Thesis: Pricing VIX Options with Wishart Matrix Affine Jump Diffusions ...

Pricing VIX Options with Wishart Matrix Affine Jump Diffusions while Preserving Consistency with SPX Options

Master’s Thesis

Submitted in partial fulfillment of the requirements for the degree of Master of Science in Quantitative Finance

Author

Pascal Marco Caversaccio born on February 27, 1990 citizen of Switzerland

Supervisor

Prof. Dr. Markus Leippold Hans Vontobel Professor of Financial Engineering Department of Banking and Finance University of Zurich

c Zurich, Switzerland, January 8, 2014

This thesis is dedicated to my parents for their love, endless support and encouragement.

Task Assignment Topic: ”Pricing VIX Options with Wishart Matrix Affine Jump Diffusions while Preserving Consistency with SPX Options” By Prof. Dr. Markus Leippold: ”After the financial crisis, the notion of liquidity risk (should have) caught the awareness of all financial market participants. Since options on the Chicago Board Options Exchange (CBOE) volatility index (VIX) belong to the class of most liquid options around the world, they are well suited for hedging purposes and require therefore an accurate pricing model. Due to the negative correlation between the VIX and the Standard & Poor’s 500 (SPX) index, a VIX call can also be seen as a put option on the SPX. This (negative) linkage of the two markets needs to be incorporated into an integrated pricing model. Accordingly, this thesis faces the challenge of pricing VIX options while preserving consistency with the SPX options. We choose a highly flexible and tractable affine framework which has gained significant attention in the finance literature and in the theory of stochastic processes over the recent years. The thesis should satisfy the following criteria: • • • • •

An overview of the related literature. Analysis of the stylized facts of the VIX and SPX market data. The model setup and transform analysis. A specific section on the numerical implementation and its design. Provide a detailed description of the estimations, mathematical proofs and data treatment in the thesis. These should either be included in the main body of the text or in the appendix.

You are invited to adopt some line of proceeding for yourself. The issues raised above serve as guidelines and some may be added or discarded upon discussion with your supervisor.”

Information Title: Pricing VIX Options with Wishart Matrix Affine Jump Diffusions while Preserving Consistency with SPX Options Author: Pascal Marco Caversaccio E-Mail: [email protected] Mobile: +41 79 604 88 69 Student ID: 09-100-371 Supervisor: Prof. Dr. Markus Leippold c Date: January 8, 2014 Department of Banking and Finance University of Zurich Plattenstrasse 14, 8032 Zurich, Switzerland http://www.bf.uzh.ch/ I

Acknowledgements First of all, I want to thank my supervisor Professor Dr. Markus Leippold for giving me the opportunity to work on this extremely interesting subject and agreeing to supervise this thesis. I also owe him an invaluable debt for his support, comments and fruitful discussions during the writing of this thesis. Moreover, Professor Leippold’s course on financial engineering was among the most valuable and rewarding lectures in my M.Sc. curriculum and enriched, fostered and facilitated my interest and knowledge in this particular field. The foundation of this work can therefore, almost surely, be traced back to this particular lecture. I’m also very grateful for the possible opportunity to undertake my Ph.D. at Professor Leippold’s chair which spans my research interests on a wide range. Furthermore, I would like to thank Chris Bardgett, Ph.D. student at the Swiss Finance Institute, for various very helpful discussions and comments which have considerably improved the content of this thesis. Even though he was not officially assigned to assist me, he provided tremendously important answers to my questions and appeared encumbrances. I also would like to thank him for providing me access to OptionMetrics which facilitated the empirical data investigation. I would also like to mention all my fellow students, in particular Laurent Oberholzer, Tom Noppe and Ryan Kurniawan, with whom I enjoyed very interesting discussions on numerical mathematics and financial engineering. Further, a particular mention goes to my girlfriend Raquel who always tolerated my busy weekends and nights, and always stood behind me. Finally, I’m infinitely indebted to my parents who unrelentingly supported me and provided the framework to my educational development, and, additionally of course, for their continuous encouragement. Everything that has a beginning comes to an end. — Marcus Fabius Quintilianus (A.D. 35 – 100), Roman Rhetorician

II

Abstract This thesis addresses the challenge of pricing volatility index (VIX) options while preserving consistency with the Standard & Poor’s 500 index (SPX) options. We choose a highly flexible and tractable Wishart matrix affine jump diffusion framework for the joint SPX-VIX2 covariance dynamics and integrate the SPX dynamics via a time-changed Lévy process. Our model design involves enough flexibility to account for the different empirical evidence such as the volatility smile, fat tails of the return distribution, the power-law scaling property of return moments and the seminal leverage effect of returns. Further, the jump-extended Wishart process allows for the coexistence of multifactor volatility, stochastic correlation, stochastic skewness, and jumps in returns or second moments. By means of transform methods, we obtain semi-closed expressions for SPX– and VIX options using the fast Fourier transform (FFT) and the Fourier-cosine series expansion (COS method). We show the high model flexibility by a daily estimation of the VIX option pricing formulae without jumps which performs substantially better than the Heston (1993) model in the replication of the implied volatility surface slice shapes and their levels.

Keywords: Matrix affine jump diffusions, jump-extended Wishart process, time-changed Lévy process, multifactor volatility, stochastic correlation, option pricing, SPX and VIX joint modeling, fast Fourier transform, Fourier-cosine series expansion. JEL classification: C51, G12, G13.

III

Executive Summary Problem Description After the financial crisis, the notion of liquidity risk (should have) caught the awareness of all financial market participants. Since volatility index (VIX) options, introduced by the Chicago Board Options Exchange (CBOE) in 2006, belong to the class of the most liquid options around the world, they are well suited for hedging purposes and require therefore an accurate pricing model. Due to the negative correlation between the VIX and the Standard & Poor’s 500 index (SPX), a VIX call can also be seen as a put option on the SPX. This (negative) linkage of the two markets need to be incorporated into an integrated pricing model. Accordingly, this thesis addresses the challenge of pricing VIX options while preserving consistency with the SPX options. We choose a highly flexible and tractable affine framework which has gained significant attention in the finance literature and in the theory of stochastic processes over the recent years.

Model Design For the specification of the theoretical model, we assume, due to its generality, a time-changed Lévy process for the SPX dynamics. The motivation of this approach is twofold. First, Lévy processes cannot capture stochastic volatility, stochastic risk reversal (skewness) and stochastic correlation. These drawbacks can be resolved, to some extent, by considering time-changed Lévy processes for which it is possible to generate distributions which vary over time. Second, by specifying a suitable time change for the Brownian motion part, we can integrate the VIX dynamics via a stochastic activity rate and can therefore preserve consistency between the markets and their options. In particular, we specify the 2 × 2-dimensional joint SPX-VIX2 covariance dynamics under the risk-neutral measure Q, which can be seen equivalently as the stochastic activity rate for the Brownian motion component, by a jump-extended Wishart process. Hence, we obtain a double jump matrix-variate stochastic volatility model in which we allow for multifactor volatility, stochastic correlation, stochastic skewness, and jumps in returns and second moments. Summarizing, our model design accounts for the different empirical evidence such as the volatility smile, fat tails of the return distribution, the power-law scaling property of return moments and the seminal leverage effect of returns.

Methodology and Results Since we preserve the affine structure, we can solve our financial pricing problem by means of transform methods. In order to capture the well-known leverage effect we introduce a correlation between the Brownian motion in the SPX dynamics without time change and the Brownian motion component in the activity rate of the corresponding time change. By using the leverage-neutral measure change method and the fast Fourier transform (FFT) technique we obtain a semi-closed expression for SPX options. For VIX options the Fourier-cosine series expansion (COS method) similarly yields a semi-closed expression. We estimate a simplified version of the VIX option pricing formulae, namely with a zero-jump component, on February 27, 2009, and demonstrate that the Heston (1993) model fails to fit the implied volatility surface (IVS) slice for each time-to-maturity where our model however performs well, due to the high flexibility even without jumps, in the replication of the IVS slice shapes and their levels. IV

Contents 1 Introduction

1

2 Analytically Pricing VIX– and SPX Options 2.1 Probabilistic Setting and Notation . . . . . . . . . . . . 2.2 Affine Processes . . . . . . . . . . . . . . . . . . . . . . . 2.3 Data Analysis and Stylized Facts of the VIX and SPX . 2.3.1 The Volatility Index . . . . . . . . . . . . . . . . 2.3.2 Empirical Data Evidence . . . . . . . . . . . . . 2.4 Preliminary Outline on the S&P 500 Dynamics . . . . . 2.4.1 Preliminaries on Lévy Processes . . . . . . . . . 2.4.2 Non-Homogenous Lévy Process . . . . . . . . . . 2.4.3 Specification of the Lévy Density . . . . . . . . . 2.5 The Joint Covariance Dynamics . . . . . . . . . . . . . . 2.5.1 Matrix Affine Jump Diffusion Wishart Process . 2.5.2 Gamma-Type Lévy Density . . . . . . . . . . . . 2.6 SPX Price– and Joint SPX-VIX2 Covariance Dynamics 2.7 Transform Analysis . . . . . . . . . . . . . . . . . . . . . 2.7.1 Pricing SPX Options . . . . . . . . . . . . . . . . 2.7.2 Pricing VIX Options . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

6 6 7 11 13 14 21 22 30 34 35 36 37 39 44 44 51

3 Numerical Investigation 55 3.1 Parameter Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 3.2 Pricing Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 4 Conclusion

61

Appendices A The Ft -Conditional Characteristic Function of the CIR Process

64

B Proof of Proposition 2.8

66

C The Fourier Transform of the Call Option c (k)

68

D Proof of Corollary 2.2

70

Index

79

V

List of Figures 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9

Joint Evolution of the SPX and VIX . . . . . . . . . . . . SPX Implied Volatility Skews . . . . . . . . . . . . . . . . VIX Implied Volatility Skews . . . . . . . . . . . . . . . . Joint Evolution of the VIX and VXST . . . . . . . . . . . Evolution of the Daily Difference between VXST and VIX Symmetric α-Stable Distributions . . . . . . . . . . . . . . Skewed α-Stable Distributions . . . . . . . . . . . . . . . . Variance-Gamma Density . . . . . . . . . . . . . . . . . . Gamma-Type Lévy Density . . . . . . . . . . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

15 17 18 20 20 29 30 35 40

3.1

Fitted VIX Implied Volatility Skews . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

List of Tables 2.1 2.2 2.3 2.4

Correlation Coefficients for the SPX and VIX . . . . . . . . Descriptive Statistics for the SPX Log-Returns and the VIX Implied Volatility Quantiles for SPX– and VIX Options . . Parameter Overview Gamma-Type Distribution . . . . . . .

