799
2802
2776
0.305743
0.406065
0.497784
2827.984087
2763.381425
2778.130966
2015-06-12
2015-06-15
2015-06-16
1.000014
1.000011
1.000008
2827
2912
2858
0.604188
0.689681
0.730901
2748.494268
2788.339034
2856.790440
2015-06-17
2015-06-18
1.000005
1.000003
2764
2689
0.800957
0.903601
2792.748076
2705.047722
2015-06-19
1.000000
2607
1.048771
2629.033294
sigma**2
t
\
As an example, assume that a trader buys on June 9, intraday 100,000 notional vega at a traded
implied volatility level of 26. The number of futures she has bought then is
VegaNotional
T
⋅
2 ⋅ 𝜎ti
T −t
100000
15
=
⋅
2 ⋅ 26 15 − 7
= 3605.77
futures =
273
Trading and Settlement
In [16]: futures = 100000. /(2 * 26) * T / (T - data['t']['2015-06-09'])
In [17]: futures
Out[17]: 3605.769230769231
The intraday futures price for the variance futures contract is:
In [18]: F_ti = data['F_ti']['2015-06-09']
In [19]: F_ti
Out[19]: 2827.9840866529353
The traded variance strike is
𝜎t2 =
2 + (T − t) ⋅ 𝜎 2 (t, T)
t ⋅ 𝜎0,t
i
T
7 ⋅ 13.542 + (15 − 7) ⋅ 262
=
15
= 446.08
In [20]: sigma_t2 = (data['t']['2015-06-09'] * data['sigma**2']['2015-06-09']
+ (T - data['t']['2015-06-09'] ) * 26 ** 2) / T
....:
....:
In [21]: sigma_t2
Out[21]: 446.0840897506356
The traded futures price then is
(
)
Ft = DFt ⋅ 𝜎t2 − 𝜎i2 (t, T) − ARMV Mt + 3000 = 2776.47
In [22]: F_t = (data['DF_t']['2015-06-09'] *
....:
(sigma_t2 - data['V6I1']['2015-06-01'] ** 2)
....:
....:
- data['ARMVM_t']['2015-06-09'] + 3000)
In [23]: F_t
Out[23]: 2776.4648130015858
274
LISTED VOLATILITY AND VARIANCE DERIVATIVES
Consequently, the trade has a value of:
In [24]: P_ti = futures * F_t
In [25]: P_ti
Out[25]: 10011291.393034564
11.4 TRADE MATCHING
At the end of the day, the trade in notional vega and volatility is confirmed. Notional vega and
volatility are, however, converted into variance futures at the variance futures settlement price.
To this end, the intraday transaction gets cancelled and replaced by a confirmed transaction
with the same number of futures but with the variance futures settlement price of the day.
In the example, we have:
In [26]: F_tS = data['F_tS']['2015-06-09']
In [27]: F_tS
Out[27]: 2799
Consequently, the transaction booked has a value of:
In [28]: P_t = futures * F_tS
In [29]: P_t
Out[29]: 10092548.076923078
The first margin payment given these numbers then is:
In [30]: P_t - P_ti
Out[30]: 81256.683888513595
This is roughly the same as vega notional times the difference in settlement and traded
volatility:
margin = VegaNotional ⋅ (𝜎i − 𝜎i (t, T))
In [31]: 100000 * (data['V6I1']['2015-06-09'] - 26)
Out[31]: 80199.999999999956
275
Trading and Settlement
11.5 DIFFERENT TRADED VOLATILITIES
Let us now see what impact different traded volatilities have on the first day margining of the
same trade as before. The necessary parameters and values can be calculated in vectorized
fashion. First, the array with the traded volatilities.
In [32]: import numpy as np
In [33]: trad_vols = np.arange(15, 25.01, 1)
In [34]: trad_vols
Out[34]: array([15., 16., 17., 18., 19., 20., 21., 22., 23., 24., 25.])
Second, the traded variance strikes.
In [35]: sigma_t2 = (data['t']['2015-06-09'] * data['sigma**2']['2015-06-09']
+ (T - data['t']['2015-06-09']) * trad_vols ** 2) / T
....:
....:
In [36]: sigma_t2
Out[36]:
array([205.55075642,
222.08408975,
239.68408975,
258.35075642,
278.08408975,
298.88408975,
320.75075642,
343.68408975,
367.68408975,
392.75075642,
418.88408975])
Third, the traded futures prices.
In [37]: F_t = (data['DF_t']['2015-06-09']
*(sigma_t2 - data['V6I1']['2015-06-01'] ** 2)
- data['ARMVM_t']['2015-06-09'] + 3000)
....:
....:
....:
In [38]: F_t
Out[38]:
array([ 2535.92620765,
2608.46113077,
2552.45990336,
2629.26158666,
2570.06028912,
2651.12873261,
2698.06309463,
2723.13031071,
2749.26421683])
Fourth, the first day margins (P&L).
2588.72736492,
2674.06256859,
276
LISTED VOLATILITY AND VARIANCE DERIVATIVES
In [39]: margins = futures * (F_tS - F_t)
In [40]: margins
Out[40]:
array([ 948583.3858882 ,
758194.59764436,
533189.66608347,
273568.59120552,
888966.69461993,
687039.19193707,
450495.54593716,
825503.76520532,
612037.54808344,
363955.18764451,
179335.75662019])
Let us visualize the results. The margins (P&L) are linear in traded volatility with the slope in
Figure 11.1 being equal to the vega notional of 100,000.
In [41]: import seaborn as sns; sns.set()
In [42]: import matplotlib
In [43]: matplotlib.rcParams['font.family'] = 'serif'
In [44]: results = pd.DataFrame({'margins' : margins}, index=trad_vols)
In [45]: results.plot(figsize=(10, 6));
FIGURE 11.1
Variance futures margins.
Trading and Settlement
277
11.6 AFTER THE TRADE MATCHING
What happens after the trade matching until maturity? To find out, we can calculate the subsequent margins as follows. We start with the differences between settlement prices.
In [46]: F_diffs = (data['F_tS'] - data['F_tS'].shift(1))
In [47]: F_diffs
Out[47]:
Date
2015-06-01
NaN
2015-06-02
-83.0
2015-06-03
2015-06-04
-42.0
-15.0
2015-06-05
2015-06-08
32.0
-29.0
2015-06-09
-64.0
2015-06-10
2015-06-11
2015-06-12
3.0
-26.0
51.0
2015-06-15
2015-06-16
2015-06-17
85.0
-54.0
-94.0
2015-06-18
2015-06-19
-75.0
-82.0
Name: F_tS, dtype: float64
With these differences, we can calculate the daily margins until maturity.
In [48]: margin_t = futures * F_diffs[F_diffs.index >= '2015-06-09']
In [49]: margin_t
Out[49]:
Date
2015-06-09
-230769.230769
2015-06-10
10817.307692
2015-06-11
2015-06-12
2015-06-15
-93750.000000
183894.230769
306490.384615
2015-06-16
2015-06-17
2015-06-18
-194711.538462
-338942.307692
-270432.692308
2015-06-19 -295673.076923
Name: F_tS, dtype: float64
278
LISTED VOLATILITY AND VARIANCE DERIVATIVES
The following code generates a pandas DataFrame object with the initial margins given the
traded volatility strike and the subsequent margins until maturity.
In [50]: results = pd.DataFrame(np.tile(margin_t, (len(trad_vols), 1)).T,
index=margin_t.index,
columns=trad_vols)
....:
....:
....:
In [51]: results.loc['2015-06-09', :] = margins
# setting the first
day margins
In [52]: np.round(results)
Out[52]:
15.0
16.0
17.0
18.0
19.0
20.0
948583.0
10817.0
-93750.0
888967.0
10817.0
-93750.0
825504.0
10817.0
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758195.0
10817.0
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687039.0
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612038.0
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\
Date
2015-06-09
2015-06-10
2015-06-11
2015-06-12 183894.0 183894.0 183894.0 183894.0 183894.0 183894.0
2015-06-15 306490.0 306490.0 306490.0 306490.0 306490.0 306490.0
2015-06-16 -194712.0 -194712.0 -194712.0 -194712.0 -194712.0 -194712.0
2015-06-17 -338942.0 -338942.0 -338942.0 -338942.0 -338942.0 -338942.0
2015-06-18 -270433.0 -270433.0 -270433.0 -270433.0 -270433.0 -270433.0
2015-06-19 -295673.0 -295673.0 -295673.0 -295673.0 -295673.0 -295673.0
21.0
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2015-06-09
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533190.0
10817.0
450496.0
10817.0
363955.0
10817.0
273569.0
10817.0
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Date
2015-06-16 -194712.0 -194712.0 -194712.0 -194712.0 -194712.0
2015-06-17 -338942.0 -338942.0 -338942.0 -338942.0 -338942.0
2015-06-18 -270433.0 -270433.0 -270433.0 -270433.0 -270433.0
2015-06-19 -295673.0 -295673.0 -295673.0 -295673.0 -295673.0
Figure 11.2 visualizes the cumulative P&L (assuming zero interest rates) for the different
traded volatility strikes.
In [53]: results.cumsum().plot(figsize=(10, 6), title='cumulative P&L');
279
Trading and Settlement
FIGURE 11.2
Cumulative P&L for variance future.
11.7 FURTHER DETAILS
This section briefly discusses some further details of importance for trading in Eurex variance
futures.
11.7.1 Interest Rate Calculation
ARMVM is calculated based on the Eonia rate as settled at 7 pm CET on the previous day. The
discount factors are calculated using the Euribor rates fixed at 11:00 am CET and interpolated
to the respective maturities of the variance futures. The interpolation is done in linear fashion
using the following formula:
ri = r(Ti ) =
TK+1 − Ti
Ti − TK
r(TK ) +
r(TK+1 )
TK+1 − TK
TK+1 − TK
where TK+1 is the maturity of the Euribor rate later than the futures maturity Ti and TK is the
maturity of the Euribor rate before the futures maturity Ti .
280
LISTED VOLATILITY AND VARIANCE DERIVATIVES
11.7.2 Market Disruption Events
There are three major market disruption events:
STOXX fails to provide a market closing level for the EURO STOXX 50
Eurex Exchange fails to open for trading during scheduled trading hours
Other market disruption events according to the European OTC standard.
11.8 CONCLUSIONS
The major difference between listed variance futures and OTC variance swaps is that the former
can be traded intraday. This chapter illustrates the major aspects of importance when it comes
to the trading and settlement of these instruments which are even liquid intraday (during the
trading day). Major concepts are the traded variance strike and the traded futures price. At
the end of the day, the original intraday trade gets cancelled and the traded futures price gets
replaced by the settlement price – however, the number of futures remains the same.
PART
Four
DX Analytics
Listed Volatility and Variance Derivatives: A
Python-based Guide
By Dr. Yves J. Hilpisch
© 2017 Yves Hilpisch
CHAPTER
12
DX Analytics – An Overview
12.1 INTRODUCTION
Although Python is arguably a good programming language and ecosystem for financial analytics (see chapter 1, Derivatives, Volatility and Variance or chapter 1 of Hilpisch (2014)),
dedicated libraries for finance are not that common. This is even more true when it comes to
derivatives analytics as a sub-discipline. One exception in this regard is DX Analytics (the “dx
library”), written by the author of this book, which has a major focus on advanced derivatives and risk analytics. The central resource to get started with the library is the website
http://dx-analytics.com.
This chapter provides an overview of the relevant parts of the library for the purposes of the
case studies which follow. The development of the library is guided by two central principles:
global valuation approach: in practice, this approach translates into the non-redundant
modeling of all risk factors (e.g. option underlyings like equity indexes) and the valuation
of all derivative instruments by a unique, consistent numerical method – which is Monte
Carlo simulation in the case of DX Analytics
unlimited computing resources: Monte Carlo simulation is computationally and memory intensive and has therefore often been dismissed as an adequate numerical method to
implement, for example, front-office analytics libraries; in 2016, the technical infrastructures available to even smaller players in the financial industry have reached performance
levels that 10 years ago seemed impossible or at least not financially feasible; in that sense
“unlimited resources” is not to be understood literally but rather as the guiding principle
that hardware and computing resources generally are no longer a bottleneck
Among others, DX Analytics provides the following features:
models: models for risk factors include simple ones like geometric Brownian motion as
well as more sophisticated ones like stochastic volatility jump diffusion models
derivatives: derivatives models include single risk factor as well as multi risk factor models, both with European and American (Bermudan) exercise
283
284
LISTED VOLATILITY AND VARIANCE DERIVATIVES
portfolios: derivatives portfolios can be arbitrarily complex with multiple, correlated
risk factors and multiple, diverse derivative instruments (single risk and multi risk);
simulations and valuations are implemented in such a way that both value and risk aggregations are consistent for each Monte Carlo path.
The following sections illustrate the use of the library by means of a simple, yet still realistic,
example with two correlated risk factors and two different options.
12.2 MODELING RISK FACTORS
First, some necessary imports and in particular the import of the dx library.
In [1]: import dx
In [2]: import numpy as np
In [3]: np.random.seed(1000)
In [4]: import pandas as pd
In [5]: import datetime as dt
In [6]: import seaborn as sns; sns.set()
In [7]: import matplotlib as mpl;
Usually, the first step is to define a model for the risk-neutral discounting since all valuations
are based on the risk-neutral (or martingale) pricing approach (see Björk (2009)). Throughout,
we will work with a constant short rate model although DX Analytics also provides deterministic yield curve and stochastic short rate models.
In [8]: r = dx.constant_short_rate('r', 0.01)
The next step is to define a market environment containing the parameter specifications needed.
Several objects might have different market environments but they might also share market
environments. The first risk factor to be modeled is a geometric Brownian motion (BlackScholes-Merton (1973) model). The following market environment object contains all parameters needed for this model. Comments in the code explain the single elements.
# instantiation of market environment object
In [9]: me_1 = dx.market_environment('me', dt.datetime(2016, 1, 1))
285
DX Analytics – An Overview
# starting value of simulated processes
In [10]: me_1.add_constant('initial_value', 100.)
# volatility factor
In [11]: me_1.add_constant('volatility', 0.2)
# horizon for simulation
In [12]: me_1.add_constant('final_date', dt.datetime(2016, 6, 30))
# currency of instrument
In [13]: me_1.add_constant('currency', 'EUR')
# frequency for discretization (here: weekly)
In [14]: me_1.add_constant('frequency', 'W')
# number of paths
In [15]: me_1.add_constant('paths', 25000)
# short rate model for discount curve
In [16]: me_1.add_curve('discount_curve', r)
Equipped with this object, the model object for the risk factor can be instantiated.
In [17]: gbm_1 = dx.geometric_brownian_motion('gbm_1', me_1)
In [18]: gbm_1
Out[18]:
The get_instrument_values() method initiates a Monte Carlo simulation and delivers
back the simulated paths – given the parametrizations from the market environment object –
as a NumPy ndarray object.
In [19]: gbm_1.get_instrument_values()
Out [19]:
array([[ 100.
,
100.
,
[ 103.99970675,
100.
, 100.
, ..., 100.
,
100.
],
100.94138063, 101.87289049, ..., 101.57080458,
100.82390334,
[ 101.93603717,
99.98530091,
98.15531921],
94.66240389,
102.49930045],
...,
[ 94.34847745,
111.87828236,
109.03063645,
107.8625328 ],
91.81649108, ..., 101.20745936,
105.88943893, ...,
89.84492969,
286
LISTED VOLATILITY AND VARIANCE DERIVATIVES
[
91.01721535,
104.17599152,
113.83465258, 102.59432299, ...,
101.67756176],
92.09515398,
[
90.56511589,
100.6385713 ,
113.20730366, 100.79160449, ...,
101.76583508]])
90.91965416,
Via the time_grid attribute one can access the date-time information for the time series
data.
In [20]: gbm_1.time_grid[:10]
Out[20]:
array([datetime.datetime(2016, 1, 1, 0, 0),
datetime.datetime(2016, 1, 3, 0, 0),
datetime.datetime(2016, 1, 10, 0, 0),
datetime.datetime(2016, 1, 17, 0, 0),
datetime.datetime(2016, 1, 24, 0, 0),
datetime.datetime(2016, 1, 31, 0, 0),
datetime.datetime(2016, 2, 7, 0, 0),
datetime.datetime(2016, 2, 14, 0, 0),
datetime.datetime(2016, 2, 21, 0, 0),
datetime.datetime(2016, 2, 28, 0, 0)], dtype=object)
Combining both arrays to a single pandas DataFrame object makes plotting straightforward
(see Figure 12.1).
FIGURE 12.1
Simulated paths for the risk factor based on geometric Brownian motion.
DX Analytics – An Overview
287
In [21]: pdf_1 = pd.DataFrame(gbm_1.get_instrument_values(),
index=gbm_1.time_grid)
In [22]: pdf_1.ix[:, :10].plot(legend=False, figsize=(10, 6));
Next, we define a second risk factor, again based on geometric Brownian motion. We use the
market environment information from the first risk factor and only overwrite the volatility
value.
# instantiate new market environment object
In [23]: me_2 = dx.market_environment('me_2', me_1.pricing_date)
# add complete environment
In [24]: me_2.add_environment(me_1)
# overwrite volatility value
In [25]: me_2.add_constant('volatility', 0.5)
Using the updated market environment, define the second risk factor as follows.
In [26]: gbm_2 = dx.geometric_brownian_motion('gbm_2', me_2)
The plot in Figure 12.2 illustrates the higher volatility of the second risk factor graphically.
In [27]: pdf_2 = pd.DataFrame(gbm_2.get_instrument_values(),
index=gbm_2.time_grid)
In [28]: ax = pdf_1.ix[:, :10].plot(legend=False, figsize=(10, 6),
style=11 * ['b']);
In [29]: pdf_2.ix [:, :10].plot(legend=False, style=11 * ['r'], ax=ax);
12.3 MODELING DERIVATIVES
Based on the risk factors, we can define derivatives models for valuation. To this end, we need
to add at least one (the maturity), in general two (maturity and strike), parameters to
the market environment(s).
288
LISTED VOLATILITY AND VARIANCE DERIVATIVES
FIGURE 12.2
Simulated paths for the two risk factors; solid lines = low volatility, dashed lines =
high volatility.
# instantiation of market environment object for option
In [30]: me_opt = dx.market_environment('me_opt', me_1.pricing_date)
# add complete market environment
In [31]: me_opt.add_environment(me_1)
# add maturity date for option
In [32]: me_opt.add_constant('maturity', dt.datetime(2016, 6, 30))
# add strike for option
In [33]: me_opt.add_constant('strike', 110.)
The first derivative is an American put option on the first risk factor gbm_1.
In [34]: am_put = dx.valuation_mcs_american_single(
....:
name='am_put',
....:
underlying=gbm_1,
# name of the option as string
....:
mar_env=me_opt,
....:
payoff_func='np.maximum(strike - instrument_values, 0)')
# the risk factor object
# the market environment
....:
Let us calculate a Monte Carlo present value estimate and estimates for the Greeks of the
American put.
289
DX Analytics – An Overview
In [35]: am_put.present_value()
# Monte Carlo estimator
Out[35]: 11.799
In [36]: am_put.delta()
# delta of the option
Out[36]: -0.6809
In [37]: am_put.gamma()
# gamma of the option
Out[37]: 0.0149
In [38]: 0.5 * am_put.gamma() * am_put.underlying.initial_value ** 2
# dollar gamma
Out[38]: 74.5
In [39]: am_put.vega()
Out[39]: 23.8208
# vega of the option
In [40]: am_put.theta()
Out[40]: -3.81
In [41]: am_put.rho()
Out[41]: -30.113
# theta of the option
# rho of the option
The second derivative is a European call option on the second risk factor gbm_2. It has the
same strike and maturity as the American put option.
In [42]: eur_call = dx.valuation_mcs_european_single(
....:
name='eur_call',
....:
underlying=gbm_2,
....:
....:
....:
mar_env=me_opt,
payoff_func='np.maximum(maturity_value - strike, 0)')
The major statistics for this option are:
In [43]: eur_call.present_value()
Out[43]: 10.364663
In [44]: eur_call.delta()
Out[44]: 0.4174
In [45]: eur_call.gamma()
Out[45]: 0.0121
290
LISTED VOLATILITY AND VARIANCE DERIVATIVES
In [46]: 0.5 * eur_call.gamma() * eur_call.underlying.initial_value ** 2
Out[46]: 60.5
In [47]: eur_call.vega()
Out[47]: 27.6147
In [48]: eur_call.theta()
Out[48]: -14.1996
In [49]: eur_call.rho()
Out[49]: 18.0684
Note that all these values might vary to a greater or lesser extent with the parameters chosen
for the Monte Carlo simulation.
To conclude this section, let us analyze the European call option in a bit more detail. We
want to estimate and collect the Greeks for different strikes. The following code implements
the necessary steps:
In [50]: k_list = np.arange(80., 120.5, 2.5)
In [51]: pv = []; de = []; ve = []; th = []; rh = []; ga = []
In [52]: for k in k_list:
....:
eur_call.update(strike=k)
....:
....:
....:
pv.append(eur_call.present_value())
de.append(eur_call.delta())
ve.append(eur_call.vega())
....:
....:
....:
th.append(eur_call.theta())
rh.append(eur_call.rho())
ga.append(eur_call.gamma())
....:
Figure 12.3 shows the results graphically.
In [53]: dx.plot_option_stats_full(k_list, pv, de, ga, ve, th, rh)
12.4 DERIVATIVES PORTFOLIOS
The previous sections show how convenient and flexible it is to model single derivatives with
DX Analytics. The numerical methods used and the API of the library mimic working with the
Black-Scholes-Merton closed option pricing formula when it comes to Greeks although the
derivative itself might be much more complex than a plain vanilla European call or put option
DX Analytics – An Overview
FIGURE 12.3
Greeks of the European call option for different strikes.
291
292
LISTED VOLATILITY AND VARIANCE DERIVATIVES
(e.g. it might have American exercise and an exotic payoff). However, the area which differentiates DX Analytics most from other derivatives analytics libraries is the global valuation
approach for derivatives portfolios. How it works is explained in this section.
12.4.1 Modeling Portfolios
In a portfolio context, we need to add information about the model class(es) to be used by the
market environments of the risk factors.
In [54]: me_1.add_constant('model', 'gbm')
In [55]: me_2.add_constant('model', 'gbm')
To compose a portfolio consisting of the two options, we need to define derivatives positions.
Note that this step is independent from the risk factor model and option model definitions. We
only use the market environment data and some additional information needed (e.g. payoff
functions).
In [56]: put = dx.derivatives_position(
....:
....:
....:
name='put', # name as string
quantity=2, # number of options in the portfolio
underlyings=['gbm_1'], # the underlying(s) as list object
....:
....:
....:
mar_env=me_opt, # the market environment object
otype='American single', # the option type
payoff_func='np.maximum(strike - instrument_values, 0)')
# the payoff
....:
In [57]: call = dx.derivatives_position(
....:
name='call',
....:
....:
....:
quantity=3,
underlyings=['gbm_2'],
mar_env=me_opt,
....:
....:
....:
otype='European single',
payoff_func='np.maximum(maturity_value - strike, 0)')
In a portfolio context, we also need to define the market. It consists of the risk factors, the
correlation between them, the derivatives positions as well as the valuation environment.
In [58]: risk_factors = {'gbm_1': me_1, 'gbm_2': me_2}
# as dictionary
In [59]: correlations = [['gbm_1', 'gbm_2', 0.4]]
# as list
In [60]: positions = {'put' : put, 'call' : call}
# as dictionary
293
DX Analytics – An Overview
The valuation environment (technically another market environment) contains all those parameters shared by all derivatives positions. This might imply that certain parameters from
the market environments of the derivatives get replaced for the portfolio simulations and
valuations.
In [61]: val_env = dx.market_environment('general', dt.datetime(2016, 1, 1))
In [62]: val_env.add_constant('frequency', 'W')
In [63]: val_env.add_constant('paths', 25000)
In [64]: val_env.add_constant('starting_date', val_env.pricing_date)
In [65]: val_env.add_constant('final_date', val_env.pricing_date)
In [66]: val_env.add_curve('discount_curve', r)
These new objects are needed to instantiate a portfolio object.
In [67]: port = dx.derivatives_portfolio(
....:
name='portfolio',
....:
positions=positions,
# name as string
....:
val_env=val_env,
# derivatives positions
# valuation environment
....:
risk_factors=risk_factors, # relevant risk factors
....:
correlations=correlations,
....:
parallel=False)
# correlation between risk factors
# parallel valuation True/False
....:
12.4.2 Simulation and Valuation
Simulation and valuation are now as straightforward as in the single option case.
In [68]: port.get_values()
Total
pos_value
# get all present values
54.103966
dtype: float64
Out[68]:
0
1
position
put
call
0
pos_value
23.608000
1
30.495966
name quantity
otype risk_facts
value currency
put
2 American single
[gbm_1] 11.804000
EUR
call
3 European single
[gbm_2] 10.165322
EUR
\
294
LISTED VOLATILITY AND VARIANCE DERIVATIVES
In [69]: port.get_statistics()
Totals
pos_value
pos_delta
pos_vega
# get major statistics
54.1030
0.0082
130.9992
dtype: float64
Out[69]:
0
1
position
put
call
name quantity
otype risk_facts
put
2 American single
[gbm_1]
call
3 European single
[gbm_2]
0
pos_value
23.608
pos_delta
-1.4192
pos_vega
48.7344
1
30.495
1.4274
82.2648
value currency
11.804
EUR
10.165
EUR
\
12.4.3 Risk Reports
A strength of DX Analytics and the global valuation approach is that one can easily generate
consistent risk reports. By this we mean that single parameters are shocked and the effect on
the portfolio value is estimated. Think of a larger portfolio containing multiple options on the
S&P 500 equity index. By changing the spot value of the index, DX Analytics can estimate in
a single step what the impact is on the overall portfolio value (and not only on a single option
as in the case of a delta calculation).
When calling the get_port_risk() method you need to define which parameter will
be shocked. You get back all hypothetical portfolio values and the benchmark value without
shock. The following estimates show the “portfolio deltas.”
In [70]: deltas, benchvalue = port.get_port_risk(Greek='Delta')
gbm_1
0.8 0.9 1.0 1.1 1.2
gbm_2
0.8 0.9 1.0 1.1 1.2
In [71]: benchvalue
Out[71]: 54.103966
In [72]: deltas
Out[72]:
Dimensions: 2 (items) x 5 (major_axis) x 2 (minor_axis)
Items axis: gbm_1_Delta to gbm_2_Delta
Major_axis axis: 0.8 to 1.2
Minor_axis axis: factor to value
295
DX Analytics – An Overview
There is a convenience function called risk_report() in DX Analytics to nicely print the
results.
In [73]: dx.risk_report (deltas)
# gives the resulting values ...
gbm_1_Delta
factor
value
0.8
0.9
1.0
1.1
1.2
80.00 90.00 100.0 110.00 120.00
90.51 70.89 54.1 42.19 35.37
gbm_2_Delta
factor
value
0.8
0.9
1.0
1.1
1.2
80.00 90.00 100.0 110.00 120.00
33.28 41.94 54.1 69.74 88.39
In [74]: dx.risk_report(deltas.ix[:, :, 'value'] - benchvalue,
....:
gross=False) # ... as net changes
....:
gbm_1_Delta
0.8
36.41
gbm_2_Delta
-20.83
0.9
1.0
1.1
16.79
0.00
-11.92
-12.16
0.00
15.63
1.2
-18.74
34.28
“Portfolio vegas” are calculated in the same way.
In [75]: vegas, benchvalue = port.get_port_risk(Greek='Vega', step=0.05)
gbm_1
0.8 0.85 0.9 0.95 1.0 1.05 1.1 1.15 1.2
gbm_2
0.8 0.85 0.9 0.95 1.0 1.05 1.1 1.15 1.2
In [76]: dx.risk_report(vegas)
gbm_1_Vega
factor
value
0.80
0.85
0.90
0.95 1.00
1.05
1.10
1.15
1.20
0.16
0.17
0.18
0.19
0.21
0.22
0.23
0.24
0.2
52.40 52.85 53.20 53.64 54.1 54.59 55.05 55.52 56.01
gbm_2_Vega
factor
value
0.80
0.85
0.90
0.95 1.00
1.05
1.10
1.15
1.20
0.40
0.43
0.45
0.48
0.53
0.55
0.58
0.60
0.5
45.92 47.95 50.00 52.05 54.1 56.16 58.22 60.28 62.34
296
LISTED VOLATILITY AND VARIANCE DERIVATIVES
In [77]: dx.risk_report(vegas.ix[:, :, 'value'] - benchvalue, gross=False)
gbm_1_Vega
gbm_2_Vega
0.80
-1.70
-8.19
0.85
-1.26
-6.15
0.90
-0.91
-4.11
0.95
-0.47
-2.05
1.00
0.00
0.00
1.05
0.49
2.06
1.10
0.95
4.12
1.15
1.41
6.18
1.20
1.90
8.24
12.5 CONCLUSIONS
This chapter provides a quick start with DX Analytics, a Python-based financial analytics
library with a focus on derivatives and risk analytics. The library offers many more features
than are covered in this brief chapter. It is recommended to check out the main page http://dxanalytics.com and to work through the different parts of the documentation which are all based
on executable Jupyter Notebooks.