3.1 3.2

Fitted Parameter Values of the Wishart Option Pricing Model . . . . . . . . . . . . . . . 57 Fitted Parameter Values of the Heston Option Pricing Model . . . . . . . . . . . . . . . . 58

VI

. . . . Levels . . . . . . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

15 16 19 39

NOTATION AND ABBREVIATIONS

VII

Notation and Abbreviations We shall use the standard notations and abbreviations. Further symbols and acronyms are context– or content-specific and defined in the text. Due to consistency, we deliberately avoid the use of one symbol denoting different objects. 1A

Indicator function of the set A

a.s.

almost surely

A

Infinitesimal generator of a semigroup

B (X )

Borel σ-algebra of a Borel set X ⊆ Rk Complex numbers

C C

k

c` adl` ag

Set of functions with continuous derivatives up to the k-th order continue à droite, limite à gauche

cf.

confer

dx

Multivariate differential for x ∈ Rk

d, h, i, j, k, l, `, n, nv det (X) diag (X)

Positive natural numbers Determinant of the matrix X Diagonal matrix of X Expectation operator

E E ⊗F

σ (E × F) , smallest σ-algebra generated by two σ-algebras E and F

e.g.

exempli gratia

exp

Exponential function e

F

σ-algebra

F = {Ft : t ∈ [0, T ]} FX = FtX : t ∈ [0, T ]

Filtration

fx (x) , ∂x f (x) , f 0 (x) g˙ (t) Ik i i.e. iff i.i.d. Im (k)

Natural filtration of the process X Mathematical derivatives of the function f (x) Time derivative of the function g (t) k × k identity matrix √ −1 id est if and only if independent and identically distributed Imaginary part of the complex number k

inf

infimum

1

Set of all integrable random variables

L

L (X) N N0

Law of the random variable X Natural numbers \ {0} N ∪ {0}

P

Historical probability measure

Q

Risk-neutral probability measure

Pascal Marco Caversaccio

c 2014

NOTATION AND ABBREVIATIONS

R R≥0 R+ Re (k) resp. Sk+ sgn (x) sup T tr (X) W = {Wt : t ∈ [0, T ]} x·y x∧y x∨y

VIII Real numbers Non-negative real numbers Positive real numbers Real part of the complex number k respectively Positive cone of symmetric positive semi-definite k × k matrices Signum function of x supremum Maturity date Trace of the matrix X Standard Brownian motion Scalar product of the vectors x, y ∈ Rk inf (x, y) sup (x, y)

Xt−

Left limit of the process X at t

X−1

Inverse of the matrix X

hx, yi c [X]t

[X]t δx σ (X) Ω (Ω, F, P) (Ω, F, F, P) d

Canonical inner product of the vectors x, y ∈ Rk Quadratic variation for the continuous part of a semimartingale X Quadratic variation for an absolutely continuous semimartingale X Dirac measure at x σ-algebra generated by the random variable X State space Probability space Filtered probability space

=

Equality in distribution

>

Transpose of a matrix or vector

4

End of a definition

End of a proof

J ⊥

End of a remark End of an example

#I


Cardinality of the set I

c 2014

Chapter

1

Introduction It seems that scientists are often attracted to beautiful theories in the way that insects are attracted to flowers — not by logical deduction, but by something like a sense of smell. — Steven Weinberg, Nobel Prize in Physics 1979

The inception of option-like contracts can be traced back to the seminal story of the pre-Socratic Greek mathematician and philosopher Thales of Miletus who lived from around 625 to 550 B.C. in Miletus. According to (Aristotle, 350 B.C., Chapter XI), he knew by his knowledge of the stars, while it was yet winter, that there would be a great crop of olives in the coming year. He seized the opportunity to negotiate with the olive press owners the right, but not the obligation, to hire all the olive presses in the region for the following autumn. To secure his right but having little money he gave cash deposits for the use of all the olive presses in Chios and Miletus which he could hire at a low price because no one bid against him. When the harvest time came, many olive presses were wanted all at once and he let them out at any rate which he pleased and made a quantity of money. Thus, he showed the world that philosophers can easily be rich if they like, but that their ambition is of another sort. This historical anecdote also illustrates remarkably the potential for profits that could be made from anticipating the supply and demand. Hence, this story manifests that the notion of derivatives has its roots in the ancient times and is not only a modern creation. The first options were issued in the 12th century by the Counts of Champagne with food and materials as underlyings. In Antwerp, Belgium, the first modern financial markets appeared at the beginning of the 16th century, and in Amsterdam during the 17th century. One of the first records of an organized market for derivatives trading comes from Osaka and dates back to the 17th century. Japanese landlords shipped their rice surplus to storage warehouses in some major cities and issued tickets for the promised future delivery of the rice. These tickets were traded at first in 1730 on the Dojima rice market near Osaka. Another worth mentioning event consists of the first financial crisis in the Netherlands around 1636, which was referred to as tulipomania due to the irrational prices of tulips at this time. The broker John Castaing, operating out of Jonathan’s Coffee House, established the origin of the London stock exchange by posting regular lists of stock and commodity prices. In 1848, a first derivatives exchange, the Chicago Board of Trade (CBOT) which is nowadays the oldest still operating organized futures market, was 1

1. INTRODUCTION

2

created in Chicago, United States. The continuously amendments on the trading environment has led on February 8, 1971, to the world’s first electronic stock exchange NASDAQ.1 For further very interesting insights and stories on the history of derivatives, we refer to the recent document of the Swiss State Secretariat for Economic Affairs (SECO) written by Kummer and Pauletto (2012). The main purpose of derivatives is to transfer risk from one person or firm to another. In other words, to provide an insurance. Indeed, there is also the notion of speculation for using options, but this can be seen as the counterpart of the hedging party which in turn transfers risks by its derivatives transactions. This particular potential of contingent claims offers large financial– and non-financial cooperations as well as individuals the possibility to build simultaneously long– and short positions with different instruments within their portfolios and immunize the risk according to their preferences. As a matter of fact, it is a great tool for the risk management department which nowadays stands in a dynamically changing environment of regulation (e.g. Basel III for banks, solvency 2 for insurance firms that operate in the EU, or the Swiss solvency test (SST) for Swiss insurance firms) and asks continuously for more sophisticated instruments to hedge their huge and complex portfolios whereupon suiting the regulations (e.g. CoCo bonds). One way to deal with this issue is by trading contingent claims on volatility. These volatility– and variance derivatives are a very attractive class of financial products for risk management because they can be used as protection against market crashes, i.e. to diversify equity risk, and they provide a direct exposure to the variance without the need to delta-hedge the underlying stock exposure. According to Michael Weber, now with J.P. Morgan, the first volatility derivative appears to have been an OTC variance swap dealt in 1993 by him at the Union Bank of Switzerland (UBS) (cf. Carr and Lee (2009)). Between 1993 and 1998, both variance– and volatility swaps on indices were traded sporadically. The reason why variance swaps are nowadays much more popular is that they exhibit a higher robustness of hedges in contrast to volatility swaps. In 1998, several hedge funds built huge short positions in variance swaps, i.e. sold the realized variance, due to the basic idea that selling realized variance has positive alpha. This strategy led to the deterioration of different hedge funds during the recent financial crisis. As next, variance swaps on individual stocks were introduced. The ongoing demand for more sophisticated instruments on volatility and variance increased over the years and led to several innovations such as options on the realized variance, conditional– and corridor variance swaps on indices or timer options (also known as mileage option) on the volatility. Since these transactions were executed OTC, options exchanges were supposed to react in particular due to the liquidity– and settlement risk. This led to the introduction of the Chicago Board Options Exchange (CBOE) volatility index (VIX) in 1993. Nowadays, the VIX market consists of options and futures on the CBOE VIX (cf. Section 2.3.1 for the detailed outline of the VIX). These European-style options are cash-settled at their expiry. For a more in-depth analysis of the history of volatility derivatives, we refer in particular to Carr and Lee (2009), on which this paragraph is partly based. After the financial crisis, the notion of liquidity risk (should have) caught the awareness of all financial market participants. Since VIX options belong to the class of the most liquid options around the world, they are well suited for hedging purposes and require therefore an accurate pricing model. Due to the negative correlation between the VIX and the Standard & Poor’s 500 index (SPX), a VIX call can also be seen as a put option on the SPX. This (negative) linkage of the two markets need to be incorporated into an integrated pricing model. Accordingly, this thesis addresses the challenge of pricing VIX options while preserving consistency with the SPX options. We choose a highly flexible and tractable affine framework which has gained significant attention in the finance literature and in the theory of stochastic processes over the recent years. 1

NASDAQ is an acronym for ”National Association of Securities Dealers Automated Quotations”.


c 2014

1. INTRODUCTION

3

”The shortest path between two truths in the real domain passes through the complex domain.” This famous quote of Jacques Hadamard, a French mathematician, self-explains the rationale why affine processes, which make use of characteristic functions, are used in financial applications. Affine Markov models have been applied in finance since decades, and they have found growing interest due to their computational tractability as well as their capability to capture empirical evidence from financial time series such as the implied volatility smile/skew or the leverage effect of returns.2 In particular, according to Duffie et al. (2000), an affine based option pricing framework allows us to efficiently solve our financial pricing problem by means of transform methods. Therefore, we can circumvent the problem of numerical solutions without losing the flexibility induced by affine models. The main applications of affine processes lie in the theory of term structure of interest rates, stochastic volatility option pricing and the modeling of credit risk. Duffie et al. (2003) provide as first a complete characterization of regular affine processes v on the state space Rn≥0 × Rd for (nv , d) ∈ N2 . Subsequently, over the recent years many mathematicaloriented papers solved the challenge of regularity on different state spaces. We refer, among others, to Keller-Ressel et al. (2011), and Keller-Ressel et al. (2013). Furthermore, Keller-Ressel (2008) and Filipović and Mayerhofer (2009) provide analytical solutions for the pricing of bond– and stock options with affine models. Recent research has also introduced highly flexible and tractable matrix affine jump diffusion (MAJD) to model asset returns, in which multifactor volatility, stochastic correlation, stochastic skewness, and jumps in returns or second moments can naturally coexist (cf. Leippold and Trojani (2010)). The Wishart pure diffusion process of Bru (1991), which belongs to the class of matrix affine diffusion (MAD), is studied in Gouriéroux et al. (2009) and Gouriéroux and Sufana (2010) to price derivatives. Leippold and Trojani (2010) extend this framework and allow for a rich structure of jumps in the dynamics. Barndorff-Nielsen and Stelzer (2007) introduce some pure jump matrix Ornstein-Uhlenbeck subordinators, an extension of the one-dimensional approach developed in Barndorff-Nielsen and Shephard (2001), which can be used as a multivariate stochastic volatility model. The Wishart process is a multivariate mean-reverting process, hereby implying, if we apply stochastic volatility to the general stock dynamics, to model the volatility dynamics by a jump-extended version of the Wishart process to control for sharp upward (respectively downward) jumps in the instantaneous covariance process. Moreover, Buraschi et al. (2008) consider for the joint modeling of market prices of risk and conditional second moments of risk factors a Wishart process which is governing the risk factor dynamics. Another application of MAD models lie in the field of dynamic portfolio choice for the case of a stochastic covariance matrix of returns. Buraschi et al. (2010), for instance, show that the hedging demand in the matrix-variate framework is typically four to five times larger than in univariate models. Crameri et al. (2010) propose a general family of multivariate portfolio time-changed Lévy processes based on a gammatype, tempered– and inverse Gaussian matrix subordination approach. Further, it is a well documented fact that single-factor stochastic volatility models with jumps can replicated the implied volatility smiles and –skews but fail to capture the variability in the smile dynamics over time. This problem is obviously linked to the stochastic skewness of returns. Gruber et al. (2010) propose a modeling setting for the valuation of European options in which dynamic short– and long volatility components drive the smile dynamics and where the model state dynamics is driven by a MAJD of the Wishart type. Eventually, for a mathematical rigorous treatment of affine processes on (symmetric) positive semi-definite matrices we refer to Cuchiero et al. (2011a) and Cuchiero et al. (2011b). This thesis incorporates the high analytical tractability as well as the enormous flexibility of Wishart processes to model the joint covariance dynamics of the SPX and the VIX2 . Particularly, we are able to introduce via the co-volatility component of the process a wide degree of variation in the jump-driven– and diffusive skewness. 2

The leverage effect arises from the negative correlation between the stock price and its volatility. When the value of a stock drops, the volatility of its returns tends to increase. Indeed, if the stock price decreases, the debt-to-equity ratio increases and the risk of the firm therefore increases, which translates into a higher volatility for the firm.


c 2014

1. INTRODUCTION

4

The CBOE VIX reflects the market’s expectation of the 30-day forward SPX volatility and serves as a proxy for the investor sentiment. Hence, given the direct exposure to the volatility, we have seen an extended use of VIX options in the risk management because they provide a direct exposure to variance without the need to delta-hedge the underlying stock exposure. The increased demand for volatility derivatives and in particular VIX options over the recent years, due to the aforementioned rationale, claims of course for a consistent pricing of this particular class of derivatives while preserving consistency with the underlying index. The notion of volatility indices and –derivatives can be traced back to Brenner and Galai (1989) who propose indices on realized volatility with corresponding futures– and option contracts. The original VIX, which was based on the average of S&P 100 option implied volatilities, is described in Fleming et al. (1995) while Whaley (1993) as first proposes derivative contracts with the VIX as underlying. Thenceforward, an extensive collection of research has been arising for the pricing of volatility derivatives. To our best knowledge, the paper of Gr¨ unbichler and Longstaff (1996) is the first attempt to price volatility futures and –options in a continuous time setting. In particular, they model the instantaneous variance rate by a continuous time generalized autoregressive conditional heteroscedasticity (GARCH) process. Further early research on volatility derivatives include Detemple and Osakwe (2000), Heston and Nandi (2000), Howison et al. (2004), Daouk and Guo (2004), Bergomi (2004, 2005), Friz and Gatheral (2005) and Zhang and Zhu (2006), among others. In particular we emphasize that the first attempt of pricing VIX futures is provided in Zhang and Zhu (2006), where the SPX is described by the stochastic volatility model of Heston (1993). Since the introduction of VIX futures in 2004 and plain vanilla option contracts on the VIX in 2006, many research papers faced the challenge of pricing this particular class of products. Particularly, VIX option values are by definition dependent on the SPX through its options (see Definition 2.4) and therefore, due to the sake of consistent modeling, one needs to incorporate the SPX dynamics into the pricing setting. The paper of Lin and Chang (2010) seemed to solve this problem. They model the SPX by a stochastic volatility process with simultaneous jumps in both, the asset return and volatility process. Moreover, the provide a closed-form VIX option formula. Nevertheless, Cheng et al. (2012) show the incorrectness of this formulae, especially they prove that the characteristic function of the pricing equation cannot be exponentially affine, as proposed by Lin and Chang (2010). Two years before, Sepp (2008b) already provides an exact analytical formula for VIX– and SPX options under the assumption of a stochastic volatility process with volatility jumps but no jumps in the asset return. Another interesting approach is taken in Albanese et al. (2009), where they develop a pricing framework that can simultaneously handle European options, forward-starts, options on the realized variance and options on the VIX. To do so, they make use of spectral methods. The recent paper of Lian and Zhu (2013) try to remove the inflexibility for the asset return dynamics in many papers (e.g. Sepp (2008b)) and present an analytical solution for the price of VIX options under stochastic volatility with simultaneous jumps in the asset price and volatility processes. Nonetheless, they are restricted to independent Poisson processes in the jump components and do not allow for general Lévy densities. Bardgett et al. (2013) develop a three-factor affine model which yields semi-closed expressions for SPX– and VIX options. Their two-factor stochastic volatility model with jumps offers a high flexibility but lacks of stochastic correlation and co-volatility structures. Furthermore, we refer, among others, to Sepp (2008a), Bergomi (2008), Cont and Kokholm (2011), Kallsen et al. (2011) and Drimus and Farkas (2013) for recent papers covering volatility– and particularly VIX options, respectively. Concerning the research papers which have used the wrong formula of Lin and Chang (2010), e.g. Chung et al. (2011), the results need to be reexamined. Moreover, Menc´ıa and Sentana (2013) conduct an extensive empirical analysis of VIX derivative valuation models before, during, and after the 2008–2009 financial crisis, where their results indicate that stochastic volatility plays a much more important role for VIX options than for VIX futures. Pascal Marco Caversaccio

c 2014

1. INTRODUCTION

5

Another very interesting investigation is conducted in Lian and Zhu (2013). The authors show, using variance swap prices (i.e. VIX2 , cf. Section 2.3.1 for an explanation of this analogy), that affine jump diffusion models exhibit not only a misspecified diffusion term but also lack of correctly specified affine drift, –jump and –risk premiums. Concluding, one of the main remaining challenges in the option pricing theory is the specification of an analytical tractable and highly flexible VIX option pricing model while preserving consistency with the SPX and allowing for parsimonious calibration. To design our theoretical model, we assume, due to its generality, a time-changed Lévy process for the SPX dynamics. The motivation of this approach is twofold. First, Lévy processes cannot capture stochastic volatility, stochastic risk reversal (skewness) and stochastic correlation. These drawbacks can be resolved, to some extent, by considering time-changed Lévy processes for which it is possible to generate distributions which vary over time. Second, by specifying a suitable time change for the Brownian motion part, we can integrate the VIX dynamics via a stochastic activity rate and can therefore preserve consistency between the markets and their options. In particular, we specify the 2 × 2-dimensional joint SPX-VIX2 covariance dynamics under the risk-neutral measure Q, which can be seen equivalently as the stochastic activity rate for the Brownian motion component, by a jump-extended Wishart process. Hence, we obtain a double jump matrix-variate stochastic volatility model with co-volatilities, i.e. the off-diagonal elements model the volatility of volatility between the SPX and VIX, which can replicate the stochastic skewness and –correlation in the markets. Since we preserve the affine structure, we can solve our financial pricing problem by means of transform methods. Semi-closed expressions are thereupon obtained for SPX– and VIX options. Finally for the sake of a numerical illustration, we show the high model flexibility by a daily estimation of the VIX option pricing formulae without jumps which performs substantially better than the Heston (1993) model in the replication of the implied volatility surface (IVS) slice shapes and their levels. This thesis is structured as follows. The Chapter 2 consists of 7 interconnected sections and subsections which are briefly outlined in the following. Section 2.1 presents our probabilistic setup and some notation. We shall use the standard notations and abbreviations, and deliberately avoid the use of one symbol denoting different objects. Section 2.2 summarizes the fundamental properties of affine processes. To illustrate the calculation approach applied in the affine setting, we also provide a specific example. Section 2.3 provides a preliminary data analysis of the VIX, SPX and their options which induces the model design. Particularly, in Section 2.3.1 we outline the VIX followed by Section 2.3.2 which illustrates the empirical data evidence and the stylized facts of the VIX and SPX (option) markets. The recently released CBOE S&P 500 short-term volatility index (VXST) is also presented. The general multivariate Lévy dynamics and its deployment to the SPX dynamics under the risk-neutral measure Q is elaborated in Section 2.4. In particular, Section 2.4.1 provides a mathematical introduction into Lévy processes. Moreover, Section 2.4.2 presents the time-changed Lévy model with the corresponding Lévy density for the univariate case, described in Section 2.4.3, governing the jumps in the Lévy process. Section 2.5 is devoted to the joint covariance dynamics of the SPX and VIX2 . Indeed, Section 2.5.1 presents the new class of flexible and tractable MAJD with the corresponding matrix-variate Lévy density sketched in Section 2.5.2. Section 2.6 provides the model specification and the explicit martingale drift condition for the SPX dynamics. In Section 2.7 we work out the Laplace transforms in detail. Particularly, Section 2.7.1 and Section 2.7.2 derive analytical solutions for SPX– and VIX options, respectively. The Chapter 3 investigates the empirical performance of the VIX option pricing model specified previously in Section 2.7.2 by setting, for simplicity’s sake, the jump component to zero. In Section 3.1 the detailed description of the estimation procedure conducted for the numerical part is provided. Further, Section 3.2 examines the pricing performance of the VIX option pricing model by means of implied volatilities. Finally, the Chapter 4 concludes.


c 2014

Chapter

2

Analytically Pricing VIX– and SPX Options A mathematical theory is not to be considered complete until you have made it so clear that you can explain it to the first man whom you meet on the street. — David Hilbert (1862-1943), German Mathematician

This chapter corresponds to the heart of this thesis and will be guiding you properly through the different sections. We start with preliminary results on affine processes in Section 2.2 and continue thereupon with the outline of the VIX and its stylized facts, together with the empirical data evidence of the SPX, in Section 2.3. Thereafter, Section 2.4 and Section 2.5 lay the theoretical basis for the comprehension of our model design in Section 2.6. Eventually, we obtain semi-closed solutions for VIX– and SPX options in Section 2.7 by means of transform methods. Furthermore, to enhance the grasp of the theory, we enrich the sections on preliminary results with specific examples. To be able to work in a mathematical rigorous way, let us first fix the probabilistic setup and notation as next.

2.1

Probabilistic Setting and Notation

Throughout this thesis, let W : [0, T ] × Ω → R be a standard Brownian motion starting at 0 a.s., defined on a filtered probability space (Ω, F, F, P), satisfying the usual conditions of right-continuity and completeness, where the filtration F = {Ft : t ∈ [0, T ]} is the P-augmentation of the natural filtration FX = FtX : t ∈ [0, T ] generated by {Wu : 0 ≤ u ≤ t}. Moreover, we assume that F0 is P-trivial and work on the finite time horizon [0, T ] , T < ∞. We fix P as the historical measure and Q defines the pricing (risk-neutral) measure. Let N be the collection of all natural numbers and N0 := N ∪ {0}. Besides, let Rk be an Euclidean k-space, where k ∈ N and R := R1 , and furthermore, the set of all complex numbers is denoted by Ck , with k ∈ N and C := C1 . By R≥0 we denote the non-negative real numbers, i.e. R≥0 := [0, ∞), by R+ the positive real numbers, i.e. R+ := (0, ∞), by Sk+ the positive cone of symmetric 6

2.2. AFFINE PROCESSES

7

positive semi-definite k × k matrices, and let d, h, i, j, k, l, `, n, nv ∈ N. Henceforth, we denote by x · y the > scalar product of the vectors x, y ∈ Rk and use dx, where x := (x1 , . . . , xk ) , to emphasize the Rk -value dependence.3 We abbreviate almost surely, if and only if and independent and Pkidentically distributed by a.s., iff and i.i.d., respectively, the operator tr : Rk×k → R, B 7→ tr [B] = i=1 Bii is the trace of a k × k-matrix B and B−1 denotes its corresponding inverse, L (X) stands for the law of the random variable X and let C k be the set of functions with continuous derivatives up to the k-th order, i.e. k ∈ N0 . The term c` adl` ag means continue ` a droite, limite ` a gauche, i.e. right continuous with left limits. d Betimes and if appropriate, resp. denotes the abbreviation of respectively. By = we denote the equality in distribution, > stands for the transpose of a matrix (or vector) and Re (·) and √ Im (·) represent the real– and imaginary part of a complex number k = Re (k) + iIm (k), with i := −1. For the smallest σ-algebra on two spaces E and F generated by two σ-algebras E and F, σ (E × F), we write E ⊗ F. Recall that for a subset A of E, the indicator function 1A (x) is defined to be ( 1, if x ∈ A, 1A (x) := 0, if x 6∈ A, and the identity matrix can be defined by Ik := diag (1, 1, . . . , 1) with diag (·) the diagonal matrix. The Dirac measure x 7→ δ (x) is defined such that it is almost everywhere zero except at x = 0. Furthermore, R f (x) δ (dx) = f (0) for a continuous function f with compact support. We write L1 for the set of R all integrable random variables. The operator sgn (x) := 1(0,∞) (x) − 1(−∞,0) (x) denotes the signum function, det (·) corresponds to the determinant of a matrix and exp stands for the exponential function e. Occasionally, we also write ∞ instead of +∞ and, for the mathematical derivatives, fx (x) instead of ∂x f (x) or f 0 (x). In case of a time-dependent function g (t), we shall write occasionally g˙ (t) for the c corresponding (time) derivative. We call h·, ·i the canonical inner product on Rk , [·]t (resp. [·]t ) denotes the quadratic variation for the continuous part of a semimartingale (resp. for an absolutely continuous semimartingale), sup (resp. inf) stands for the supremum (resp. infimum), and x ∧ y (resp. x ∨ y) represents inf (x, y) (resp. sup (x, y)). Given a set I, we denote the corresponding cardinality by #I. Finally, definitions, proofs, remarks and examples are ended by 4, , J and ⊥, respectively, and we abbreviate confer, id est and exempli gratia by cf., i.e. and e.g., respectively.