The focus of this chapter is on the basic tool set and a basic understanding of the API to
have a good foundation for the case studies in the two subsquent chapters. The case studies
use DX Analytics to model the VSTOXX volatility index by the square-root diffusion model
from chapter 6, Valuing Volatility Derivatives as well as the square-root jump diffusion model
from chapter 7, Advanced Modeling of the VSTOXX Index. The major goal of the case studies
is to analyze how well the two models perform over time in replicating the market quotes of
traded VSTOXX options.
Listed Volatility and Variance Derivatives: A
Python-based Guide
By Dr. Yves J. Hilpisch
© 2017 Yves Hilpisch
CHAPTER
13
DX Analytics – Square-Root Diffusion
13.1 INTRODUCTION
This chapter uses DX Analytics to model the VSTOXX volatility index by a square-root diffusion process as proposed in Grünbichler and Longstaff (1996) and discussed in chapter 6
Valuing Volatility Derivatives. It implements a study over a time period of three months to
analyze how well the model performs in replicating market quotes for VSTOXX options.
13.2 DATA IMPORT AND SELECTION
The data we are working with is for the first quarter of 2014. The complete data set is contained
in the online resources accompanying this book. As usual, some imports first.
In [1]: import numpy as np
In [2]: import pandas as pd
In [3]: import datetime as dt
Next, we read the data from the source into pandas DataFrame objects.
In [4]: h5 = pd.HDFStore('./source/data/vstoxx_march_2014.h5', 'r')
In [5]: vstoxx_index = h5['vstoxx_index']
# data for the index itself
In [6]: vstoxx_futures = h5['vstoxx_futures']
# data for the futures
In [7]: vstoxx_options = h5['vstoxx_options']
# data for the options
In [8]: h5.close()
297
298
LISTED VOLATILITY AND VARIANCE DERIVATIVES
Inspecting the data sub-set for the VSTOXX index itself, we see that we are dealing with 63
trading days.
In [9]: vstoxx_index.info()
DatetimeIndex: 63 entries, 2014-01-02 to 2014-03-31
Data columns (total 9 columns):
V2TX
63 non-null float64
V6I1
V6I2
V6I3
57 non-null float64
63 non-null float64
61 non-null float64
V6I4
V6I5
V6I6
63 non-null float64
63 non-null float64
62 non-null float64
V6I7
V6I8
63 non-null float64
63 non-null float64
dtypes: float64(9)
memory usage: 4.9 KB
In [10]: vstoxx_index.tail()
Out[10]:
V2TX
V6I1
Date
2014-03-25 18.2637 18.2303
2014-03-26 17.5869 17.4810
V6I2
V6I3
V6I4
V6I5
V6I6
18.3078
17.7009
19.0371
18.4499
19.8378
19.4150
20.3065
19.9961
18.1063
20.2562
2014-03-27
2014-03-28
2014-03-31
17.7608
17.2864
17.6879
18.6249
18.3281
18.5689
19.4860
19.3032
19.4285
20.0477
19.8332
20.0430
20.1078
20.1371
19.9823
17.6397
17.0324
17.6639
V6I7
17.5032
16.6849
17.6087
\
V6I8
Date
2014-03-25
2014-03-26
20.8292
20.4541
21.2046
20.8563
2014-03-27
2014-03-28
2014-03-31
20.4865
20.3808
20.4448
20.9449
20.8210
20.8994
Per trading day, there are eight futures quotes for the eight different maturities of the VSTOXX
futures contract. This makes for a total of 504 futures quotes.
In [11]: vstoxx_futures.info()
Int64Index: 504 entries, 0 to 503
Data columns (total 5 columns):
DATE
504 non-null datetime64[ns]
299
DX Analytics – Square-Root Diffusion
EXP_YEAR
EXP_MONTH
504 non-null int64
504 non-null int64
PRICE
504 non-null float64
MATURITY
504 non-null datetime64[ns]
dtypes: datetime64[ns](2), float64(1), int64(2)
memory usage: 23.6 KB
In [12]: vstoxx_futures. tail()
Out[12]:
DATE
499 2014-03-31
EXP_YEAR
2014
EXP_MONTH
7
PRICE
MATURITY
20.40 2014-07-18
500 2014-03-31
501 2014-03-31
502 2014-03-31
2014
2014
2014
8
9
10
20.70 2014-08-15
20.95 2014-09-19
21.05 2014-10-17
503 2014-03-31
2014
11
21.25 2014-11-21
By far the biggest data sub-set is the one for the VSTOXX options. For each trading day there
are market quotes for puts and calls for eight different maturities and a multitude of different
strike prices. This makes for a total of 46,960 option quotes for the first quarter of 2014.
In [13]: vstoxx_options.info()
Int64Index: 46960 entries, 0 to 46959
Data columns (total 7 columns):
DATE
EXP_YEAR
EXP_MONTH
46960 non-null datetime64[ns]
46960 non-null int64
46960 non-null int64
TYPE
STRIKE
PRICE
46960 non-null object
46960 non-null float64
46960 non-null float64
MATURITY
46960 non-null datetime64[ns]
dtypes: datetime64[ns](2), float64(2), int64(2), object(1)
memory usage: 2.9+ MB
In [14]: vstoxx_options.tail()
Out[14]:
DATE
46955 2014-03-31
EXP_YEAR
2014
EXP_MONTH TYPE
11
P
46956 2014-03-31
46957 2014-03-31
46958 2014-03-31
2014
2014
2014
11
11
11
46959 2014-03-31
2014
11
STRIKE
85.0
PRICE MATURITY
63.65 2014-11-21
P
P
P
90.0
95.0
100.0
68.65 2014-11-21
73.65 2014-11-21
78.65 2014-11-21
P
105.0
83.65 2014-11-21
Maturity-wise we are dealing with a total of eleven dates. This is due to the fact that at any
given time eight maturities for the VSTOXX futures and options contracts are available and
we are looking at data for three months.
300
LISTED VOLATILITY AND VARIANCE DERIVATIVES
In [15]: third_fridays = sorted(set(vstoxx_futures['MATURITY']))
In [16]: third_fridays
Out[16]:
[Timestamp('2014-01-17 00:00:00'),
Timestamp('2014-02-21 00:00:00'),
Timestamp('2014-03-21 00:00:00'),
Timestamp('2014-04-18 00:00:00'),
Timestamp('2014-05-16 00:00:00'),
Timestamp('2014-06-20 00:00:00'),
Timestamp('2014-07-18 00:00:00'),
Timestamp('2014-08-15 00:00:00'),
Timestamp('2014-09-19 00:00:00'),
Timestamp('2014-10-17 00:00:00'),
Timestamp('2014-11-21 00:00:00')]
When it comes to the calibration of the square-root diffusion model for the VSTOXX,
it is necessary to work with a selection from the large set of option quotes. The following function implements such a selection procedure, using different conditions to generate a sensible set of option quotes around the forward at-the-money level. The function
srd_get_option_selection() is used in what follows to select the right sub-set of
option quotes for each day during the calibration.
def srd_get_option_selection(pricing_date, maturity, tol=tol):
''' Function to select option quotes from data set.
Parameters
==========
pricing_date: datetime object
date for which the calibration shall be implemented
maturity: datetime object
maturity date for the options to be selected
tol: float
moneyness tolerance for OTM and ITM options to be selected
Returns
=======
option_selection: DataFrame object
selected options quotes
forward: float
futures price for maturity at pricing_date
'''
forward = vstoxx_futures[(vstoxx_futures. DATE == pricing_date)
& (vstoxx_futures.MATURITY == maturity)]['PRICE'].values[0]
DX Analytics – Square-Root Diffusion
301
option_selection = \
vstoxx_options[(vstoxx_options.DATE == pricing_date)
& (vstoxx_options.MATURITY == maturity)
& (vstoxx_options.TYPE == 'C') # only calls
& (vstoxx_options.STRIKE > (1 - tol) * forward)
& (vstoxx_options.STRIKE < (1 + tol) * forward)]
return option_selection, forward
13.3 MODELING THE VSTOXX OPTIONS
The previous chapter illustrates how European options are modeled with DX Analytics based
on a geometric Brownian motion model (dx.geometric_brownian_motion()). To
model the VSTOXX options for the calibration, we just need to replace that model with the
square-root diffusion model dx.square_root_diffusion(). The respective market
environment then needs some additional parameters.
All the code used for the calibration is found in the Python script dx_srd_
calibration.py (see sub-section 14.6.1 dx_srd_calibration.py). After some imports, the
script starts by defining some general parameters and curves for the market environment. During the calibration process, some of these get updated to reflect the current status of the optimization procedure.
import dx
import time
import numpy as np
import pandas as pd
import datetime as dt
import scipy.optimize as spo
import matplotlib.pyplot as plt
import seaborn as sns; sns.set()
import matplotlib
matplotlib.rcParams['font.family'] = 'serif'
# importing the data
h5 = pd.HDFStore('../data/vstoxx_march_2014.h5', 'r')
vstoxx_index = h5['vstoxx_index']
vstoxx_futures = h5['vstoxx_futures']
vstoxx_options = h5['vstoxx_options']
h5.close()
# collecting the maturity dates
third_fridays = sorted(set(vstoxx_futures['MATURITY']))
# instantiation of market environment object with dummy pricing date
me_vstoxx = dx.market_environment('me_vstoxx', dt.datetime(2014, 1, 1))
me_vstoxx.add_constant('currency', 'EUR')
me_vstoxx.add_constant('frequency', 'W')
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LISTED VOLATILITY AND VARIANCE DERIVATIVES
me_vstoxx.add_constant('paths',
5000)
# constant short rate model with somewhat arbitrary rate
csr = dx.constant_short_rate('csr', 0.01)
me_vstoxx.add_curve('discount_curve', csr)
# parameters to be calibrated later, dummies only
me_vstoxx.add_constant('kappa', 1.0)
me_vstoxx.add_constant('theta', 20)
me_vstoxx.add_constant('volatility', 1.0)
# payoff function for all European call options
payoff_func = 'np.maximum(maturity_value - strike, 0)'
tol = 0.2
# otm & itm moneyness tolerance
The function srd_get_option_models() creates valuation models for all options in a
given selection of option quotes.
def srd_get_option_models(pricing_date, maturity, option_selection):
''' Function to instantiate option pricing models.
Parameters
==========
pricing_date: datetime object
date for which the calibration shall be implemented
maturity: datetime object
maturity date for the options to be selected
option_selection: DataFrame object
selected options quotes
Returns
=======
vstoxx_model: dx.square_root_diffusion
model object for VSTOXX
option_models: dict
dictionary of dx.valuation_mcs_european_single objects
'''
# updating the pricing date
me_vstoxx.pricing_date = pricing_date
# setting the initial value for the pricing date
initial_value = vstoxx_index['V2TX'][ pricing_date]
me_vstoxx.add_constant('initial_value', initial_value)
# setting the final date given the maturity date
me_vstoxx.add_constant('final_date', maturity)
# instantiating the risk factor (VSTOXX) model
DX Analytics – Square-Root Diffusion
303
vstoxx_model = dx.square_root_diffusion('vstoxx_model', me_vstoxx)
# setting the maturity date for the valuation model(s)
me_vstoxx.add_constant('maturity', maturity)
option_models = {}
# dictionary collecting all models
for option in option_selection.index:
# setting the strike for the option to be modeled
strike = option_selection['STRIKE'].ix[option]
me_vstoxx.add_constant('strike', strike)
# instantiating the option model
option_models[option] = \
dx.valuation_mcs_european_single(
'eur_call_%d' % strike,
vstoxx_model,
me_vstoxx,
payoff_func)
return vstoxx_model, option_models
13.4 CALIBRATION OF THE VSTOXX MODEL
Calibration of a parametrized model usually boils down to using global and local optimization
algorithms to find parameters that minimize a given target function. This process is discussed
in detail in Hilpisch (2015, ch. 11). For the calibration process to follow, we use the helper
function srd_calculate_model_values() to calculate “at once” the model values
for the VSTOXX options at hand. The function parameter p0 is a tuple since this is what the
optimization functions provide as input.
def srd_calculate_model_values(p0):
''' Returns all relevant option values.
Parameters
===========
p0: tuple/list
tuple of kappa, theta, volatility
Returns
=======
model_values: dict
dictionary with model values
'''
# read the model parameters from input tuple
kappa, theta, volatility = p0
# update the option market environment
vstoxx_model.update(kappa=kappa,
theta=theta,
volatility=volatility)
304
LISTED VOLATILITY AND VARIANCE DERIVATIVES
# estimate and collect all option model present values
results = [option_models[option].present_value(fixed_seed=True)
for option in option_models]
# combine the results with the option models in a dictionary
model_values = dict(zip(option_models, results))
return model_values
The target function to be minimized during the calibration is the mean-squared
error of the model values given the market quotes of the VSTOXX options. Again,
refer to Hilpisch (2015, ch. 11) for details and alternative formulations. The function srd_mean_squared_error() implements this concept and uses the function
srd_calculate_model_values() for the option model value calculations.
def srd_mean_squared_error(p0, penalty=True):
''' Returns the mean-squared error given
the model and market values.
Parameters
===========
p0: tuple/list
tuple of kappa, theta, volatility
Returns
=======
MSE: float
mean-squared error
'''
# escape with high value for non-sensible parameter values
if p0[0] < 0 or p0[1] < 5. or p0[2] < 0 or p0[2] > 10.:
return 1000
# define/access global variables/objects
global option_selection, vstoxx_model, option_models, first, last
# calculate the model values for the option selection
model_values = srd_calculate_model_values(p0)
option_diffs = {} # dictionary to collect differences
for option in model_values:
# differences between model value and market quote
option_diffs[option] = (model_values[option]
- option_selection['PRICE'].loc[option])
# calculation of mean-squared error
MSE = np.sum(np.array(option_diffs.values()) ** 2) / len(option_diffs)
if first is True:
# if in global optimization, no penalty
penalty = 0.0
else:
# if in local optimization, penalize deviation from previous
305
DX Analytics – Square-Root Diffusion
# optimal parameter combination
penalty = (np.sum((p0 - last) ** 2)) / 100
if penalty is False:
return MSE
return MSE + penalty
Equipped with the target function to be minimized, we can define the function for
the global and local calibration routine itself. The calibration takes place for one
or multiple maturities over the pricing date range defined. For example, the function
srd_get_parameter_series() can calibrate the model (separately) for the two maturities May and June 2014 over the complete first quarter 2014.
def srd_get_parameter_series(pricing_date_list, maturity_list):
''' Returns parameter series for the calibrated model over time.