2.2

Affine Processes

We now introduce the basic definition of an affine process (cf. Filipović and Mayerhofer (2009)) and present some corresponding characteristics. The section tries to summarize the fundamental properties needed to understand our model setup in the subsequent sections (see Section 2.4, Section 2.5 and Section 2.6). We refer, among others, to Duffie et al. (2003), Keller-Ressel (2008), Filipović and Mayerhofer (2009), and Keller-Ressel et al. (2011, 2013) for a mathematical more rigorous and comprehensive treatment of affine processes. First, fix a dimension k ≥ 1 and a closed state space X ⊂ Rk with non-empty interior. Second, we define the characteristic function, due to its use in the sequel, as next. Definition 2.1. Let X be a random variable in Rk . The characteristic function of X, µ b : Rk → Ck , 3

For notational simplicity, we only apply the bold face to the differential dx. Hence, for the readers convenience, one only needs to check the differential to distinguish between multi– and one-dimensionality.


c 2014

2.2. AFFINE PROCESSES

8

is defined as µ b (u) := E eiu·X =

Z

eiu·x dFX (x) =

Rk

Z

eiu·x fX (x) dx,

(2.1)

Rk

where FX (x) is the cumulative distribution function (CDF), fX (x) is the probability density function (PDF), and u ∈ Rk . 4 For the sake of consistency, we define the characteristic function in terms of scalar products, as in the Lévy-Khintchine representation in Theorem 2.2. Remark 2.1. The characteristic function fully describes the probability structure of an option pricing model and is sufficient in order to compute option prices by a single integration. This feature explains the wide application of this particular transform method in the finance literature. J By conditioning on Ft in (2.1), we get to the seminal affine processes. Definition 2.2. We call X an affine process if the Ft -conditional characteristic function of XT is exponential affine in Xt , ∀t ≤ T . That is, there exist C– and Ck -valued functions φ(T − t, z) and ψ(T − t, z), respectively, with jointly continuous t-derivatives such that X = X x satisfies h > i > (2.2) E ez XT Ft = eφ(T −t,z)+ψ(T −t,z) Xt ∀z ∈ iRk , t ≤ T and x ∈ X .

4

k k Remark 2.2.k The set iR denotes the purely imaginary numbers in C , therefore we can define it as k := iR z ∈ C : Re (z) = 0 . J

To illustrate the calculation steps for a characteristic function approach in a option pricing framework, we borrow the following example from Cheng et al. (2012), which includes the case of a one-dimensional Markov process dXt = µ (Xt ) dt + σ (Xt ) dWt + dJt , X0 = x ∈ X , where W denotes, as usual, a P-Brownian motion, J is a pure-jump process with jump arrival intensity {λ (St ) : t ∈ [0, T ]} for some function λ : R → [0, ∞), and the jump sizes Z1 , Z2 , . . . , are i.i.d and independent of W .4 Example 2.1. We target to elaborate the following expectation5 f (Xt , t) = E eXT Xt ,

(2.3)

2

where we assume the subsequent affine dependence for µ, σ and the jump intensity λ on X: µ (x) := a + bx, 2 σ (x) := cx, λ (x) := l0 + l1 x,

where a, b, c, l0 , l1 ∈ R.

We can apply Itˆ o’s formulae for jump diffusions (see, for instance, Cont and Tankov (2004a)), under the assumption f (x, t) ∈ C 2,1 (X × [0, ∞)), which yields to Z t Z t X f (Xt , t) = f (X0 , t) + (Af ) (Xs− , s) ds + fx (Xs− , s) dWs + [f (Xs , s) − f (Xs− , s)] , 0 4 5

0

0

is assumed to be continuous in x ∈ X and W denotes a k-dimensional P-Brownian motion. Further on, we assume there exist a unique solution X = X x of (2.8). Then, the diffusion matrix Q (x) and b (x) are affine in x. That is, k X xi αi , Q (x) = a + i=1

b (x) = b +

k X

(2.9) xi βi = b + Bx,

i=1

for some k × k-matrices a and αi , and k-vectors b and βi , where we denote by B := (β1 , . . . , βk ) > the k × k-matrix with i-th column vector βi , 1 ≤ i ≤ k. Moreover, φ and ψ := (ψ1 , . . . , ψk ) solve the system of Riccati equations

1 > ψ (t, z) aψ (t, z) + b> ψ (t, z) , 2 φ (0, z) = 0, 1 > ∂t ψi (t, z) = ψ (t, z) αi ψ (t, z) + βi> ψ (t, z) , 2 ψ (0, z) = z. ∂t φ (t, z) =


(2.10) 1 ≤ i ≤ k,

c 2014

2.3. DATA ANALYSIS AND STYLIZED FACTS OF THE VIX AND SPX

11

In particular, φ is determined by ψ via simple integration: Z t 1 > > φ (t, z) = ψ (s, z) aψ (s, z) + b ψ (s, z) ds. 2 0 Conversely, suppose the diffusion matrix Q (x) and drift b (x) are affine of the form (2.9) and suppose > there exist a solution (φ, ψ) of the Riccati equations (2.10) such that φ (t, z) + ψ (t, z) x has negative k real part for all t ∈ [0, T ], z ∈ iR and x ∈ X . Then X is affine with conditional characteristic function (2.2). Proof. Idea of the proof: Apply Itˆ o’s lemma, separately to the real– and imaginary part, of the complexvalued Itˆ o process Mt = eφ(T −t,z)+ψ(T −t,z)Xt and use the martingale property to imply the affine dependence of Q (x) and b (x). See (Filipović and Mayerhofer, 2009, Section 2) for the complete proof. Remark 2.3. In 1723, Count Jacopo Riccati (1676-1764) studied the differential equation x˙ (t) = a (t) − c (t) x2 (t) for particular choices of the time-dependent coefficients a (t) and c (t). According to Watson (1958), Jean-le-Rond D’Alembert was the first, in 1763, to refer to a more general differential equation of the form x˙ (t) = a (t) + b (t) x (t) − c (t) x2 (t) as Riccati’s generalized equation. J We eventually present an affine process which is extensively used in finance to model the dynamics of interest rates or stochastic volatility. Example 2.2. For every δ ∈ R≥0 and x ∈ R≥0 , the unique strong solution to the equation Z ρt = x + δt + 2

t

√

ρs dWs ,

ρt ∈ R≥0 ,

0

is called a squared Bessel process with dimension δ, starting at x, which is denoted by BESQδx and belongs to the class of affine processes. ⊥ This thesis faces the challenge of pricing VIX options while preserving consistency with the SPX options in a MAJD framework. As starting point we outline the VIX and its stylized facts, together with the empirical data evidence of the SPX, as next.

2.3

Data Analysis and Stylized Facts of the VIX and SPX

Every empirical analysis depends strongly on the corresponding data treatment and therefore should not be omitted due to the sake of transparency. This particular and crucial issue is treated as first in the following. We obtain the SPX– and VIX option data quotes, covering a wide range of strikes and maturities, from the OptionMetrics database. For the continuously compounded risk-free interest rates we take the zero-coupon yield curve, also available on OptionMetrics, covering various maturities. The data set spans the period from January 3, 2007, to January 31, 2013, covering six years of data. Since it is really Pascal Marco Caversaccio

c 2014


12

difficult to predict accurately future dividends on the SPX, and additionally following the discussion in Section 2.6 which explains why the true underlying of VIX options is the corresponding VIX futures contract, we define the moneyness in terms of futures prices. The standardized log-moneyness is defined, for i ∈ {S, V}, by ln K/Fti (T ) √ , (2.11) m := ATMIV (t, T ) T − t where K is the strike level of the European-style option, Fti (T ) denotes the closing SPX (resp. VIX) futures price if i = S (resp. i = V) today at time t with maturity T , ATMIV (t, T ) stands for the at-the-money (ATM) implied volatility quote of the closing option price at time t with maturity T , and T − t is the option’s time-to-maturity quoted in fractional years where the convention of 360 days is used. We infer the value of the SPX– and VIX futures, respectively, at closing by backing out the value using the ATM forward put-call parity in (2.33). Moreover, we follow the standard convention in the literature and take the mid-price, defined by the average of the best bid– and offer market quotes, to calculate the implied volatilities. To avoid noise in the data, three additional exclusionary criteria are applied. First, we delete all non-traded and therefore illiquid options. Thus, we remove all options which have zero open interest or were not traded for some time, i.e. volume = 0. Second, we follow A¨ıt-Sahalia and Lo (1998) and delete observations under a price level of 0.10$. This can be justified by the rationale that it is not possible to give an arbitrary decimal price due to the minimum tick for option prices. Third, we delete all in-the-money (ITM) options if there exist corresponding liquid out-of-the-money (OTM) options, since OTM options contain usually more information due to their high liquidity. For the particular case that the OTM options are not sufficiently liquid, we continue working with the most liquid one of the OTM– and ITM option. Concerning the implied volatilities, we use a hybrid algorithm, consisting of the (efficient) Newton-Raphson algorithm and the bisection method, for the calculations. We have to admit at this point that it is impossible to obtain a perfectly cleaned up data set, since there are some issues which cannot be resolved, at least not with daily data.11 For instance, the trading time of options may not be the closing time, which means that the closing price of the underlying value does not correspond to the underlying value at the time of trade. One way to deal with this issue is to consider futures prices backed out from the ATM put-call parity (see (2.33)) with highly liquid options as we conduct it in this thesis. If such high liquid options does not exist, we are back to the problem of non-synchronized prices. Nonetheless, we think that our data treatment removes most of the noise and allows an empirical data analysis. Remark 2.4. One can also obtain implied volatility estimates from OptionMetrics. However, there exist different approaches for the construction of an IVS. In particular, OptionMetrics computes first the implied volatility for each option, and in a second step the IVS is reconstructed by a Gaussian kernel smoothing with empirically adjusted widths. Since the data treatment and the computational method is different compared to OptionMetrics, our IVS slices in Figure 2.2 and Figure 2.3 below do not correspond to the IVS slices obtained by OptionMetrics. We refer to Homescu (2011) for a survey of the existing literature on the construction of an IVS. J We provide, in the following section, a brief (mathematical) review of the basic facts of the VIX.

11

High-frequency data, for example, offer more flexibility but also requires special attention. Therefore, it is a trade-off between ordinary frequented data and high-frequency data which can however open a Pandora’s box how to partition the data.


c 2014


2.3.1

13

The Volatility Index

The CBOE VIX reflects the market’s expectation of the 30-day forward S&P 500 index volatility and serves as a proxy for the investor sentiment. This is why the VIX is also often referred to as fear gauge. A nice characteristics of the VIX methodology is that it is calculated in a model-free manner as a weighted sum of OTM option prices across all available strikes on the SPX. This particular virtue is depicted in the official definition stated in the CBOE white study.12 Definition 2.4. The volatility index is defined, by means of VIX2 at time t, by 2 ! 1 Ft (30) 2 X ∆Ki rτ 2 e Q(Ki ) − × 1002 , VIXt := −1 τ i Ki2 τ K0

(2.12)

where 30 τ ≡ 365 , Ft (30) is the time-t 30-day forward SPX level, K0 is the first strike below the forward SPX level, Ki is the strike price of the i-th OTM option written on the SPX, ∆Ki denotes the interval between strike prices, Q(Ki ) denotes the time-t midquote price of the OTM option at strike Ki , and r denotes the time-t risk-free rate with maturity τ .

4 Expression (2.12) can be interpreted, with mathematical simplification, as the risk-neutral expectation of the log contract (see, for instance, Lin (2007) and Zhu and Lian (2012)).13 Definition 2.5. Expression (2.12) can be represented in terms of the risk-neutral expectation of the log-contract at time t by St+τ 2 Q 2 Ft × 1002 , VIXt := − E ln τ Ft (30) where Ft (30) = St erτ denotes the 30-day forward price of the underlying S&P 500 index with a risk-free interest rate r. 4 Under the assumption that the underlying does not jump, the VIX2 represents the risk-neutral expectation of the future realized variance of the SPX returns over the next 30 days, which implies that the VIX2 can be seen as an approximation of the strike price of the one month variance swap, i.e. we have at time t " 2 # N Sti 1X 2 VS Q ln × 1002 . K u VIXt = E lim N →∞ τ S t i−1 i=1 Remark 2.5. If we consider possible discontinuities, namely jumps, the VIX2 differs from the one month realized variance of the underlying SPX. J 12 13

Consider the white study of the VIX, available on http://www.cboe.com/micro/vix/vixwhite.pdf. The log-contract is a European-style contingent claim. It can be statically hedged using European options (cf. Carr and Lee (2009)).


c 2014


14

Remark 2.6. There also exist over-the-counter (OTC) traded volatility swaps since investors like to think in terms of volatility (and not variance). However, because volatility swaps are much harder to hedge, they are much less traded than variance swaps. J After this short technical summery we now discuss in detail the stylized facts observed in the VIX– and SPX market data. The results will govern our model design in Section 2.4.2, Section 2.4.3, Section 2.5 and Section 2.6.

2.3.2

Empirical Data Evidence

The VIX market consists of options and futures on the CBOE VIX. On the February 24, 2006, approximately two years after the launch of VIX futures, VIX options were introduced. In less than five years, the combined trading activity in VIX options and futures has grown to more than 100’000 contracts per day.14 Due to the increased exposure to volatility in financial markets, VIX options have become an extremely useful derivative in risk management for hedging purposes. Volatility and variance derivatives are a very attractive class of financial products for risk management because they can be used as protection against market crashes, i.e. to diversify equity risk, and they provide a direct exposure to the variance without the need to delta-hedge the underlying stock exposure. A peculiarity is given by the underlying for VIX options, which is the forward value of the VIX at expiration, rather than the spot VIX value. We elaborate this particular feature in a more rigorous way at the beginning of Section 2.6. Figure 2.1 displays the joint evolution of the SPX and VIX. The figures imply a strong negative correlation, i.e. a drop in the S&P 500 index is followed by upward moves in the VIX and vice versa. Furthermore, we can deduce a mean-reverting behavior in the VIX dynamics. Moreover, there are sharp upward movements in the VIX, indicating an intense positive jump component, and more regularly decreasing downward movements, indicating less strong negative jumps. The reverse is true for the SPX dynamics. Hence, for modeling purposes, one needs to add positive and negative jump components to both movements, SPX and VIX, but with sharp upward amplitudes in the case of the VIX and strong downward jumps in the SPX. As explained in Section 2.4.3 and Section 2.5 below, this can be obtained by a different scaling of the Lévy densities. To prove the above fact of a negative linkage between the SPX and VIX, we need to take a closer look at the correlation coefficients. We depict the correlation coefficients in Table 2.1 for different date intervals, covering the pre-crisis–, crisis–, and post-crisis periods. The sample period consists of the period 03/01/2005 – 02/08/2013. At first, the correlation coefficients are as expected, in five out of six cases, strongly negative. In the period 03/01/2005 – 02/05/2008 we observe an outlier with a positive correlation. This anomaly can be explained by the fact that before the financial crisis the volatility movements were very low and the returns were continuously increasing as can be seen from Figure 2.1 and Table 2.2 below. Furthermore, the negative correlation coefficients induce the leverage effect between the VIX and SPX since negative returns are often observed together with a rise of volatility. Table 2.1 implies another stylized fact, namely the time-varying correlation coefficient, indicating different covolatility structures in time. This rationale yields to the multivariate jump-extended Wishart process (see Section 2.5.1 and Assumption 2.5), for which stochastic correlation and stochastic skewness can naturally coexist. To infer more peculiarities we need to delve into the most crucial descriptive statistical measures for the log-returns of the Standard & Poor’s 500 index and the volatility index levels. 14

See http://www.cboe.com/micro/vix/vixwhite.pdf.


c 2014


15

S&P 500 Index Dynamics 1800

SPX Level

1600 1400 1200 1000 800 600 Jan−05

Jan−06

Feb−07

Mar−08

Apr−09 Date

May−10

Jun−11

Jul−12

Aug−13

Jun−11

Jul−12

Aug−13

CBOE VIX Dynamics 100

VIX Level

80 60 40 20 0 Jan−05

Jan−06

Feb−07

Mar−08

Apr−09 Date

May−10

Figure 2.1 – The evolution of the Standard & Poor’s 500 index (blue solid line) and the CBOE volatility index (black solid line) during the sample period 03/01/2005 – 02/08/2013. The y-axes correspond to the index levels.

In Table 2.2 the first four moments of the SPX log-returns and the VIX levels, over three periods of time, are depicted. The first period spans approximately the pre-crisis period, i.e. from January 3, 2005, until May 2, 2008. The second period covers the crisis from May 5, 2008, up to July 1, 2010. The third period coincides with the post-crisis period from July 2, 2010, till August 1, 2013. We see overall that the log-returns of the SPX exhibit a lower skewness but a higher excess kurtosis, which implies a leptokurtic behavior indicating rare but sharp movements in the tails. One way to model this issue is to let the jump intensity in the (discounted) logarithmic price process be stochastic (cf. Assumption 2.1 and Assumption 2.4). Further, the VIX levels follow a skewed but not highly leptokurtic distribution which can be modeled via a general Lévy density using a different scaling for up– and down jumps (cf. Section 2.5.2).

Correlation Coefficients between SPX– and VIX Levels 03/01/2005 – 02/08/2013 -0.6539

05/07/2007 – 02/08/2013 -0.7261

20/07/2009 – 02/08/2013 -0.6670

03/01/2005 – 02/05/2008 0.3259

05/05/2008 – 01/07/2010 -0.6905

02/07/2010 – 02/08/2013 -0.6971

Table 2.1 – Correlation coefficients for the Standard & Poor’s 500 index and the volatility index for different date intervals, covering the pre-crisis–, crisis–, and post-crisis periods. The sample period consists of the period 03/01/2005 – 02/08/2013.


c 2014


16

Descriptive Statistics

Log-Returns SPX VIX

Mean -0.0002 0.1543

03/01/2005 – 02/05/2008 Volatility Skewness Excess Kurtosis 0.0087 0.1457 2.5872 0.0509 1.2577 0.4770

Log-Returns SPX VIX

Mean 0.0006 0.3126


Log-Returns SPX VIX

Mean -0.0007 0.1983


Table 2.2 – We provide a summary of the most crucial descriptive statistical measures for the log-returns of the Standard & Poor’s 500 index and the volatility index levels, where we standardized the VIX by 100. In particular, the mean, the volatility (measured by the standard deviation), the skewness and the excess kurtosis are calculated and depicted. By choosing three different date intervals, we can cover the pre-crisis–, crisis–, and post-crisis periods. The sample period consists of the period 03/01/2005 – 02/08/2013.

Finally we take a look at the implied volatility skews of SPX– and VIX options. The results above indicate very different shapes of the IVS slice for a fixed time-to-maturity. Indeed, as depicted in Figure 2.2, the backed out implied volatilities of the SPX options are negatively skewed, i.e. decreasing with increasing standardized log-moneyness m / 1.5, and for m ' 1.5 shortly increasing. The main driver of this behavior can be traced back to the leverage effect arising from the negative correlation between the SPX log-returns and its volatility. Moreover, the risk-averse investor pays more for a lowstrike put option than for a high-strike call option, making the implied volatility more skewed for lower strikes. The reverse is true for VIX options. Depending on the economic state, the IVS slices are in most cases increasing with increasing standardized log-moneyness m for a fixed time-to-maturity as can be seen from Figure 2.3. For the particular cases where the IVS slices are shaped like a hockey stick, the main explanation is the increasing implied volatility part of the corresponding IVS slice of the SPX options. Translated into economic terms, investors also believe that an upwards move in the SPX levels is possible for a given time-to-maturity, hence they buy call options on the SPX or put options on the VIX since they expect a lower volatility due to the leverage effect.15 Furthermore, in Table 2.3 we depict six different quantiles of implied volatility values for SPX– and VIX options for the four different dates in Figure 2.2 and Figure 2.3. We see that the VIX implied volatilities are in general substantially higher than the corresponding SPX implied volatilities. One exception is observed for the high quantiles 0.9, 0.95, 0.975 and 0.99 on February 29, 2009. On this day two main events coincided and boosted the anxiety and uncertainty in the investor’s sentiment, explaining this behavior. First, the U.S. Commerce Department released the information that the U.S. gross domestic product fell 6.2% in the final fiscal quarter of 2008. Second, the U.S. federal government announced the increase of its equity stake in Citigroup to 36%. A closer look into the data also reveals that this high quantiles are caused by very short-term SPX put options. A natural explanation is that very short-term oriented investors prefer 15

See Remark 2.7 below for a comment on the different information content of the VIX– and SPX option market.


c 2014


17

SPX put options rather than VIX call options since the leverage effect takes place with some time lags. A further stylized fact which is worth mentioning is the inversely proportional put-call trading ratio for SPX– and VIX options, i.e. we observe almost twice as many puts as calls traded daily in the S&P 500 options market and the reverse is true for the VIX market (see Bardgett et al. (2013)). SPX: 22/08/2007

SPX: 27/02/2009

0.4

0.8 τ = 0.0861 τ = 0.1639 τ = 0.2417 τ = 0.3389 τ = 0.5917 τ = 1.35 τ = 2.3611

0.3 0.25 0.2 0.15

0.6 0.5 0.4 0.3

0.1 0.05 −2

τ = 0.0611 τ = 0.1389 τ = 0.2167 τ = 0.3139 τ = 0.5667 τ = 0.8194

0.7 Implied Volatility

Implied Volatility

0.35

−1

0 1 Moneyness, m

2

0.2 −6

3

−4

SPX: 10/10/2011

4

0.4 τ = 0.1111 τ = 0.1889 τ = 0.225 τ = 0.2861 τ = 0.4417 τ = 0.6944 τ = 1.2194

0.4

τ = 0.0611 τ = 0.0806 τ = 0.1222 τ = 0.1556 τ = 0.2194 τ = 0.2972 τ = 0.3944 τ = 0.6472 τ = 0.9



2

SPX: 31/01/2013

0.6

0.3 0.2

0.3 0.25 0.2 0.15 0.1

0.1 −2

−2 0 Moneyness, m

−1

0 Moneyness, m

1

2

0.05 −4

−2

0 Moneyness, m

2

4

Figure 2.2 – The SPX implied volatility skews of four different dates as a function of the standardized logln(K/F S (T )) moneyness m := ATMIV(t,Tt )√T −t , where K is the strike level of the European-style option, FtS (T ) denotes the closing SPX futures price today at time t with maturity T , ATMIV (t, T ) stands for the at-the-money implied volatility quote, and τ := T − t is the option’s time-to-maturity which is depicted in fractional years in the plots. The data for which the volume or open interest is zero are not considered. Additionally, options with prices below 0.1$ are also not taken into considerations due to their lack of accuracy. To back out the implied volatilities, a hybrid algorithm, consisting of the Newton-Raphson algorithm and the bisection method, is used.