Parameters
==========
pricing_date_list: pd.DatetimeIndex
object with relevant pricing dates
maturity_list: list
list with maturities to be calibrated
Returns
=======
parameters: pd.DataFrame
DataFrame object with parameter series
'''
# define/access global variables/objects
global option_selection, vstoxx_model, option_models, first, last
parameters = pd.DataFrame() # object to collect parameter series
for maturity in maturity_list:
first = True
for pricing_date in pricing_date_list:
# select relevant option quotes
option_selection, forward = srd_get_option_selection
(pricing_date, maturity)
# instantiate all model given option selection
vstoxx_model, option_models = srd_get_option_models
(pricing_date, maturity, option_selection)
if first is True:
# global optimization to start with
opt = spo.brute(srd_mean_squared_error,
((1.25, 6.51, 0.75), # range for kappa
(10., 20.1, 2.5), # range for theta
(0.5, 10.51, 2.5)),
finish=None)
# range for volatility
306
LISTED VOLATILITY AND VARIANCE DERIVATIVES
# local optimization
opt = spo.fmin(srd_mean_squared_error, opt,
maxiter=550, maxfun=650,
xtol=0.0000001, ftol=0.0000001);
# calculate MSE for storage
MSE = srd_mean_squared_error(opt, penalty=False)
# store main parameters and results
parameters = parameters.append(
pd.DataFrame(
{'date' : pricing_date,
'maturity' : maturity,
'initial_value' :vstoxx_model.initial_value,
'kappa' : opt [0],
'theta' : opt [1],
'sigma' : opt [2],
'MSE' : MSE},
index = [0]), ignore_index=True)
first = False # set to False after first iteration
last = opt # store optimal parameters for reference
print ("Maturity %s" % str(maturity)[:10]
+ " | Pricing Date %s" % str(pricing_date)[:10]
+ " | MSE %6.5f " % MSE)
return parameters
The final step is to start the calibration and collect the calibration results. The calibration we
implement is for the April 18, 2014 maturity.
if _ _name_ _ is '_ _main_ _ ':
t0 = time.time()
# define the dates for the calibration
pricing_date_list = pd.date_range('2014/1/2', '2014/3/31', freq='B')
# select the maturities
maturity_list = [third_fridays[3]]
# only 18. April 2014 maturity
# start the calibration
parameters = srd_get_parameter_series(pricing_date_list, maturity_list)
# plot the results
for mat in maturity_list:
fig1, ax1 = plt. subplots()
to_plot = parameters [parameters.maturity ==
maturity_list[0]].set_index('date')[
['kappa', 'theta', 'sigma', 'MSE']]
to_plot.plot(subplots=True, color='b', figsize=(10, 12),
title='SRD | ' + str(mat)[:10], ax=ax1)
plt.savefig('../images/dx_srd_cali_1_ %s.pdf' % str(mat)[:10])
# plotting the histogram of the MSE values
DX Analytics – Square-Root Diffusion
fig2,
307
ax2 = plt. subplots()
dat = parameters.MSE
dat.hist(bins=30, ax=ax2)
plt.axvline(dat.mean(), color='r', ls='dashed',
lw=1.5, label='mean = %5.4f' % dat.mean())
plt.legend()
plt.savefig('../images/dx_srd_cali_1_hist_%s.pdf' % str(mat)[:10])
# measuring and printing the time needed for the script execution
print "Time in minutes %.2f" % ((time.time() - t0) / 60)
A visualization of the calibration results tells the whole story. Figure 13.1 shows the three
square-root diffusion parameters over time and the resulting MSE values.
As we can see throughout, the results are quite good given the low MSE values. The mean
MSE value is below 0.01 as seen in Figure 13.2.
FIGURE 13.1
Square-root diffusion parameters and MSE values from the calibration to a single
maturity (April 18, 2015).
308
LISTED VOLATILITY AND VARIANCE DERIVATIVES
FIGURE 13.2
Histogram of the mean-squared errors for the calibration of the
square-root diffusion model to a single maturity (April 18, 2015).
13.5 CONCLUSIONS
This chapter uses DX Analytics to model the VSTOXX volatility index by a square-root diffusion process. In similar vein, it is used to model traded European call options on the index
to implement a calibration of the VSTOXX model over time. The results show that when calibrating the model to a single options maturity only, the model performs quite well yielding
rather low MSE values throughout. The parameter values also seem to be in sensible regions
throughout (e.g. theta between 15 and 18) and they evolve rather smoothly.
There exist closed form solutions for the price of a European call option in the square-root
diffusion model of Grünbichler and Longstaff (1996) as shown in chapter 6, Valuing Volatility
Derivatives. For our analysis in this chapter we have nevertheless used the Monte Carlo valuation model of DX Analytics since this approach is more general in that we can easily replace
one model by another, maybe more sophisticated, one. This is done in the next chapter where
the same study is implemented based on the square-root jump diffusion model presented in
chapter 7, Advanced Modeling of the VSTOXX Index. The only difference is that a few more
parameters need to be taken care of.
13.6 PYTHON SCRIPTS
13.6.1 dx srd calibration.py
#
# Calibration of Grünbichler and Longstaff (1996)
# Square-Root Diffusion (SRD) model to
DX Analytics – Square-Root Diffusion
# VSTOXX call options with DX Analytics
#
# All data from www.eurexchange.com
#
# (c) Dr. Yves J. Hilpisch
# Listed Volatility and Variance Derivatives
#
import dx
import time
import numpy as np
import pandas as pd
import datetime as dt
import scipy.optimize as spo
import matplotlib.pyplot as plt
import seaborn as sns; sns. set()
import matplotlib
matplotlib.rcParams['font.family'] = 'serif'
# importing the data
h5 = pd.HDFStore('../data/vstoxx_march_2014.h5', 'r')
vstoxx_index = h5['vstoxx_index']
vstoxx_futures = h5['vstoxx_futures']
vstoxx_options = h5['vstoxx_options']
h5.close()
# collecting the maturity dates
third_fridays = sorted(set(vstoxx_futures['MATURITY']))
# instantiation of market environment object with dummy pricing date
me_vstoxx = dx.market_environment('me_vstoxx', dt.datetime(2014, 1, 1))
me_vstoxx.add_constant('currency', 'EUR')
me_vstoxx.add_constant('frequency', 'W')
me_vstoxx.add_constant('paths', 5000)
# constant short rate model with somewhat arbitrary rate
csr = dx.constant_short_rate('csr', 0.01)
me_vstoxx.add_curve ('discount_curve', csr)
# parameters to be calibrated later, dummies only
me_vstoxx.add_constant('kappa', 1.0)
me_vstoxx.add_constant('theta', 20)
me_vstoxx.add_constant('volatility', 1.0)
# payoff function for all European call options
payoff_func = 'np.maximum(maturity_value - strike, 0)'
tol = 0.2
# otm & itm moneyness tolerance
309
310
LISTED VOLATILITY AND VARIANCE DERIVATIVES
def srd_get_option_selection (pricing_date, maturity, tol=tol):
''' Function to select option quotes from data set.
Parameters
==========
pricing_date: datetime object
date for which the calibration shall be implemented
maturity: datetime object
maturity date for the options to be selected
tol: float
moneyness tolerace for OTM and ITM options to be selected
Returns
=======
option_selection: DataFrame object
selected options quotes
forward: float
futures price for maturity at pricing_date
'''
forward = vstoxx_futures[(vstoxx_futures.DATE == pricing_date)
& (vstoxx_futures.MATURITY == maturity)]
['PRICE'].values[0]
option_selection = \
vstoxx_options[(vstoxx_options.DATE == pricing_date)
&(vstoxx_options.MATURITY == maturity)
&(vstoxx_options.TYPE == 'C') # only calls
&(vstoxx_options.STRIKE > (1 - tol) * forward)
&(vstoxx_options.STRIKE <(1 + tol) * forward)]
return option_selection, forward
def srd_get_option_models (pricing_date, maturity, option_selection):
''' Function to instantiate option pricing models.
Parameters
==========
pricing_date: datetime object
date for which the calibration shall be implemented
maturity: datetime object
maturity date for the options to be selected
option_selection: DataFrame object
selected options quotes
Returns
=======
vstoxx_model: dx.square_root_diffusion
model object for VSTOXX
option_models: dict
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DX Analytics – Square-Root Diffusion
dictionary of dx.valuation_mcs_european_single objects
'''
# updating the pricing date
me_vstoxx.pricing_date = pricing_date
# setting the initial value for the pricing date
initial_value = vstoxx_index['V2TX'][pricing_date]
me_vstoxx.add_constant('initial_value', initial_value)
# setting the final date given the maturity date
me_vstoxx.add_constant('final_date', maturity)
# instantiating the risk factor (VSTOXX) model
vstoxx_model = dx.square_root_diffusion('vstoxx_model', me_vstoxx)
# setting the maturity date for the valuation model(s)
me_vstoxx.add_constant('maturity', maturity)
option_models = {} # dictionary collecting all models
for option in option_selection.index:
# setting the strike for the option to be modeled
strike = option_selection['STRIKE'].ix[option]
me_vstoxx.add_constant('strike', strike)
# instantiating the option model
option_models[option] = \
dx.valuation_mcs_european_single(
'eur_call_%d' % strike,
vstoxx_model,
me_vstoxx,
payoff_func)
return vstoxx_model, option_models
def srd_calculate_model_values(p0):
''' Returns all relevant option values.
Parameters
===========
p0: tuple/list
tuple of kappa, theta, volatility
Returns
=======
model_values: dict
dictionary with model values
'''
# read the model parameters from input tuple
kappa, theta, volatility = p0
# update the option market environment
vstoxx_model. update(kappa=kappa,
theta=theta,
volatility=volatility)
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LISTED VOLATILITY AND VARIANCE DERIVATIVES
# estimate and collect all option model present values
results = [option_models[option].present_value(fixed_seed=True)
for option in option_models]
# combine the results with the option models in a dictionary
model_values = dict(zip(option_models, results))
return model_values
def srd_mean_squared_error(p0, penalty=True):
''' Returns the mean-squared error given
the model and market values.
Parameters
===========
p0: tuple/list
tuple of kappa, theta, volatility
Returns
=======
MSE: float
mean-squared error
'''
# escape with high value for non-sensible parameter values
if p0[0] < 0 or p0[1] < 5. or p0[2] < 0 or p0[2] > 10.:
return 1000
# define/access global variables/objects
global option_selection, vstoxx_model, option_models, first, last
# calculate the model values for the option selection
model_values = srd_calculate_model_values(p0)
option_diffs = {} # dictionary to collect differences
for option in model_values:
# differences between model value and market quote
option_diffs[option] = (model_values[option]
- option_selection['PRICE'].loc[option])
# calculation of mean-squared error
MSE = np.sum(np.array(option_diffs.values()) ** 2) / len(option_diffs)
if first is True:
# if in global optimization, no penalty
penalty = 0.0
else:
# if in local optimization, penalize deviation from previous
# optimal parameter combination
penalty = (np.sum((p0 - last) ** 2)) / 100
if penalty is False:
return MSE
return MSE + penalty
DX Analytics – Square-Root Diffusion
def srd_get_parameter_series(pricing_date_list, maturity_list):
'''Returns parameter series for the calibrated model over time.
Parameters
==========
pricing_date_list: pd.DatetimeIndex
object with relevant pricing dates
maturity_list: list
list with maturities to be calibrated
Returns
=======
parameters: pd.DataFrame
DataFrame object with parameter series
'''
# define/access global variables/objects
global option_selection, vstoxx_model, option_models, first, last
parameters = pd.DataFrame() # object to collect parameter series
for maturity in maturity_list:
first = True
for pricing_date in pricing_date_list:
# select relevant option quotes
option_selection, forward = srd_get_option_selection
(pricing_date, maturity)
# instantiate all model given option selection
vstoxx_model, option_models = srd_get_option_models
(pricing_date, maturity, option_selection)
if first is True:
# global optimization to start with
opt = spo.brute(srd_mean_squared_error,
((1.25, 6.51, 0.75), # range for kappa
(10., 20.1, 2.5), # range for theta
(0.5, 10.51, 2.5)), # range for volatility
finish=None)
# local optimization
opt = spo.fmin(srd_mean_squared_error, opt,
maxiter=550, maxfun=650,
xtol=0.0000001, ftol=0.0000001);
# calculate MSE for storage
MSE = srd_mean_squared_error(opt, penalty=False)
# store main parameters and results
parameters = parameters.append(
pd.DataFrame(
{'date' : pricing_date,
'maturity' : maturity,
'initial_value' : vstoxx_model.initial_value,
'kappa' : opt[0],
313
314
LISTED VOLATILITY AND VARIANCE DERIVATIVES
'theta' : opt[1],
'sigma' : opt[2],
'MSE' : MSE},
index=[0]), ignore_index=True)
first = False # set to False after first iteration
last = opt # store optimal parameters for reference
print ("Maturity %s" %str(maturity)[:10]
+ " | Pricing Date %s" % str(pricing_date)[:10]
+ " | MSE %6.5f " % MSE)
return parameters
if _ _name_ _ is '_ _main_ _':
t0 = time.time()
# define the dates for the calibration
pricing_date_list = pd.date_range('2014/1/2', '2014/3/31', freq='B')
# select the maturities
maturity_list = [third_fridays[3]] # only 18. April 2014 maturity
# start the calibration
parameters = srd_get_parameter_series(pricing_date_list,
maturity_list)
# plot the results
for mat in maturity_list:
fig1, ax1 = plt.subplots()
to_plot = parameters[parameters.maturity ==
maturity_list[0]].set_index('date')[
['kappa', 'theta', 'sigma', 'MSE']]
to_plot.plot(subplots=True, color='b', figsize=(10, 12),
title='SRD | ' + str(mat)[:10], ax=ax1)
plt.savefig('../images/dx_srd_cali_1_%s.pdf' % str(mat)[:10])
# plotting the histogram of the MSE values
fig2, ax2 = plt. subplots()
dat = parameters.MSE
dat.hist(bins=30, ax=ax2)
plt.axvline(dat.mean(), color='r', ls='dashed',
lw=1.5, label='mean = %5.4f' %dat.mean())
plt.legend()
plt.savefig('../images/dx_srd_cali_1_hist_%s.pdf' % str(mat)[:10])
# measuring and printing the time needed for the script execution
print "Time in minutes %.2f" % ((time.time()- t0)/ 60)
Listed Volatility and Variance Derivatives: A
Python-based Guide
By Dr. Yves J. Hilpisch
© 2017 Yves Hilpisch
CHAPTER
14
DX Analytics – Square-Root
Jump Diffusion
14.1 INTRODUCTION
Similarly to the previous chapter, this chapter again uses DX Analytics to model the VSTOXX
index, but this time by the square-root jump diffusion (SRJD) process as introduced in chapter 7, Advanced Modeling of the VSTOXX Index. The study this chapter implements is actually the same as in the previous one using the very same data set. However, the challenge is
increased in that we require multiple VSTOXX option maturities to be calibrated simultaneously and over time.