Remark 2.7. The information content of the SPX– and VIX option market are completely different as pointed out in Bardgett et al. (2013). Since the VIX reflects the risk-neutral expectation of the future realized volatility of the SPX returns (see Section 2.3.1), VIX options with maturity T in fractional years 30 incorporate information about the SPX volatility between T and T + τ days with τ ≡ 360 , where on the other hand SPX options with maturity T embed information about the SPX volatility up to time T . J Pascal Marco Caversaccio

c 2014


VIX: 22/08/2007

VIX: 27/02/2009

3

2

2

Implied Volatility

τ = 0.0778 τ = 0.1556 τ = 0.2528 τ = 0.5056 τ = 0.7583


18

1.5 1

τ = 0.0528 τ = 0.1306 τ = 0.2278 τ = 0.3056

1.5

1

0.5

0.5 0 −2

−1

0 Moneyness, m

1

0 −1

2

0

1 2 Moneyness, m

VIX: 10/10/2011 2 τ = 0.025 τ = 0.103 τ = 0.2 τ = 0.278 τ = 0.356 τ = 0.453

Implied Volatility

Implied Volatility

2

4

VIX: 31/01/2013

3 2.5

3

1.5 1

1.5

τ = 0.0361 τ = 0.1333 τ = 0.2111 τ = 0.3083 τ = 0.3861 τ = 0.4639

1

0.5

0.5 0 −2

0 2 Moneyness, m

4

0 −4

−2

0 Moneyness, m

2

4

Figure 2.3 – The VIX implied volatility skews of four different dates as a function of the standardized logln(K/FtV (T )) √ moneyness m := ATMIV(t,T , where K is the strike level of the European-style option, FtV (T ) denotes ) T −t the closing VIX futures price today at time t with maturity T , ATMIV (t, T ) stands for the at-the-money implied volatility quote, and τ := T − t is the option’s time-to-maturity which is depicted in fractional years in the plots. The data for which the volume or open interest is zero are not considered. Additionally, options with prices below 0.1$ are also not taken into considerations due to their lack of accuracy. To back out the implied volatilities, a hybrid algorithm, consisting of the Newton-Raphson algorithm and the bisection method, is used.

After this in-depth analysis of the VIX, SPX and their options, we would like to emphasize that very recently on October 1, 2013, the CBOE released the new CBOE S&P 500 short-term volatility index, abbreviated by VXST.16 The VXST provides a new market-based gauge of the expected 9-day volatility, making it particularly responsive to changes in the SPX. Based on the VIX methodology, the VXST is calculated using nearby– and second nearby options with at least 1 day left to expiration. Since this calculation involves SPX weekly options, it is especially useful for targeting short-term movements in the SPX volatility or particular spreading strategies. According the latest news, the CBOE and 16

See http://www.cboe.com/micro/vxst/pdf/VXST.pdf.


c 2014


Quantiles 0.25 0.5 0.9 0.95 0.975 0.99

22/08/2007 0.1469 0.1834 0.2749 0.4657 0.6561 1.0773

SPX 27/02/2009 10/10/2011 0.3318 0.2289 0.3812 0.2685 1.5586 0.4384 2.3247 0.6792 3.284 0.9935 5.7732 1.7618

31/01/2013 0.1037 0.1195 0.2155 0.3137 0.4835 0.5845

Quantiles 0.25 0.5 0.9 0.95 0.975 0.99

22/08/2007 0.8805 1.0294 2.1289 3.5588 4.7641 5.9199

VIX 27/02/2009 10/10/2011 0.7187 0.7973 0.8686 0.9331 1.3746 2.6488 1.9428 3.4821 3.2996 5.9773 4.419 7.7593

31/01/2013 0.6261 0.7703 1.608 3.6591 5.7652 7.8278

19

Table 2.3 – We calculate six different quantiles of implied volatility values for SPX– and VIX options for the four different dates in Figure 2.2 and Figure 2.3. The data for which the volume or open interest is zero are not considered. Additionally, options with prices below 0.1$ are also not taken into considerations due to their lack of accuracy.

CBOE Futures Exchange (CFE) are exploring the possibility of introducing VXST cash-settled options and futures contracts. This opens a new challenge for academic research by developing an analytical tractable and parsimonious model for the simultaneous pricing of VIX and VXST options. In Figure 2.4 and Figure 2.5, respectively, we plot the joint evolution of the VIX and VXST and its daily difference, defined by VXST – VIX, during the sample period 03/01/2011 – 01/10/2013. The VXST data were calculated back from the CBOE using nearby– and second nearby options with at least 1 day left to expiration. On April 20, 2011, the calculation of the VXST was based solely on 30-day SPX options since weekly SPX options were not available on that date. We immediately see that the VXST exhibits a higher historical volatility and bigger one-day movements. This high fluctuations are due to the higher implied volatilities of weekly SPX options compared with monthly SPX options which are part of the VIX calculations. From an investor’s view, one can efficiently take advantage of market events, such as earnings, government reports and Fed announcements, by trading weekly options. The average daily volume of weekly SPX options has reached nearly 200’000 contracts in September 2013, which justifies in a reasonable way the introduction of the VXST. Let us remark that there also exists the CBOE 3-month volatility index (VXV), which measures the expected 3-month volatility of the SPX. By using VXST, VIX and VXV together, one can obtain a useful insight into the term structure of SPX options implied volatilities. We continue in the next section by the modeling setup of the S&P 500 index. The former empirical findings will be incorporated via introducing jump components, governed by Lévy densities, and a timechanged Browian motion with a MAJD Wishart activity rate (see Section 2.5.1).


c 2014


20

Joint CBOE VIX−VXST Dynamics 70

60

VIX / VXST Level

50

40

30

20

10 Jan−11

May−11

Sep−11

Jan−12

May−12 Date

Sep−12

Jan−13

May−13

Sep−13

Figure 2.4 – The evolution of the CBOE volatility index (black solid line) and the Standard & Poor’s 500 short-term volatility index (red solid line) during the sample period 03/01/2011 – 01/10/2013. The VXST data were calculated back from the CBOE using nearby– and second nearby options with at least 1 day left to expiration. On April 20, 2011, the calculation of the VXST was based solely on 30-day SPX options since weekly SPX options were not available on that date. The y-axis corresponds to the index levels. Daily Difference Between VXST and VIX 20

VXST − VIX Level

15

10

5

0

−5 Jan−11

May−11

Sep−11

Jan−12

May−12 Date

Sep−12

Jan−13

May−13

Sep−13

Figure 2.5 – The evolution of the daily difference between the CBOE volatility index and the Standard & Poor’s 500 short-term volatility index during the sample period 03/01/2011 – 01/10/2013. The difference is calculated by VXST − VIX. The VXST data were calculated back from the CBOE using nearby– and second nearby options with at least 1 day left to expiration. On April 20, 2011, the calculation of the VXST was based solely on 30-day SPX options since weekly SPX options were not available on that date. The y-axis corresponds to the difference level.


c 2014

2.4. PRELIMINARY OUTLINE ON THE S&P 500 DYNAMICS

2.4

21

Preliminary Outline on the S&P 500 Dynamics

Every option pricing model which is an extension of the classical Black-Scholes framework is based on some particular empirical findings. We provide a brief (mathematical rigorous) introduction to different models with increasing flexibility. The reader with a solid background on local volatility–, stochastic volatility–, and jump diffusion models can skip this part. These foundations build the empirical rationale behind the time-changed Lévy model in Section 2.4.2. Moreover, in Section 2.4.1 we provide a preliminary mathematical introduction into Lévy processes. The reader which is familiar with the mathematical foundations of Lévy processes can also skip this part. We start our analysis by depicting the general theoretical framework of geometric Brownian motions. The diffusion framework from Black and Scholes (1973) and Merton (1973) is subsequently extended to stochastic volatility and jumps which allow to incorporate fat tails and skewness into the S&P 500 dynamics (see Figure 2.2 and Table 2.2). The arbitrage-free price of a European-style option written on k underlying assets can be represented i h RT V (t, x) = EQ e− t r(Xs− )ds ξ (XT ) Xt = x > where X := X 1 , . . . , X k is a Rk -valued stochastic process, r ∈ C 0 Rk ; R≥0 is the deterministic interest rate, ξ : Rk → R≥0 denotes the payoff of the option and the expectation operator E is taken under the risk-neutral measure Q. As next, we define the general stock price process of the k assets by d X Σij (Xt ) dWtj , X0i = Z i ∈ R, i = 1, . . . , k, (2.13) dXti = bi (Xt ) dt +

by

j=1

with the coefficients b : Rk → Rk , Σ : Rk → Rk×d and, for j = 1, . . . , d, the R-valued (possibly correlated) Brownian motion W j . As we only consider the S&P 500 index, we can set k = 1 in (2.13). To accurately represent the volatility smile and to be able to better capture its dynamics than local volatility models,17 we introduce the large class of stochastic volatility (SV) models18 in which the volatility is modeled as a function of at least one additional stochastic process Y 1 , . . . , Y nv , nv ≥ 1.19 > This results in a (nv + 1)-dimensional process X, Y 1 , . . . , Y nv . By assumption (k = 1), let the S&P 500 index price process, for µ ∈ R, be given by dXt = µXt dt + σt Xt dWt1 ,

X0 = x ∈ R≥0 ,

(2.14)

where we define the volatility process σ = {σt : t ∈ [0, T ]} by v σt := ζ Yt1 , . . . , Ytnv , ζ Y01 , . . . , Y0nv = y ∈ Rn≥0 , for some function ζ : Rnv → R≥0 and the Rnv -valued diffusion process Y := Y 1 , . . . , Y nv we assume nv ≥ 1. 17

18 19

>

, for which

Consider, for example, Cox and Ross (1976), Beckers (1980), Schroder (1989), Dumas et al. (1998) and Davydov and Linetsky (2001) for an overview of the parametric local volatility model constant elasticity of variance (CEV). Analytic formulas can be found in Hsu et al. (2008). SV models were introduced by Hull and White (1987). See also Heston (1993), Derman and Kani (1998), Britten-Jones and Neuberger (2000) and Fouque et al. (2000) for particular extensions and discussions. The following facts on SV models are partly based on Hilber et al. (2013), which also provide a comprehensive glance on finite element methods for derivatives pricing.


c 2014


22

√ Remark 2.8. For instance, in Heston (1993), the function ζ is given by ζ (y) = y with nv = 1 and Y following the seminal Cox-Ingersoll-Ross (CIR) process introduced by Cox et al. (1985). Extensions to multi-scale models, assuming Ornstein-Uhlenbeck (OU) dynamics (cf. Uhlenbeck and Ornstein (1930)), are analyzed by Fouque et al. (2000) for nv = 1 (the particular case of the Scott (1987)– and Stein and Stein (1991) model can be carried out of their model) and for nv = 2 by Fouque et al. (2003) in which they model contemporaneous fast– and slow scale volatility factors. J Summarizing, we can describe the dynamics of the vector process Z := X, Y 1 , . . . , Y nv by the following system of SDEs dZt = b (Zt ) dt + Σ (Zt ) dWt ,

>

∈ Rnv +1

Z0 = z ∈ Rnv +1 ,

with the coefficients b : Rnv +1 → Rnv +1 , Σ : Rnv +1 → R(nv +1)×(nv +1) and the Rnv +1 -valued Brownian motion W . A particular drawback of SV models is their weak short-term leptokurtic behavior, inducing the replication of a too flat implied volatility skew for short maturities. Therefore, researchers have tended to add a jump component to the dynamics in (2.14) to ensure a better fit of the short-term volatility smile.20,21 This kind of approach is abbreviated by stochastic volatility with jumps in the underlying asset (SVJ) model. Further modifications in the literature are given by adding an additional jump component to the stochastic variance process which are referred to as stochastic volatility with random jumps in both the underlying asset and variance process (SVJJ) model, or without any jumps in the underlying asset denoted by stochastic volatility with jumps in the variance process (SVVJ) model. The former deliberations will lead to a rich structure of jumps in the stock and volatility dynamics of the SPX. However, before stating the general stock price dynamics of the S&P 500 which will involve Lévy densities and jumps, we provide a short introduction into Lévy processes.

2.4.1

Preliminaries on L´ evy Processes

Lévy processes represent a generalization of the classical Black-Scholes framework by allowing the stock prices to exhibit jumps while preserving the independence and stationarity of returns. The particular class of exponential Lévy models are capable to capture the leptokurticity–, skewness– and aggregational Gaussianity 22 of returns and possible discontinuities in the stock dynamics. In the framework of derivative pricing, for which the underlyings follow different Lévy processes, the mathematical rigor becomes more involved and also increases the complexity for numerical solutions due to the solving of partial integro-differential equations. Nevertheless, the general class of Lévy processes exhibit a high level of flexibility with analytical tractability, since their characteristic function is known analytically, and contain most processes suggested as realistic models for log-returns. In spite of these advantages, one particular shortcoming is featured by the limitation of independent increments avoiding capturing the volatility clustering effect. Hence, Lévy processes cannot generate distributions that vary over time. 20

21

22

Jump diffusion processes were historically introduced shortly after the seminal paper of Black and Scholes (1973) by Merton (1976). Consider also, for instance, Kou (2002) and Kou (2004) or Carr et al. (2003) where they develop an exponential L´ evy model (see Section 2.4.1 for an introduction into L´ evy processes) with SV via time change. A¨ıt-Sahalia and Jacod (2012) introduce a theory of increment power variation for semimartingales, using high frequency data, for which the limiting behavior exhibit jumps or a continuous part in the stock price dynamics. Hence, they prove that option data consist of both, jumps and a continuous part, hereby implying the necessity of modeling jumps. Generally speaking, if one increases the time scale over which returns are calculated, their distribution converge to a normal distribution.


c 2014


23

If we model returns by Lévy processes, they generate option IVSs that stay the same over time. Thus, Lévy processes cannot capture stochastic volatility, stochastic risk reversal (skewness) and stochastic correlation. These drawbacks can be resolved, at least to some extent, by considering time-changed Lévy processes for which it is possible to generate distributions which vary over time. If the return innovation is modeled by a Brownian motion, we can let the instantaneous variance be stochastic (cf. Heston (1993) and Bates (1996) for instance) and are able to create dependence of the return increments.23 We apply this particular technique in Section 2.4.2 to transform a non-homogenous Lévy process into a homogenous one. We first give the general definition of a Lévy process and continue accordingly by some general characteristics, representations and seminal formulaes. We mostly follow Applebaum (2009), Jeanblanc et al. (2009) and partly Hilber et al. (2013). By no means is the following section a comprehensive treatment of Lévy processes. We refer to Bertoin (1996), Sato (1999), Cont and Tankov (2004a) and Applebaum (2009) for further details. Definition 2.6. An adapted, c` adl` ag stochastic process X = {Xt : t ∈ [0, T ]} on (Ω, F, F, P) with values in Rk such that X0 = 0 a.s. is called a L´ evy process if it has the following properties: (i) Independent increments: For any sequence t0 < · · · < tn the Rk -valued random variables Xt0 , Xt1 − Xt0 , . . . , Xtn − Xtn−1 are independent; (ii) Stationary increments: The law of Xt+h − Xt does not depend on t, i.e. L (Xt+h − Xt ) = L (Xh ) where L (Xt ) is the law of Xt for t ∈ [0, T ]; (iii) Stochastic continuity: ∀ε ∈ R+ , limh→0 P (|Xt+h − Xt | ≥ ε) = 0. 4 This definition can also be found in (Applebaum, 2009, Chapter 1, Section 1.3) or (Hilber et al., 2013, Chapter 10, Section 10.1). Note that for an adapted, càdlàg stochastic process (Xt )0≤t≤T starting at X0 = 0 on (Ω, F, F, P) (i) and (ii) imply (iii). Remark 2.9. The particular class of Lévy processes taking values in [0, ∞[ (equivalently, which have increasing paths) are called subordinators. In Section 2.5.1 we define a matrix-valued subordinator on the positive cone of symmetric positive semi-definite k × k matrices. Other examples of subordinators include the gamma– and Poisson process or the first passage time of a Brownian motion. J An important tool in the framework of Lévy processes is the the jump measure. First we give the general definition of a random measure which will be linked to the jump measure in a further step. Definition 2.7. Let (E, E) be a measurable space and (Ω, F, P) a probability space. A random measure ϑ on the space Rk≥0 × E is a family of positive measures (ϑ (ω; dt, dx) ; ω ∈ Ω) defined on Rk≥0 × E such that, for [0, t] × A ∈ B ⊗ E, the map ω 7→ ϑ (ω; [0, t] , A) is F-measurable, and satisfying ϑ (ω; {0} × E) = 0. 4 This definition of a random measure can be found in (Jeanblanc et al., 2009, Chapter 8, Section 8.8). We can associate to the Lévy process X = {Xt : t ∈ [0, T ]} the jump measure J on [0, T ] × Λ, where

23

Carr and Wu (2004), for instance, introduced a time-changed L´ evy model to better capture the leverage effect. See also Footnote 2 for the description of the leverage effect.


c 2014

2.4. PRELIMINARY OUTLINE ON THE S&P 500 DYNAMICS ¯ where Λ ¯ is the closure of Λ, by Λ ∈ Rk is a bounded Borel set, such that 0 6∈ Λ, X JX (ω, ·) = 1Λ (∆Xt ) .

24

(2.15)

t∈[0,T ]

Literally speaking, for any measurable subset Λ ⊂ Rk , JX ([0, t] × Λ) counts the number of jumps of the process X which take values in Λ up to time t. Next we define the seminal Lévy measure (cf. (Jeanblanc et al., 2009, Chapter 11, Section 11.1) and link it to the random jump measure. Definition 2.8. A L´ evy measure on Rk is a positive measure ν on Rk \ {0} such that Z 1 ∧ |x|2 ν (dx) < ∞,

(2.16)

Rk \{0}

i.e.

Z

Z

|x|2 ν (dx) < ∞.

ν (dx) < ∞ and |x|>1

0 0, the random variable Xt is infinitely divisible.


c 2014

2.4. PRELIMINARY OUTLINE ON THE S&P 500 DYNAMICS Proof. By using the decomposition Xt =

Pn

k=1

X kt − X (k−1)t n

25

we can write

n

t Xt = X nt + X 2t − X + · · · + X − X . (n−1)t t n n n

Straightforward we observe, due to the independent and stationary increments of a Lévy process, the infinitely divisible character of any variable Xt . The following two theorems are essential tools for calculations within the Lévy framework. Theorem 2.2 (L´ evy-Khintchine representation). Let X be a Lévy process taking values in Rk . Then, for each t, the random variable Xt is infinitely divisible and its characteristic function is given, for u ∈ Rk , by the Lévy-Khintchine formula: !! Z u · Au iu·x E (exp (iu · Xt )) = exp t iu · m − + e − 1 − iu · x1|x|≤1 ν (dx) , (2.17) 2 Rk \{0} where m ∈ Rk , A is a positive semi-definite matrix, and ν is a Lévy measure on Rk \ {0}. The random variable Xt admits the characteristic triple (tm, tA, tν). We shall say in short that X is a (m, A, ν) Lévy process. Proof. The proof is depicted in detail by (Applebaum, 2009, Chapter 1, Section 1.2) for general infinitely divisible stochastic processes and by (Jeanblanc et al., 2009, Chapter 11, Section 11.2) for Lévy processes.

Note that if A vanishes, then X = {Xt : t ∈ [0, T ]} is called a pure jump process. Theorem 2.3 (L´ evy-Itˆ o decomposition). If X is a Rk -valued Lévy process, it can be decomposed into X = Y (0) + Y (1) + Y (2) + Y (3) where Y (0) is constant drift, Y (1) is a linear transform of a Brownian motion, Y (2) is a compound Poisson process with jump sizes greater than or equal to 1 and Y (3) is a Lévy process with jump sizes smaller than 1. The processes Y (i) are independent. Proof. The proof may be found in (Applebaum, 2009, Chapter 1, Section 2.4) or in (Sato, 1999, Chapter 1). The former result yields the following representation: Z tZ Z tZ fX (ds, dx) + Xt = mt + Wt + x1{|x|≤1} J x1{|x|>1} JX (ds, dx) 0 0 Z tZ X fX (ds, dx) + = mt + Wt + x1{|x|≤1} J ∆Xs 1{|∆Xs |>1} , 0

(2.18)

s≤t

where W is a Rk -valued Brownian motion with correlation matrix A, JX the random measure of the fX (dt, dx) := JX (dt, dx)−dtν (dt, dx) is the compensated jumps of X (see (2.15) for the definition), and J martingale measure. Pascal Marco Caversaccio

c 2014


26

As shown by Ané and Geman (2000), returns belong to the class of semimartingales. Hence, to be able to apply the tractability of the Lévy framework to financial modeling issues, we need to assure the semimartingale property of Lévy processes. Corollary 2.1. A Lévy process is a semimartingale. Proof. It is a consequence of the Lévy-Itˆ o decomposition in Theorem 2.3 and (2.18). We refer to (Applebaum, 2009, Chapter 2, Section 2.7) for the detailed proof. The Lévy-Khintchine representation in (2.17) provides the characteristic function of a Lévy process. Its exponent entirely determines the distribution of X1 (which is infinitely divisible) and is given a special name due to its importance. The subsequent definitions can be found in (Jeanblanc et al., 2009, Chapter 11, Section 11.2). Definition 2.10. The continuous function κ : Rk → C such that E [exp (iu · X1 )] = exp (−κ (u)) is called the characteristic exponent (sometimes the Lévy exponent) of the Lévy process X.

4

Accordingly, we can also define the cumulant function of the Lévy process X. Definition 2.11. If E eλ·X1 < ∞ for any λ with positive components, the function Ψ defined on k [0, ∞) , such that E [exp (λ · X1 )] = exp (Ψ (λ)) is called the Laplace exponent (sometimes the cumulant function) of the Lévy process X.

4

Straightforward from the former definitions, it follows that E [exp (iu · Xt )] = exp (−tκ (u)) and, if Ψ (λ) exists, E [exp (λ · Xt )] = exp (tΨ (λ)) and Ψ (λ) = −κ (−iλ) . By (2.17), the Lévy-Khintchine formulae, we get Z u · Au − eiu·x − 1 − iu · x1|x|≤1 ν (dx) , κ (u) := −iu · m + 2 Rk \{0} Z λ · Aλ Ψ (λ) := λ · m + + eλ·x − 1 − λ · x1|x|≤1 ν (dx) . 2 Rk \{0}

(2.19)

(2.20) (2.21)

In Section 2.7 we extensively use the former identities, namely (2.19), (2.20) and (2.21), and Definition 2.10 and Definition 2.11. Lévy processes can be characterized by finite– and infinite activity, respectively, and by finite– and infinite variation, respectively. As we will see, the Levy measure incorporates most of the information of Pascal Marco Caversaccio

c 2014


27

the process. The subsequent definition is partly taken from (Applebaum, 2009, Chapter 2, Section 2.4) and (Jeanblanc et al., 2009, Chapter 11, Section 11.2). We refer to (Sato, 1999, Chapter 4, Section 4.21) for a more general definition of finite variation. Definition 2.12. Let X = {Xt : t ∈ [0, T ]} be a k-dimensional Lévy process with characteristic triple (m, A, ν). If ν Rk \ {0} < ∞, then we say that X is of finite activity. If moreover A = 0, the process k X is a compound Poisson process with drift. Conversely, if ν R \ {0} = ∞, the process X is said to be R of infinite activity. Besides assume that A = 0, then we say if |x|≤1 |x| ν (dx) < ∞, then almost all R paths of X have finite variation on any finite time interval. On the other hand, if |x|≤1 |x| ν (dx) = ∞, then almost all paths of X have infinite variation on any finite time interval. 4 We conclude the theoretical part of this section by a useful result to define (complex) measure changes.