14.2 MODELING THE VSTOXX OPTIONS
DX Analytics provides a class for the deterministic shift square-root jump diffusion model. It is
called dx.square_root_jump_diffusion(). Although the calibration we are implementing in this chapter is more or less the same as in the previous one, we need nevertheless
to adjust the code significantly in many places. Therefore, we are stepping through the single
elements again in what follows. All the code used in this chapter is found in the Python script
dx_srjd_calibration.py (see sub-section 14.6.1, dx_srjd_calibration.py).
The beginning of the script is rather similar to the one implementing the calibration of the
square-root diffusion model. A major difference is that we need three additional parameters
lambda, mu and delta.
import dx
import time
import numpy as np
import pandas as pd
import datetime as dt
import scipy.optimize as spo
import matplotlib.pyplot as plt
import seaborn as sns; sns.set()
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LISTED VOLATILITY AND VARIANCE DERIVATIVES
import matplotlib
matplotlib.rcParams['font.family'] = 'serif'
from copy import deepcopy
# importing the data
h5 = pd.HDFStore('../data/vstoxx_march_2014.h5', 'r')
vstoxx_index = h5['vstoxx_index']
vstoxx_futures = h5['vstoxx_futures']
vstoxx_options = h5['vstoxx_options']
h5.close()
# collecting the maturity dates
third_fridays = sorted(set(vstoxx_futures['MATURITY']))
# instantiation of market environment object with dummy pricing date
me_vstoxx = dx.market_environment('me_vstoxx', dt.datetime(2014, 1, 1))
me_vstoxx.add_constant('currency', 'EUR')
me_vstoxx.add_constant('frequency', 'W')
me_vstoxx.add_constant('paths', 5000)
# constant short rate model with somewhat arbitrary rate
csr = dx.constant_short_rate('csr', 0.01)
me_vstoxx.add_curve('discount_curve', csr)
# parameters to be calibrated later, dummies only
# SRD part
me_vstoxx.add_constant('kappa', 1.0)
me_vstoxx.add_constant('theta', 20)
me_vstoxx.add_constant('volatility', 1.0)
# jump part
me_vstoxx.add_constant('lambda', 0.5)
me_vstoxx.add_constant('mu', -0.2)
me_vstoxx.add_constant('delta', 0.1)
# payoff function for all European call options
payoff_func = 'np.maximum(maturity_value - strike, 0)'
tol = 0.2 # otm & itm moneyness tolerance
first = True # flag for first calibration
The function srjd_get_option_selection() selects the options used for the calibration. In the current case, we can choose multiple option maturities to be included.
def srjd_get_option_selection(pricing_date, tol=tol):
' ' ' Function to select option quotes from data set.
DX Analytics – Square-Root Jump Diffusion
317
Parameters
==========
pricing_date: datetime object
date for which the calibration shall be implemented
tol: float
moneyness tolerance for OTM and ITM options to be selected
Returns
=======
option_selection: DataFrame object
selected options quotes
futures: DataFrame object
futures prices at pricing_date
'''
option_selection = pd.DataFrame()
mats = [third_fridays[3],] # list of maturity dates
# select the relevant futures prices
futures = vstoxx_futures[(vstoxx_futures.DATE == pricing_date)
& (vstoxx_futures.MATURITY.apply(lambda x: x in mats))]
# collect option data for the given option maturities
for mat in mats:
forward = futures[futures.MATURITY == mat]['PRICE'].values[0]
option_selection = option_selection.append(
vstoxx_options[(vstoxx_options.DATE == pricing_date)
& (vstoxx_options.MATURITY == mat)
& (vstoxx_options.TYPE == 'C') # only calls
& (vstoxx_options.STRIKE > (1 - tol) * forward)
& (vstoxx_options.STRIKE < (1 + tol) * forward)])
return option_selection, futures
The calibration of the SRJD model consists of two steps:
term structure calibration: using the futures prices at a given pricing date, this step
calibrates the forward rates of the model
option quote calibration: using market quotes of traded options, this step calibrates the
model parameters to optimally reflect the market quotes.
Two functions are used to implement the first step. The function srd_forward_error()
calculates the mean-squared error (MSE) for the futures term structure given a set of model
parameters.
def srd_forward_error(p0):
' ' ' Calculates the mean-squared error for the
term structure calibration for the SRD model part.
Parameters
===========
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LISTED VOLATILITY AND VARIANCE DERIVATIVES
p0: tuple/list
tuple of kappa, theta, volatility
Returns
=======
MSE: float
mean-squared error
'''
global initial_value, f, t
if p0[0] < 0 or p0[1] < 0 or p0[2] < 0:
return 100
f_model = dx.srd_forwards(initial_value, p0, t)
MSE = np.sum((f - f_model) ** 2) / len(f)
return MSE
The second function is called generate_shift_base() and calculates the single deterministic shift values to match the futures term structure perfectly – after the minimization
of the term structure MSE. Note that the perfect shift gets lost again later on when the SRD
parameters are updated during the calibration to the option quotes.
def generate_shift_base(pricing_date, futures):
' ' ' Generates the values for the deterministic shift for the
SRD model part.
Parameters
==========
pricing_date: datetime object
date for which the calibration shall be implemented
futures: DataFrame object
futures prices at pricing_date
Returns
=======
shift_base: ndarray object
shift values for the SRD model part
'''
global initial_value, f, t
# futures price array
f = list(futures['PRICE'].values)
f.insert(0, initial_value)
f = np.array(f)
# date array
t = [_.to_pydatetime() for _ in futures['MATURITY']]
t.insert(0, pricing_date)
t = np.array(t)
# calibration to the futures values
opt = spo.fmin(srd_forward_error, (2., 15., 2.))
DX Analytics – Square-Root Jump Diffusion
319
# calculation of shift values
f_model = dx.srd_forwards(initial_value, opt, t)
shifts = f - f_model
shift_base = np.array((t, shifts)).T
return shift_base
As in the SRD calibration case, the function srjd_get_option_models() creates the
valuation models for all selected options.
def srjd_get_option_models(pricing_date, option_selection, futures):
' ' ' Function to instantiate option pricing models.
Parameters
==========
pricing_date: datetime object
date for which the calibration shall be implemented
maturity: datetime object
maturity date for the options to be selected
option_selection: DataFrame object
selected options quotes
Returns
=======
vstoxx_model: dx.square_root_diffusion
model object for VSTOXX
option_models: dict
dictionary of dx.valuation_mcs_european_single objects
'''
global initial_value
# updating the pricing date
me_vstoxx.pricing_date = pricing_date
# setting the initial value for the pricing date
initial_value = vstoxx_index['V2TX'][pricing_date]
me_vstoxx.add_constant('initial_value', initial_value)
# setting the final date given the maturity dates
final_date = max(futures.MATURITY).to_pydatetime()
me_vstoxx.add_constant('final_date', final_date)
# adding the futures term structure
me_vstoxx.add_curve('term_structure', futures)
# instantiating the risk factor (VSTOXX) model
vstoxx_model = dx.square_root_jump_diffusion_plus('vstoxx_model',
me_vstoxx)
# generating the shift values and updating the model
vstoxx_model.shift_base = generate_shift_base(pricing_date, futures)
vstoxx_model.update_shift_values()
320
LISTED VOLATILITY AND VARIANCE DERIVATIVES
option_models = {} # dictionary collecting all models
for option in option_selection.index:
# setting the maturity date for the given option
maturity = option_selection['MATURITY'].ix[option]
me_vstoxx.add_constant('maturity', maturity)
# setting the strike for the option to be modeled
strike = option_selection['STRIKE'].ix[option]
me_vstoxx.add_constant('strike', strike)
# instantiating the option model
option_models[option] = \
dx.valuation_mcs_european_single(
'eur_call_%d' % strike,
vstoxx_model,
me_vstoxx,
payoff_func)
return vstoxx_model, option_models
14.3 CALIBRATION OF THE VSTOXX MODEL
The function srjd_calculate_model_values() only differs from the SRD case in
that three more parameters need to be taken care of.
def srjd_calculate_model_values(p0):
' ' ' Returns all relevant option values.
Parameters
===========
p0: tuple/list
tuple of kappa, theta, volatility, lamb, mu, delt
Returns
=======
model_values: dict
dictionary with model values
'''
# read the model parameters from input tuple
kappa, theta, volatility, lamb, mu, delt = p0
# update the option market environment
vstoxx_model.update(kappa=kappa,
theta=theta,
volatility=volatility,
lamb=lamb,
mu=mu,
delt=delt)
# estimate and collect all option model present values
DX Analytics – Square-Root Jump Diffusion
321
results = [option_models[option].present_value(fixed_seed=True)
for option in option_models]
# combine the results with the option models in a dictionary
model_values = dict(zip(option_models, results))
return model_values
The same holds true for the function srjd_mean_squared_error() which now also
penalizes certain parameter ranges for the additional parameters of the SRJD model.
def srjd_mean_squared_error(p0, penalty=True):
' ' ' Returns the mean-squared error given
the model and market values.
Parameters
===========
p0: tuple/list
tuple of kappa, theta, volatility
Returns
=======
MSE: float
mean-squared error
'''
# escape with high value for non-sensible parameter values
if (p0[0] < 0 or p0[1] < 5. or p0[2] < 0 or p0[2] > 10.
or p0[3] < 0 or p0[4] < 0 or p0[5] < 0):
return 1000
# define/access global variables/objects
global option_selection, vstoxx_model, option_models, first, last
# calculate the model values for the option selection
model_values = srjd_calculate_model_values(p0)
option_diffs = {}
# dictionary to collect differences
for option in model_values:
# differences between model value and market quote
option_diffs[option] = (model_values[option]
- option_selection['PRICE'].loc[option])
# calculation of mean-squared error
MSE = np.sum(np.array(option_diffs.values()) ** 2) / len(option_diffs)
if first is True:
# if in first optimization, no penalty
penalty = 0.0
else:
# if 2, 3, ... optimization, penalize deviation from previous
# optimal parameter combination
penalty = (np.sum((p0 - last) ** 2))
if penalty is False:
return MSE
return MSE + penalty
322
LISTED VOLATILITY AND VARIANCE DERIVATIVES
The function srjd_get_parameter_series() implementing the calibration routines
then takes on the form:
def srjd_get_parameter_series(pricing_date_list):
' ' ' Returns parameter series for the calibrated model over time.
Parameters
==========
pricing_date_list: pd.DatetimeIndex
object with relevant pricing dates
Returns
=======
parameters: pd.DataFrame
DataFrame object with parameter series
'''
# define/access global variables/objects
global initial_value, futures, option_selection, vstoxx_model,
\
option_models, first, last
parameters = pd.DataFrame()
# DataFrame object to collect parameter series
for pricing_date in pricing_date_list:
# setting the initial value for the VSTOXX
initial_value = vstoxx_index['V2TX'][pricing_date]
# select relevant option quotes
option_selection, futures = srjd_get_option_selection(pricing_date)
# instantiate all model given option selection
vstoxx_model, option_models = srjd_get_option_models(pricing_date,
option_selection,
futures)
# global optimization to start with
opt = spo.brute(srjd_mean_squared_error,
((1.25, 6.51, 0.75),
(10., 20.1, 2.5),
(0.5, 10.51, 2.5),
# range for kappa
# range for theta
# range for volatility
(0.1, 0.71, 0.3),
# range for lambda
(0.1, 0.71, 0.3),
# range for mu
(0.1, 0.21, 0.1)),
# range for delta
finish=None)
# local optimization
opt = spo.fmin(srjd_mean_squared_error, opt,
maxiter=550, maxfun=650,
xtol=0.0000001, ftol=0.0000001);
# calculate MSE for storage
MSE = srjd_mean_squared_error(opt, penalty=False)
# store main parameters and results
parameters = parameters.append(
pd.DataFrame(
{'date' : pricing_date,
323
DX Analytics – Square-Root Jump Diffusion
'initial_value' : vstoxx_model.initial_value,
'kappa' : opt[0],
'theta' : opt[1],
'sigma' : opt[2],
'lambda' : opt[3],
'mu' : opt[4],
'delta' : opt[5],
'MSE' : MSE},
index=[0]), ignore_index=True)
first = False
last = opt
# set to False after first iteration
# store optimal parameters for reference
print ("Pricing Date %s" % str(pricing_date)[:10]
+ " | MSE %6.5f" % MSE)
return parameters
Finally, the function srjd_plot_model_fit() plots the results for the last pricing date of
the calibration procedure and compares the model values to the market quotes.
def srjd_plot_model_fit(parameters):
# last pricing date
pdate = max(parameters.date)
# optimal parameters for that date and the maturity
opt = np.array(parameters[parameters.date == pdate][[
'kappa', 'theta', 'sigma', 'lambda', 'mu', 'delta']])[0]
option_selection, futures = srjd_get_option_selection(pdate, tol=tol)
vstoxx_model, option_models = srjd_get_option_models(pdate,
option_selection,
futures)
model_values = srjd_calculate_model_values(opt)
model_values = pd.DataFrame(model_values.values(),
index=model_values.keys(),
columns=['MODEL'])
option_selection = option_selection.join(model_values)
mats = set(option_selection.MATURITY.values)
mats = sorted(mats)
# arranging the canvas for the subplots
height = max(8, 2 * len(mats))
if len(mats) == 1:
mat = mats[0]
fig, axarr = plt.subplots(2, figsize=(10, height))
os = option_selection[option_selection.MATURITY == mat]
strikes = os.STRIKE.values
axarr[0].set_ylabel('%s' % str(mat)[:10])
axarr[0].plot(strikes, os.PRICE.values, label='Market Quotes')
axarr[0].plot(strikes, os.MODEL.values, 'ro', label='Model Prices')
axarr[0].legend(loc=0)
324
LISTED VOLATILITY AND VARIANCE DERIVATIVES
wi = 0.3
axarr[1].bar(strikes - wi / 2, os.MODEL.values - os.PRICE.values,
width=wi)
axarr[0].set_xlabel('strike')
axarr[1].set_xlabel('strike')
else:
fig, axarr = plt.subplots(len(mats), 2, sharex=True,
figsize=(10, height))
for z, mat in enumerate(mats):
os = option_selection[option_selection.MATURITY == mat]
strikes = os.STRIKE.values
axarr[z, 0].set_ylabel('%s' % str(mat)[:10])
axarr[z, 0].plot(strikes, os.PRICE.values, label='Market Quotes')
axarr[z, 0].plot(strikes, os.MODEL.values, 'ro', label='Model
Prices')
axarr[z, 0].legend(loc=0)
wi = 0.3
axarr[z, 1].bar(strikes - wi / 2,
os.MODEL.values - os.PRICE.values, width=wi)
axarr[z, 0].set_xlabel('strike')
axarr[z, 1].set_xlabel('strike')
plt.savefig('../images/dx_srjd_cali_1_fit.pdf')
The final step is to start the calibration, collect the calibration results and to
plot them. Remember that the maturities in this case are selected in the function
srjd_get_option_selection().
if _ _name_ _ is '_ _main_ _':
t0 = time.time()
# selecting the dates for the calibration
pricing_date_list = pd.date_range('2014/3/1', '2014/3/31', freq='B')
# conducting the calibration
parameters = srjd_get_parameter_series(pricing_date_list)
# storing the calibation results
date = str(dt.datetime.now())[:10]
h5 = pd.HDFStore('../data/srjd_calibration_%s_%s_%s' %
(me_vstoxx.get_constant('paths'),
me_vstoxx.get_constant('frequency'),
date.replace('-', '_')), 'w')
h5['parameters'] = parameters
h5.close()
# plotting the parameter time series data
fig1, ax1 = plt.subplots(1, figsize=(10, 12))
to_plot = parameters.set_index('date')[
['kappa', 'theta', 'sigma',
'lambda', 'mu', 'delta', 'MSE']]
DX Analytics – Square-Root Jump Diffusion
325
to_plot.plot(subplots=True, color='b', title='SRJD', ax=ax1)
plt.savefig('../images/dx_srjd_cali_1.pdf')
# plotting the histogram of the MSE values
fig2, ax2 = plt.subplots()
dat = parameters.MSE
dat.hist(bins=30, ax=ax2)
plt.axvline(dat.mean(), color='r', ls='dashed',
lw=1.5, label='mean = %5.4f' % dat.mean())
plt.legend()
plt.savefig('../images/dx_srjd_cali_1_hist.pdf')
# plotting the model fit at last pricing date
srjd_plot_model_fit(parameters)
# measuring and printing the time needed for the script execution
print "Time in minutes %.2f" % ((time.time() - t0) / 60)
14.4 CALIBRATION RESULTS
This section presents calibration results for three different calibration runs. The first run implements a calibration to a single maturity, the second to two maturities simultaneously while the
third run does the same for five maturities of the VSTOXX options. The final run shows the
effects of not using penalties for deviations from previous optimal parameters which in general
would be used to get smoother parameter time series.
14.4.1 Calibration to One Maturity
First, we calibrate the SRJD model to a single maturity, April 18, 2014. The calibration takes
place for all trading days in the first quarter of 2014. Figure 14.1 shows the time series data
for the parameters of the square-root jump diffusion over time and the resulting MSE values.
The MSE values are all quite low in general with a mean of about 0.1 and one outlier as
Figure 14.2 illustrates.
Figure 14.3 shows the calibration results for the last pricing date, i.e. on March 31, 2014.
14.4.2 Calibration to Two Maturities
Let us have a look at the results for the calibration to two maturities where the VSTOXX
futures term structure comes into play. Figure 14.4 shows the parameter time series data for
this case.
In this case, the resulting MSE values are even slightly lower on average with a mean of
about 0.05 as seen in Figure 14.5.
Figure 14.6 shows the calibration results for the last pricing date, i.e. on March 31, 2014.
14.4.3 Calibration to Five Maturities
Finally, let us have a look at the calibration to five maturities simultaneously. This is the largest
number of maturities for which there is data over the whole time range from January to March
2014. Figure 14.7 shows the results for the model parameters and the MSE values.
326
FIGURE 14.1
maturities.
LISTED VOLATILITY AND VARIANCE DERIVATIVES
Square-root jump diffusion parameters and MSE values from the calibration to two
DX Analytics – Square-Root Jump Diffusion
FIGURE 14.2
327
Histogram of MSE values for SRJD calibration to two
maturities.
FIGURE 14.3
Model values from the SRJD calibration vs. market quotes as well as pricing errors
(bars) on March 31, 2014 (one maturity).
328
FIGURE 14.4
maturities.
LISTED VOLATILITY AND VARIANCE DERIVATIVES
Square-root jump diffusion parameters and MSE values from the calibration to five
DX Analytics – Square-Root Jump Diffusion
FIGURE 14.5
329
Histogram of MSE values for SRJD calibration to five
maturities.
FIGURE 14.6
Model values from the SRJD calibration vs. market quotes as well as pricing errors
(bars) on March 31, 2014 (two maturities).
330
FIGURE 14.7
maturities.
LISTED VOLATILITY AND VARIANCE DERIVATIVES
Square-root jump diffusion parameters and MSE values from the calibration to five
DX Analytics – Square-Root Jump Diffusion
FIGURE 14.8
331
Histogram of MSE values for SRJD calibration to five maturities.
The resulting MSE values are again slightly lower on average. The mean MSE value is
about 0.035 (see Figure 14.8).
The final plot (see Figure 14.9) again shows the calibration results at the last pricing date,
March 31, 2014. Given that the calibration includes options with different moneyness levels
over five different maturity months, the performance of the SRJD model is satisfactory.
14.4.4 Calibration without Penalties
It might be surprising that the highest average MSE value is observed for the calibration case
with one maturity only. This is mainly due to the fact that we penalize deviations from previous
optimal parameter values quite heavily and also to the existence of one outlier. The advantage
of this is rather smooth time series for the single parameters, i.e. without too much variation.
This is usually desirable, for example, when such a model is used for hedging purposes and
hedge positions (indirectly) depend on the parameter values.
In this sub-section we therefore want to illustrate what changes when we do not penalize
deviations from the previous optimal parameter values. Figure 14.10 shows the results for the
model parameters and the MSE values from a calibration to one maturity (April 18, 2014)
without penalization. Inspection of the figure reveals how erratically the parameter time series
can behave in this case.
However, the advantage here is that there are excellent model fits to the market data with
a mean MSE value of 0.0007 only (see Figure 14.11).
332
LISTED VOLATILITY AND VARIANCE DERIVATIVES
FIGURE 14.9
Model values from the SRJD calibration vs. market quotes as well as pricing errors
(bars) on March 31, 2014 (five maturities).
14.5 CONCLUSIONS
This chapter calibrates the square-root jump diffusion (SRJD) model to both the VSTOXX
futures term structure and for multiple maturities for the VSTOXX options. It uses DX Analytics as in the previous chapter which provides flexible modeling capabilities for volatilitybased derivatives based, among other things, on square-root diffusions and square-root jump
diffusions.
The results we achieve are quite good in that the typical mean-squared error values are
relatively low for all cases covered – and this over the complete three month period for which
we are updating the calibration with only a few outliers.
DX Analytics – Square-Root Jump Diffusion
333
FIGURE 14.10 Square-root jump diffusion parameters and MSE values from the calibration to one
maturity without penalization.
334
FIGURE 14.11
LISTED VOLATILITY AND VARIANCE DERIVATIVES
Histogram of MSE values for SRJD calibration to one maturity without
penalization.
14.6 PYTHON SCRIPTS
14.6.1 dx srjd calibration.py
#
# Calibration of Square–Root Jump Diffusion (SRJD)
# model to VSTOXX call options with DX Analytics
#
# All data from www.eurexchange.com
#
# (c) Dr. Yves J. Hilpisch
# Listed Volatility and Variance Derivatives
#
import dx
import time
import numpy as np
import pandas as pd
import datetime as dt
import scipy.optimize as spo
import matplotlib.pyplot as plt
import seaborn as sns ; sns.set()
import matplotlib
DX Analytics – Square-Root Jump Diffusion
matplotlib.rcParams['font.family'] = 'serif '
from copy import deepcopy
# importing the data
h5 = pd.HDFStore('../data/vstoxx_march_2014.h5', 'r')
vstoxx_index = h5['vstoxx_index']
vstoxx_futures = h5['vstoxx_futures']
vstoxx_options = h5['vstoxx_options']
h5.close()
# collecting the maturity dates
third_fridays = sorted(set(vstoxx_futures['MATURITY']))
# instantiation of market environment object with dummy pricing date
me_vstoxx = dx.market_environment('me_vstoxx', dt.datetime(2014, 1, 1))
me_vstoxx.add_constant('currency', 'EUR')
me_vstoxx.add_constant('frequency', 'W')
me_vstoxx.add_constant('paths', 5000)
# constant short rate model with somewhat arbitrary rate
csr = dx.constant_short_rate('csr', 0.01)
me_vstoxx.add_curve('discount_curve', csr)
# parameters to be calibrated later, dummies only
# SRD part
me_vstoxx.add_constant('kappa', 1.0)
me_vstoxx.add_constant('theta', 20)
me_vstoxx.add_constant('volatility', 1.0)
# jump part
me_vstoxx.add_constant('lambda', 0.5)
me_vstoxx.add_constant('mu', -0.2)
me_vstoxx.add_constant('delta', 0.1)
# payoff function for all European call options
payoff_func = 'np.maximum(maturity_value – strike, 0)'
tol = 0.2
# otm & itm moneyness tolerance
first = True
# flag for first calibration
def srjd_get_option_selection(pricing_date, tol=tol):
' ' ' Function to select option quotes from data set.
Parameters
==========
pricing_date: datetime object
date for which the calibration shall be implemented
tol: float
moneyness tolerace for OTM and ITM options to be selected
Returns
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LISTED VOLATILITY AND VARIANCE DERIVATIVES
=======
option_selection: DataFrame object
selected options quotes
futures: DataFrame object
futures prices at pricing_date
'''
option_selection = pd.DataFrame()
mats = [third_fridays[3],]
# list of maturity dates
# select the relevant futures prices
futures = vstoxx_futures[(vstoxx_futures.DATE == pricing_date)
& (vstoxx_futures.MATURITY.apply(lambda x: x in mats))]
# collect option data for the given option maturities
for mat in mats:
forward = futures[futures.MATURITY == mat]['PRICE'].values[0]
option_selection = option_selection.append(
vstoxx_options[(vstoxx_options.DATE == pricing_date)
& (vstoxx_options.MATURITY == mat)
& (vstoxx_options.TYPE == 'C')
# only calls
& (vstoxx_options.STRIKE > (1 - tol) * forward)
& (vstoxx_options.STRIKE < (1 + tol) * forward)])
return option_selection, futures
def srd_forward_error(p0):
' ' ' Calculates the mean-squared error for the
term structure calibration for the SRD model part.
Parameters
===========
p0: tuple/list
tuple of kappa, theta, volatility
Returns
=======
MSE: float
mean-squared error
'''
global initial_value, f, t
if p0[0] < 0 or p0[1] < 0 or p0[2] < 0:
return 100
f_model = dx.srd_forwards(initial_value, p0, t)
MSE = np.sum(( f - f_model) ** 2) / len(f)
return MSE
def generate_shift_base(pricing_date, futures):
' ' ' Generates the values for the deterministic shift for the
SRD model part.
DX Analytics – Square-Root Jump Diffusion
Parameters
==========
pricing_date: datetime object
date for which the calibration shall be implemented
futures: DataFrame object
futures prices at pricing_date
Returns
=======
shift_base: ndarray object
shift values for the SRD model part
'''
global initial_value, f, t
# futures price array
f = list(futures['PRICE'].values)
f.insert(0, initial_value)
f = np.array(f)
# date array
t = [_.to_pydatetime() for _ in futures['MATURITY']]
t.insert(0, pricing_date)
t = np.array(t)
# calibration to the futures values
opt = spo.fmin(srd_forward_error, (2., 15., 2.))
# calculation of shift values
f_model = dx.srd_forwards(initial_value, opt, t)
shifts = f - f_model
shift_base = np.array(( t, shifts)).T
return shift_base
def srjd_get_option_models(pricing_date, option_selection, futures):
' ' ' Function to instantiate option pricing models.
Parameters
==========
pricing_date: datetime object
date for which the calibration shall be implemented
maturity: datetime object
maturity date for the options to be selected
option_selection: DataFrame object
selected options quotes
Returns
=======
vstoxx_model: dx.square_root_diffusion
model object for VSTOXX
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LISTED VOLATILITY AND VARIANCE DERIVATIVES
option_models: dict
dictionary of dx.valuation_mcs_european_single objects
'''
global initial_value
# updating the pricing date
me_vstoxx.pricing_date = pricing_date
# setting the initial value for the pricing date
initial_value = vstoxx_index['V2TX'][pricing_date]
me_vstoxx.add_constant('initial_value', initial_value)
# setting the final date given the maturity dates
final_date = max(futures.MATURITY).to_pydatetime()
me_vstoxx.add_constant('final_date', final_date)
# adding the futures term structure
me_vstoxx.add_curve('term_structure', futures)
# instantiating the risk factor (VSTOXX) model
vstoxx_model = dx.square_root_jump_diffusion_plus('vstoxx_model',
me_vstoxx)
# generating the shift values and updating the model
vstoxx_model.shift_base = generate_shift_base(pricing_date, futures)
vstoxx_model.update_shift_values()
option_models = {}
# dictionary collecting all models
for option in option_selection.index:
# setting the maturity date for the given option
maturity = option_selection['MATURITY'].ix[option]
me_vstoxx.add_constant('maturity', maturity)
# setting the strike for the option to be modeled
strike = option_selection['STRIKE'].ix[option]
me_vstoxx.add_constant('strike', strike)
# instantiating the option model
option_models[option] =
\
dx.valuation_mcs_european_single(
'eur_call_ %d' % strike,
vstoxx_model,
me_vstoxx,
payoff_func)
return vstoxx_model, option_models
def srjd_calculate_model_values(p0):
' ' ' Returns all relevant option values.
Parameters
===========
p0: tuple/list
tuple of kappa, theta, volatility, lamb, mu, delt
DX Analytics – Square-Root Jump Diffusion
Returns
=======
model_values: dict
dictionary with model values
'''
# read the model parameters from input tuple
kappa, theta, volatility, lamb, mu, delt = p0
# update the option market environment
vstoxx_model.update(kappa=kappa,
theta=theta,
volatility=volatility,
lamb=lamb,
mu=mu,
delt=delt)
# estimate and collect all option model present values
results = [option_models[option].present_value(fixed_seed=True)
for option in option_models]
# combine the results with the option models in a dictionary
model_values = dict(zip(option_models, results))
return model_values
def srjd_mean_squared_error(p0, penalty=True):
' ' ' Returns the mean-squared error given
the model and market values.
Parameters
===========
p0: tuple/list
tuple of kappa, theta, volatility
Returns
=======
MSE: float
mean-squared error
'''
# escape with high value for non-sensible parameter values
if (p0[0] < 0 or p0[1] < 5. or p0[2] < 0 or p0[2] > 10.
or p0[3] < 0 or p0[4] < 0 or p0[5] < 0):
return 1000
# define/access global variables/objects
global option_selection, vstoxx_model, option_models, first, last
# calculate the model values for the option selection
model_values = srjd_calculate_model_values(p0)
option_diffs = {}
# dictionary to collect differences
for option in model_values:
# differences between model value and market quote
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LISTED VOLATILITY AND VARIANCE DERIVATIVES
option_diffs[option] = (model_values[option]
-
option_selection['PRICE'].loc[option])
# calculation of mean-squared error
MSE = np.sum(np.array(option_diffs.values()) ** 2) / len(option_diffs)
if first is True:
# if in first optimization, no penalty
penalty = 0.0
else:
# if 2, 3, ... optimization, penalize deviation from previous
# optimal parameter combination
penalty =(np.sum(( p0 - last) ** 2))
if penalty is False:
return MSE
return MSE + penalty
def srjd_get_parameter_series(pricing_date_list):
' ' ' Returns parameter series for the calibrated model over time.
Parameters
==========
pricing_date_list: pd.DatetimeIndex
object with relevant pricing dates
Returns
=======
parameters: pd.DataFrame
DataFrame object with parameter series
'''
# define/access global variables/objects
global initial_value, futures, option_selection, vstoxx_model,
\
option_models, first, last
parameters = pd.DataFrame()
# DataFrame object to collect
parameter series
for pricing_date in pricing_date_list:
# setting the initial value for the VSTOXX
initial_value = vstoxx_index['V2TX'][pricing_date]
# select relevant option quotes
option_selection, futures = srjd_get_option_selection
(pricing_date)
# instantiate all model given option selection
vstoxx_model, option_models = srjd_get_option_models(pricing_date,
option_selection,
futures)
# global optimization to start with
opt = spo.brute(srjd_mean_squared_error,
((1.25, 6.51, 0.75),
# range for kappa
341
DX Analytics – Square-Root Jump Diffusion
(10., 20.1, 2.5),
# range for theta
(0.5, 10.51, 2.5),
# range for volatility
(0.1, 0.71, 0.3),
# range for lambda
(0.1, 0.71, 0.3),
# range for mu
(0.1, 0.21, 0.1)),
# range for delta
finish=None)
# local optimization
opt = spo.fmin(srjd_mean_squared_error, opt,
maxiter=550, maxfun=650,
xtol=0.0000001, ftol=0.0000001);
# calculate MSE for storage
MSE = srjd_mean_squared_error(opt, penalty=False)
# store main parameters and results
parameters = parameters.append(
pd.DataFrame(
{'date' : pricing_date,
'initial_value' : vstoxx_model.initial_value,
'kappa' : opt[0],
'theta' : opt[1],
'sigma' : opt[2],
'lambda' : opt[3],
'mu' : opt[4],
'delta' : opt[5],
'MSE' : MSE},
index=[0]), ignore_index=True)
first = False
last = opt
# set to False after first iteration
# store optimal parameters for reference
print ("Pricing Date %s" % str(pricing_date)[:10]
+ " | MSE %6.5f" % MSE)
return parameters
def srjd_plot_model_fit(parameters):
# last pricing date
pdate = max(parameters.date)
# optimal parameters for that date and the maturity
opt = np.array(parameters[parameters.date == pdate][[
'kappa', 'theta', 'sigma', 'lambda', 'mu', 'delta']])[0]
option_selection, futures = srjd_get_option_selection(pdate, tol=tol)
vstoxx_model, option_models = srjd_get_option_models(pdate,
option_selection,
futures)
model_values = srjd_calculate_model_values(opt)
model_values = pd.DataFrame(model_values.values(),
index=model_values.keys(),
columns=['MODEL'])
option_selection = option_selection.join(model_values)
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LISTED VOLATILITY AND VARIANCE DERIVATIVES
mats = set(option_selection.MATURITY.values)
mats = sorted(mats)
# arranging the canvas for the subplots
height = max(8, 2 * len(mats))
if len(mats) == 1:
mat = mats[0]
fig, axarr = plt.subplots(2, figsize=(10, height))
os = option_selection[option_selection.MATURITY == mat]
strikes = os.STRIKE.values
axarr[0].set_ylabel('%s' % str(mat)[:10])
axarr[0].plot(strikes, os.PRICE.values, label='Market Quotes')
axarr[0].plot(strikes, os.MODEL.values, 'ro', label='Model Prices')
axarr[0].legend(loc=0)
wi = 0.3
axarr[1].bar(strikes - wi / 2, os.MODEL.values os.PRICE.values,
width=wi)
axarr[0].set_xlabel('strike')
axarr[1].set_xlabel('strike')
else:
fig, axarr = plt.subplots(len(mats), 2, sharex=True,
figsize=(10, height))
for z, mat in enumerate(mats):
os = option_selection[option_selection.MATURITY == mat]
strikes = os.STRIKE.values
axarr[z, 0].set_ylabel('%s' % str(mat)[:10])
axarr[z, 0].plot(strikes, os.PRICE.values, label='Market Quotes')
axarr[z, 0].plot(strikes, os.MODEL.values, 'ro', label='Model
Prices')
axarr[z, 0].legend(loc=0)
wi = 0.3
axarr[z, 1].bar(strikes - wi / 2,
os.MODEL.values - os.PRICE.values, width=wi)
axarr[z, 0].set_xlabel('strike')
axarr[z, 1].set_xlabel('strike')
plt.savefig('../images/dx_srjd_cali_1_fit.pdf')
if _ _name_ _ is '_ _main_ _ ':
t0 = time.time()
# selecting the dates for the calibration
pricing_date_list = pd.date_range('2014/3/1', '2014/3/31', freq='B')
# conducting the calibration
parameters = srjd_get_parameter_series(pricing_date_list)
# storing the calibation results
date = str(dt.datetime.now())[:10]
h5 = pd.HDFStore('../data/srjd_calibration_%s_%s_%s' %
(me_vstoxx.get_constant('paths'),
DX Analytics – Square-Root Jump Diffusion
me_vstoxx.get_constant('frequency'),
date.replace('–', '_')), 'w')
h5['parameters'] = parameters
h5.close()
# plotting the parameter time series data
fig1, ax1 = plt.subplots(1, figsize =(10, 12))
to_plot = parameters.set_index('date')[
['kappa', 'theta', 'sigma',
'lambda', 'mu', 'delta', 'MSE']]
to_plot.plot(subplots=True, color='b', title='SRJD', ax=ax1)
plt.savefig('../images/dx_srjd_cali_1.pdf')
# plotting the histogram of the MSE values
fig2, ax2 = plt.subplots()
dat = parameters.MSE
dat.hist(bins=30, ax=ax2)
plt.axvline(dat.mean(), color='r', ls='dashed',
lw=1.5, label='mean = %5.4f' % dat.mean())
plt.legend()
plt.savefig('../images/dx_srjd_cali_1_hist.pdf')
# plotting the model fit at last pricing date
srjd_plot_model_fit(parameters)
# measuring and printing the time needed for the script execution
print "Time in minutes %.2f" % (( time.time() - t0) / 60)
343
Listed Volatility and Variance Derivatives: A
Python-based Guide
By Dr. Yves J. Hilpisch
© 2017 Yves Hilpisch
Bibliography
Bennett, Collin and Miguel Gil (2012) Volatility Trading – Trading Volatility, Correlation, Term Structure
and Skew. Banco Santander.
Björk, Tomas (2009) Arbitrage Theory in Continuous Time. 3rd ed., Oxford University Press, Oxford.
Black, Fischer (1976) “The Pricing of Commodity Contracts.” Journal of Financial Economics, Vol. 3,
167–179.
Black, Fischer and Myron Scholes (1973) “The Pricing of Options and Corporate Liabilities.” Journal
of Political Economy, Vol. 81, No. 3, 637–654.
Bossu, Sebastian (2014) Advanced Equity Derivatives: Volatility and Correlation. Wiley Finance.
Bossu, Sebastien, Eva Strasser, Regis Guichard (2005) “Just What You Need To Know About Variance
Swaps.” JPMorgan.
Breeden, Douglas and Robert Litzenberger (1978) “Prices of State-Contingent Claims Implicit in Option
Prices.” Journal of Business, Vol. 51, No. 4, 621–651.
Brenner, Menachem and Dan Galai (1989) “New Financial Instruments for Hedging Changes in Volatility.” Financial Analysts Journal, Jul/Aug, Vol. 45, No. 4.
Brigo, Damiano and Fabio Mercurio (2001) “On Deterministic-Shift Extensions of Short-Rate Models.”
Working Paper, Banca IMI, Milano, www.damianobrigo.it.
Carr, Peter and Roger Lee (2009) “Volatility Derivatives.” Annual Review of Financial Economics,
Vol. 1, 1–21.
Chicago Board Options Exchange (2003) “The CBOE Volatility Index – VIX.” White Paper.
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Cox, John, Jonathan Ingersoll and Stephen Ross (1985) “A Theory of the Term Structure of Interest
Rates.” Econometrica, Vol. 53, No. 2, 385–407.
Duffie, Darrell, Jun Pan and Kenneth Singleton (2000) “Transform Analysis and Asset Pricing for Affine
Jump-Diffusions.” Econometrica, Vol. 68, No. 6, 1343–1376.
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Measure.” The Journal of Futures Markets, Vol. 15, No. 3, 265–302.
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New Jersey.
Grünbichler, Andreas and Francis Longstaff (1996) “Valuing Futures and Options on Volatility.” Journal
of Banking and Finance, Vol. 20, 985–1001.
Guobuzaite, Renata and Lionel Martellini (2012) “The Benefits of Volatility Derivatives in
Equity Portfolio Management.” EDHEC Risk Institute, http://www.eurexchange.com/exchange-en/
about-us/news/60036/.
Haenel, Valentin, Emmanuelle Gouillart and Gaël Varoquaux (2013) “Python Scientific Lecture Notes.”
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Hilpisch, Yves (2015) Derivatives Analytics with Python – Data Analysis, Models, Simulation, Calibration and Hedging. Wiley Finance, http://derivatives-analytics-with-python.com.
Hilpisch, Yves (2014) Python for Finance — Analyze Big Financial Data. O’Reilly, http://python-for
-finance.com.
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BIBLIOGRAPHY
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Berlin.
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www.ssrn.com.
McKinney, Wes (2012) Python for Data Analysis – Data Wrangling with Pandas, NumPy, and IPython.
O”Reilly, Beijing.
Merton, Robert (1973) “Theory of Rational Option Pricing.” Bell Journal of Economics and Management
Science, Vol. 4, No. 1, 141–183.
Psychoyios, Dimitris (2005) “Pricing Volatility Options in the Presence of Jumps.” Working Paper,
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Volatility Options and Futures.” Review of Quantitative Finance and Accounting, Vol. 35, No. 3,
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of Derivatives, Vol. 1, No. 1, 71–84.
Listed Volatility and Variance Derivatives: A
Python-based Guide
By Dr. Yves J. Hilpisch
© 2017 Yves Hilpisch
Index
accumulated return on modified variation
margin, 251, 256–57, 279
additivity in time, 247–50
ARMVM, 251, 256–57, 279
Arrow-Debreu security, 56
asset pricing theory, 183
bar grams, 35, 37
base data retrieval, 71–78
data set combining, 76–77
data set saving, 78
EURO STOXX 50 data, 71–74
VSTOXX data, 74–76
Bjork, T., 4, 284
Black-Scholes-Merton model, 3–5, 59–60,
235, 284
Bossu, S., 5, 235
Breeden, D., 58, 67
Brenner, M., 7
Brigo, D., 184–85, 190
calculation
interest rates, 279
intraday trading, 252, 270–74
longest drawdown, 53
maximum drawdown, 52–53
options data, 95–106
processes, 114–18, 221–22
python scripts, 118–28
realized variance, 230–34
sub-indexes, 105–14
variance futures, 258–65
VSTOXX sub-indexes, 105–14
calibration…See model calibration; term
structure calibration
call_estimator function, 137, 139–40
call options
dollar gamma formula, 60–61
estimation, 137, 139–40
implied volatility, 6
market quotes, 179–82
Monte Carlo estimation, 135–40
pricing formula, 132–35
quote calibration, 156, 317
call_price function, 132–35
Carr, P., 8
collect_option_series function,
96–97, 98, 104
compute_subindex function, 107–8
constant proportion investment strategies,
87–93
active investment strategies, 89–90
passive investment strategies, 88–89
control structures, 22–23, 25
correlation analysis, 83–87
correlation over time, 83–87
negative correlation, 85–86
rolling correlation, 87
correlation trading, 13
Cox, J., 129, 184
cx function, 132–33
daily margins
accumulated return on modified variation
margin, 255–57
after trade matching, 277–79
discount factors, 253–54
futures trading, 252–57
data analysis
base data retrieval, 71–78
base data set combining, 76–77
basic analysis procedures, 78–82
347
348
data analysis (Continued)
constant proportion investment strategies,
87–93
correlation analysis, 83–87
data_collection function, 100–101,
104
delta hedging, 4–5
derivatives modeling, 283, 287–90
derivatives portfolios, 283, 290–92
modeling, 292–93
simulation and valuation, 293–94
derivatives positions, 292
deterministic shift approach, 184, 190, 192,
209–10
discount factors, 253–54
discretization, 183, 192–93
dollar gamma derivatives, 58–65
dollar_gamma function, 60
Duffie, D., 182
DX Analytics, 283, 296
computing resources, 283
derivatives modeling, 283, 287–90
derivatives portfolios, 283, 290–92
modeling, 292–93
simulation and valuation, 293–94
development influences, 283
global valuation approach, 283
risk factor modeling, 283–87
risk reports, 294–96
square root diffusion, 297
data import and selection, 297–301
VSTOXX model calibration, 303–8
VSTOXX options modeling, 301–3
square root jump diffusion
calibration results, 325–32
calibration without penalties, 331–32
option quote calibration, 317
term structure calibration, 317
VSTOXX option modeling, 315–20
dx_square_root_diffusion, 301
dx_srd_calibration.py, 301,
308–14
dx_srjd_calibration.py, 315–20,
334–43
elementary securities, 58
Eonia rates, 255–57
error_function, 157–58
INDEX
errors
error ratios, 149–53
mean squared errors, 304
mean squared relative errors (MSRE), 155,
158–59, 161–62
plotting, 149–50
volatility derivatives, 149–53
Eurex, 8–9, 251
Eurex T7 trading system, 252
EURO STOXX 50 data
base data retrieval, 71–74
basic data analysis, 79–82
constant proportion strategies, 87–93
correlation analysis, 83–87
EURO STOXX 50 index
composition, 221
development, 9
variance futures calculation, 258–65
variance swaps, 242–47, 251
variance trading strategies, 13
first day margins, 275–76
Fleming, J., 7
forward volatility/ variance trading, 12–13
futures term structures, 184–90
shifted volatility processes, 190–91
full truncation scheme, 193–95
fungibility, 251
futures_price function, 130–32
Galai, D., 7
generate_paths function, 137–38
generate_shift_base function, 318
generator objects, 25–28
geometric Brownian motions, 284–87
get_data_from_www function, 98–100
get_data function, 97–98
get_instrument_values function,
285
get_port_risk function, 294
Goubuzaite, R., 88
Grünbichler, A., 129, 132–33, 140–41,
154–55, 163, 184, 188
HDFStore, 46, 78
hedging
delta hedging, 4–5
static hedges, 250
349
Index
histograms, 35–36, 45
historical volatility formula, 6
if-elif-else control structure, 23, 26
implied volatility formula, 6
index_collect_option_data.py,
118–23
index_option_series function, 101
index_subindex_calculation.py,
105, 123–27
index_vstoxx_calculation
function, 114–16
index_vstoxx_calculation.py,
114–16, 127–28
instantaneous volatility formula, 6
interest rates, 279
intraday trading, 252, 270–74
investment strategies
active investment strategies, 89–90
constant proportion strategies, 87–93
passive investment strategies, 88–89
jointplot function, 86
jump modeling…See SRJD modeling
lambda functions, 24
Lee, R., 8
linspace operators, 33
liquidity, 251
listed volatility, 7
Litzenberger, R., 58, 67
log contracts, 57
log_rets function, 81–82
log returns, 81–87
Longstaff, F., 129, 132–33, 140–41, 154–55,
163, 184, 188
Lord, R., 183
make_subindex function, 108–10
margin payments, 251
market, 292
market disruption events, 280
market environment, 284–85
market expectation, 11
market integrity, 251
Martellini, L., 88
matplotlib, 34–38
matrix, 35
maturity_list function, 142
mean-reverting, 11
mean squared errors, 304
mean squared relative errors, 155, 158–59,
161–62
Mercurio, F., 184–85, 190
merge_and_filter function, 98–99, 104
model calibration, 153–54
DX Analytics
calibration results, 326–32
calibration without penalties, 331–32
option quote calibration, 317
square root diffusion, 303–8
square root jump diffusion, 317–34
term structure calibration, 317
option quotes, 154–55, 161–62
procedures, 155–59
results, 159–63
results plotting, 202–3
SRJD modeling, 196–209
VSTOXX square root diffusion, 303–8
VSTOXX square root jump diffusion,
317–34
model_calibration function, 158–59
Monte Carlo simulation, 135–40, 283
automated testing, 141–44
option valuation, 191–95
storage functions, 145–46
technical implementation, 192–95
test results, 146–53
MSRE, 155, 158–59, 161–62
MultiIndex, 108–9
ndarray objects, 28–34
negative correlations, 12, 84–86
net present value (NPV)
effects, 251
variance futures, 252–57
non-tradeable assets, 11
NumPy, 17, 28–34
benefits of, 29–31, 33
linspace operators, 33
ndarray objects, 28–34
standard arrays, 29
universal functions, 31–32
vectorized operations, 30, 32
350
open source, 14
option pricing, 58
delta hedging, 4
forumula, 132–35
hedging, 3–5
implied volatilities, 67
market quotes, 179–82
risk-neutral expectations, 4
spanning with options, 56–57
option quotes
market pricing, 179–82
model calibration, 154–55, 161–62
option valuation
Monte Carlo simulation, 191–95
pandas, 17, 38
DataFrame objects, 39–45, 61
data retrieval functions, 47–53
Input-Output operations, 45–47
longest drawdown calculation, 53
maximum drawdown calculation,
52–53
parameterization, 130
parse_data function, 98–100
par variance strikes, 258
passive investment strategies, 88–89
paths_list function, 142
plot_calibration_results, 159
plot_calibration_results
function, 202–3
plot_error_ratio function, 149
plotting
bar charts, 35, 37
calibration results, 159, 202–3
different time series, 83–85
errors, 149–50
histograms, 35–36, 45
with matplotlib, 34–38
plot lines, 34–35
seaborn plotting library, 86
polyfit, 37
polyval, 37
portfolio deltas, 294
portfolio objects, 293
portfolios…See also derivatives portfolios
contents, 292
portfolio vegas, 295
positive jumps, 12
INDEX
pricing
asset pricing theory, 183
futures pricing
formula, 130–32
futures settlement prices, 258
traded futures prices, 257, 273, 275
Monte Carlo simulation, 135–40
automoted tests, 141–44
storage functions, 145–46
test results, 146–53
option pricing
delta hedging, 4
elementary securities, 58
formula, 132–35
hedging, 3–5
implied volatilities, 67
market quotes, 179–82
risk-neutral expectations, 4
spanning with options, 56–57
traded futures, 257, 273, 276
print_results function, 147
PY1, 142
PY2, 142
Python
benefits of, 14, 26–27, 30
control structures, 22–23, 25
data structures, 20–22
data types, 17–20
generator objects, 25–28
interactivity of, 14
lambda functions, 24
matplotlib, 34–38
ndarray objects, 28–34
NumPy, 17, 28–34
special idioms, 23–28
standard arrays, 29
Python scripts
dx_square_root_diffusion, 301
dx_srd_calibration.py, 301,
308–14
dx_srjd_calibration.py, 315–20,
334–43
index_collect_option_data.py,
118–23
index_subindex_calculation.py,
105, 123–27
index_vstoxx_calculation.py,
114–16, 127–28
Index
srd_functions.py, 163–67
srd_model_calibration.py, 155,
174–78
srd_simulation_analysis.py,
141–43, 145, 148, 167–71
srd_simulation_results.py,
145–47, 171–74
srjd_fwd_calibration.py, 186,
210–12
srjd_model_calibration.py,
196–98, 215–19
srjd_simulation.py, 192–95,
212–15
random_number_gen function, 193
random numbers, 192–93
read_select_quotes function, 156
realized variance
calculation, 230–34
formula, 6–7, 229
principles, 229–34
traded variance strikes, 257, 273
variance futures, 252
vs. volatility, 249–50
realized volatility
formula, 6
volatility trading, 12
vs. variance, 249–50
recalculation…See calculation
replication of variance
constant dollar gamma derivatives, 58–65
examples, 59–65
log contracts, 57
spanning with options, 56–57
static replication, 57–58
VSTOXX volatility index, 66–67
risk, broad measures, 250
risk factor modeling, 283–87
risk neutral discounting, 284
risk_report function, 295
rolling correlation, 87
runs function, 142
seed function, 142
shifted volatility processes, 190–91
shift values, 317–18
sim_results function, 146
351
Sinclair, E., 4, 6
S&P 500 variance futures contract, 8
square-root jump diffusion modeling…See
SRJD
srd_calculate_model_values, 303,
304–5
srd_error_function, 197–98
srd_forward_error, 317–18
srd_forwards function, 186–89
srd_functions.py, 163–67
srd_fwd_error function, 187–88
srd_get_option_selection
function, 300–301
srd_get_parameter_series
function, 305–6
srd_mean_squared error function, 304
srd_model_calibration function,
198–99
srd_model_calibration.py, 155,
174–78
srd_simulation_analysis.py,
141–43, 145, 148, 167–71
srd_simulation_results.py,
145–47, 171–74
srjd_calculate_model_values
function, 320–21
srjd_call_valuation function, 196
srjd_fwd_calibration.py, 186,
210–12
srjd_get_option_models function,
319–20
srjd_get_option_selection
function, 316–17, 324–25
srjd_get_parameter_series
function, 322–23
srjd_mean_squared_error function,
321
srjd_model_calibration.py,
196–98, 215–19
SRJD modeling, 179, 182–83
all maturities models, 206–9
DX Analytics
calibration results, 325–32
calibration without penalties, 331–32
option quote calibration, 317
term structure calibration, 317
VSTOXX option modeling, 315–20
352
SRJD modeling (Continued)
four maturities models, 203–6
futures term structure, 184–91
model calibration, 196–209
Monte Carlo valuation, 191–96
risk-neutral dynamics, 182
shifted volatility process, 190–91
short maturity, 189–201
technical implementation, 192–95
term structure calibration, 183–91
two maturities models, 201–3
srjd_plot_model_fit function, 323–24
srjd_simulation function, 193–96
srjd_simulation.py, 192–95, 212–15
srjd_valuation function, 198–99
standardization, 251
start_collecting function, 97
state prices, 58
static hedges, 250
static replication, 57–58
steps_list function, 142
strike_list function, 142
sub-indexes
algorithms, 106–14
calculation/recalculation, 105–14
functions, 105, 108–10, 123–27
syntax, 14
term structure calibration, 183–91, 317
time_grid attribute, 286
traded variance strikes, 257, 273, 275
trade matching, 274, 277–79
trading
intraday trading, 252, 270–74
trade matching, 274, 277–79
volatilities comparisons, 275–76
transparency, 251
tuples, 20
V2TX, 222
valuation, of volatility derivatives
error ratios, 149–53
framework for, 129–30
model calibration, 153–54, 174–78
option quotes, 154–55, 161–62
procedures, 155–59
results, 159–63
INDEX
Monte Carlo simulation, 135–40
automated testing, 141–44
storage functions, 145–46
test results, 146–53
pricing formulae
futures pricing, 130–32
option pricing, 132–35
valuation environments, 293
valuation_function, 156–58
variance
additivity in time, 247–50
realized, 6–7
replication
constant dollar gamma derivatives,
58–65
examples, 59–65
log contracts, 57
spanning with options, 56–57
static replication, 57–58
VSTOXX, 66–67
squared variations, 247
types, 6–7
vs. volatility, 247–50
variance futures, 5
calculation examples, 258–65
end-of-day trade conversions, 252
Eurex, 251
futures settlement prices, 258
intraday trading, 252, 270–74
key terms, 269–70
net present value, 252–57
net present value (NPV) effects, 251
par variance strikes, 258
realized variance, 252
standardization benefits, 251
traded futures prices, 257, 273, 276
traded variance strikes, 257, 273, 275
trade matching, 274
variance swaps, compared, 265–67
variance reduction, 192
variance swaps, 5
constant dollar gamma, 58–59
definition, 235
EURO STOXX, 242–47, 251
examples, 235–39
market-to-market valuation, 239–41
payoff, 235
353
Index
variance futures, compared, 265–67
vega sensitivity, 241–42
VSTOXX, 66–67
variance trading
par variance strikes, 258
strategies, 13
traded variance strikes, 257, 273, 275
var_red function, 142
VDAX volatility index, 8–9
vega notional times
trade matching, compared, 274
vega sensitivity
variance swaps, 241–42
VIX, 7–8
volatility
EU development, 8–9
importance, 7
listed volatility, generally, 7
over time, formula, 129–30
realized, 6, 12, 249–50
types, 6
US development, 7–8
vs. variance, 247–50
volatility derivatives
valuation
error ratios, 149–53
framework for, 129–30
model calibration, 153–54, 174–78
option quotes, 154–55, 161–62
procedures, 155–59
results, 159–63
Monte Carlo simulation, 135–40
automated testing, 141–44
storage functions, 145–46
test results, 146–53
pricing formulae
futures pricing, 130–32
option pricing, 132–35
volatility indexes…See also
EURO STOXX; VDAX;
VSTOXX
historical development, 7–9
volatility of volatility indexes,
9–10
volatility strikes, 235
volatility trading
characteristics, 11–13
futures
long futures, 12–13
relative value arbitrage, 13
short futures, 12–13
term structure arbitrage, 12
higher than realized volatility, 12
market expectation, 11
mean-reverting nature, 11
negative correlation, 12
non-tradable assets, 11
options
long ATM straddle, 13
long OTM calls, 13
short ATM calls, 13
positive jumps, 12
strategies, 12–13
correlation trading, 13
forward variance trading, 13
long futures, 12–13
relative value arbitrage, 13
short futures, 12–13
term structure arbitrage, 12
VSTOXX 30-day Index, 221–22
VSTOXX data
base data retrieval, 74–76
basic data analysis, 79–82
constant proportion strategies, 87–93
correlation analysis, 83–87
VSTOXX volatility index
background, 95, 221
block trade size, 224
calculation/ recalculation
options data, 95–106
processes, 114–18, 221–22
python scripts, 118–28
sub-indexes, 105–14
characteristics, 11–12, 221–22
data collection
options, 95–106
development, 9
exercise prices, 224
futures
contract terms, 223–24
options, 9, 13, 95–96
contract terms, 224–25
data collection, 95–106
market quotes, 179–82
354
VSTOXX volatility index (Continued)
options modeling
DX Analytics square root diffusion,
301–3
DX Analytics SRJD, 315–20
model calibration, 303–8
settlement dates, 224
settlement prices, 224
sub-indexes, 66, 95–96
algorithms, 106–14
calculation, 105–14
trading days, 224
INDEX
trading hours, 224
variance
replication of, 66–67
trading strategies, 12–13
vendor codes, 224
VVIX, 9–10
V-VSTOXX, 10
Whaley, R., 8
write function, 142
yield statements, 28
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