Proposition 2.3. Let X = (Xt )0≤t≤T be a k-dimensional Lévy process with the characteristic exponent κ and assume that E eiu·Xt < ∞, u ∈ Rk . Then the process M = (Mt )0≤t≤T defined as Mt :=

eiu·Xt = eiu·Xt +tκ(u) E (eiu·Xt )

is a martingale. Proof. By definition κ (u) is finite and hence E (|Mt |) is bounded since E (|Mt |) ≤ exp (tκ (u)). For 0 ≤ s < t, we have i h E eiu·(Xt −Xs )+(t−s)κ(u) Fs = e(t−s)κ(u)−(t−s)κ(u) = 1.

We use in Section 2.7.1 an analogue definition for the Laplace exponent Ψ to define an ordinary measure change. Eventually, we review briefly some Lévy processes and discuss their characteristics.

Example 2.3 (Drifted Brownian motion). Let W be a standard Brownian motion. We call the Lévy process {mt + σWt : t ∈ [0, T ]} a drifted Brownian motion with characteristic exponent κ (u) = 2 2 −ium + u 2σ and Lévy triplet (m, σ, 0). Moreover, the Lévy measure is zero, and in particular it holds that the family of Lévy processes with ν = 0 are composed of all Lévy processes with continuous sample paths of the representation {mt + σWt : t ∈ [0, T ]} for any m ∈ R, σ ∈ R+ . ⊥ Example 2.4 (Poisson process). We call the Lévy process N = {Nt : t ∈ [0, T ]} a Poisson process with intensity λ and characteristic exponent κ (u) = λ 1 − eiu . Its Lévy triplet reads (0, 0, λδ1 ), where δ1 is the Dirac function evaluated at 1 (due to the jump size of 1 for a standard Poisson process). This will be extended for compound Poisson processes. ⊥ n o PNt Example 2.5 (Compound Poisson process). The Lévy process X = Xt = k=1 Yk : t ∈ [0, T ] is called a compound Poisson process where N is a Poisson process with intensity λ and Yk are an i.i.d sequence of independent random variables with PDF f and CDF exponent i Fh . The characteristic R Xt iuy reads κ (u) = λ R\{0} 1 − e f (y) dy and yields the Lévy triple λE |Xt | < 1 , 0, ν , for Xt , where ν (dx) := λF (dx). ⊥ Pascal Marco Caversaccio

c 2014


28

Example 2.6 (Further examples). Other examples of Lévy processes include linear drifts, jump diffusion models (cf. Merton (1976)), Brownian hitting times, the compensated Poisson process, the normal inverse Gaussian (NIG) process (cf. Barndorff-Nielsen (1997)), the Meixner process (cf. Schoutens (2002)), the generalized hyperbolic (GH) process (cf. Eberlein et al. (1998)), tempered stable processes (cf. Carr et al. (2002)) and the variance-gamma (VG) process (cf. Madan et al. (1998)). ⊥ The class of Lévy processes contains also the class of stable processes inducing a stable distribution.

Definition 2.13. A real-valued random variable is stable if for any a ∈ R+ , there exist b ∈ R+ and a c ∈ R such that [b µ (u)] = µ b (bu) eicu . A random variable is strictly stable if for any a ∈ R+ , there a exists b ∈ R+ such that [b µ (u)] = µ b (bu). The characteristic function µ of an α-stable law can be written  1 2 2  for α = 2, exp imu − 2 σ u , µ b (u) = exp imu − γ |u|α 1 − iβ sgn (u) tan πα , for α 6= 1, α 6= 2, 2   exp (imu − γ |u| (1 + iβ ln |u|)) , for α = 1, where β ∈ [−1, 1] and m, γ ∈ R, σ ∈ R+ . For α 6= 2 the Lévy measure of an α-stable law is absolutely continuous with respect to the Lebesgue measure with density ( c+ x−α−1 dx, if x > 0, ν (dx) := (2.22) −α−1 c− |x| dx, if x < 0. Here, c± are positive real numbers given by αγ 1 (1 + β) 2 Γ (1 − α) cos αγ 1 c− := (1 − β) 2 Γ (1 − α) cos c+ :=

απ 2 απ 2

, ,

where Γ (·) is the gamma function defined, for w ∈ R+ , by Z +∞ Γ (w) := xw−1 exp (−x) dx. 0

Conversely, if ν is a Lévy measure of the form (2.22), we obtain the characteristic function of the law by (c+ −c− ) 4 setting β = (c+ +c− ) . We can reformulate this definition in terms of of random variables. X is stable if d (n) (n) ∀n, ∃ Xi , i ≤ n, i.i.d.; βn , γn , such that X1 + · · · + Xn(n) = βn X + γn . The former definition might also be found in (Jeanblanc et al., 2009, Chapter 11, Section 11.1). We call a random variable α-stable if, for any strictly stable random variable, the following representation holds: 1 b (a) = ka α , 0 < α ≤ 2. γ

The finite γ-moments of the α-stable random variable, i.e. E (|X| ) < ∞, are assured iff γ < α. One peculiarity is given by the fact that except for the Gaussian distribution, the second moment for stable Pascal Marco Caversaccio

c 2014


29

distributions does not exist.24 Moreover, we obtain a Cauchy distribution (resp. Laplace distribution (sometimes referred to as double exponential distribution)) by setting α = 1 (resp. α = −1). By setting β = 0 and m = 0, we get α µ b (u) = exp (−γ |α| ) , which is called symmetric stable law . We depict in Figure 2.6 the symmetric PDFs of the α-stable distribution for several values of α. An example of skewed α-stable PDFs is illustrated in Figure 2.7. One can readily see the high flexibility by this particular class of distribution of capturing different skew– and tail patterns which can therefore be ideally applied to short-term option pricing due to the implied volatility smile/smirk (see, for instance, Figure 2.2, Figure 2.3 and Table 2.2 for the empirical evidence of the implied volatility smile/skew).

Symmetric α−Stable Densities α = 0.5 α = 0.75 α = 1.0 α = 1.25 α = 1.5

0.6

0.5

f(x)

0.4

0.3

0.2

0.1

0 −5

−4

−3

−2

−1

0 x

1

2

3

4

5

Figure 2.6 – The symmetric PDFs of the α-stable distribution for different values of α. The other parameters are fixed: the skewness parameter β = 0, the scale parameter γ = 1, and the location parameter m = 0. We applied the parametrization of stable distributions used in Samoradnitsky and Taqqu (1994).

24

This would be the case iff α = 2.


c 2014


30

Skewed α−Stable Densities α = 0.5 α = 0.75 α = 1.0 α = 1.25 α = 1.5

0.5

f(x)

0.4

0.3

0.2

0.1

0 −5

−4

−3

−2

−1

0 x

1

2

3

4

5

Figure 2.7 – The skewed PDFs of the α-stable distribution for different values of α with the skewness parameter β = 0.5. The other parameters are fixed: the scale parameter γ = 1, and the location parameter m = 0. We applied the parametrization of stable distributions used in Samoradnitsky and Taqqu (1994).

2.4.2

Non-Homogenous L´ evy Process

We start our analysis now by proposing the multivariate dynamics directly under the non-unique riskneutral measure Q. The following dynamics, which is governing for the univariate case, after applying the time change technique outlined below, the S&P 500 movements in Assumption 2.4 of Section 2.6, involves enough flexibility to account for the different empirical evidence such as the volatility smile, fat tails of the return distribution and the power-law scaling property of return moments. This representation can also be used as a starting point in further research to develop a consistent option pricing model for more sophisticated options, e.g. exotics. We first choose the multivariate dynamics and carve out step by step the SPX dynamics. We define the multivariate Lévy-Itˆ o-semimartingale of the following form: Assumption 2.1 (Multidimensional price dynamics under Q). Assume > Xt := (exp (St,1 ) , . . . , exp (St,k )) ,

0 ≤ t ≤ T,

where we denote the k-dimensional (discounted) logarithmic price process by (St )0≤t≤T starting a.s. at


c 2014


31

S0 = s = 0.25 We assume the following representation of the logarithmic price process S under the non-unique risk-neutral probability measure Q: Z t q Z tZ Z t W dW Q + bSs ds + tr x JS− (ds, dx) − ηJQ− (x) dxνsJ ds St = s + Vs− s 0

0

Z tZ +

x 0

(0,∞)k

JS+

0

(ds, dx) −

ηJQ+

(−∞,0)k

(x) dxνsJ ds

(2.23)

,

√ where V W represents the unique square root on Sk+ , the space of symmetric positive semi-definite k × k matrices, of the instantaneous covariance process V W = VtW : t ∈ [0, T ] specified in Assumption 2.3, and • bS denotes a Rk -valued locally bounded process, • W Q is a Rk×k -valued Q-Brownian motion, where W Q := BP > + Z

• • • • • •

p Ik − P P > ,

with the standard Brownian motions B ∈ Rk×k and Z ∈ Rk×k , which are independent of each other, and P ∈ Rk×k is a deterministic correlation matrix such that P P > ≤ Ik is a positive semi-definite matrix, the discontinuous shocks J are split up into separable additive components, i.e. Jt := Jt+ + Jt− , ∀t, JS− (ds, dx) and JS+ (ds, dx) are the random measures associated with the upward and downward jumps of S, respectively,26 the characterization of the jump structure is given by the corresponding Lévy densities ηJQ− (x) and ηJQ+ (x), the jump size is xi , for i = 1, . . . , k, the stochastic jump intensity is denoted by ν J , and hence the arrival rate of upward (resp. downward) jumps of size x at time t is determined by ηJQ+ (x) νtJ dt (resp. ηJQ− (x) νtJ dt), ηJQ+ (x) dxνtJ dt (resp. ηJQ− (x) dxνtJ dt) defines the Q-compensator for upward (resp. downward) jumps.27

Note that we do not integrate the jump size x for zero since we obviously assume xi 6= 0, for i = 1, . . . , k. Besides, the specification in (2.23) defines a non-homogenous Lévy process under Q and is characterized by its time-dependent local characteristic triplet.28 Let us now apply the time change technique applied by Leippold and Strømberg (2012). The absolutely continuous time changes can be defined by Z t Z t W TtW := Vs− ds and TtJ := νsJ ds, (2.24) 0

0

where we separately specify the components. To be consistent with our notation, we write Z t q k X d fQ W dW Q fQ W , tr Vs− = d W = dW s TW ij,T 0

25 26 27

28

t

i,j=1

(2.25)

t,ij

This particular initial condition is required by Definition 2.6. See Definition 2.7 and (2.15) for the definition of the random measure and the jump measure, respectively. Carr and Wu (2011) propose similar jump specifications for the equity index dynamics that accommodates all three economic channels of the return volatility variation, i.e. the leverage– and volatility effect and the self-exciting behavior. Additionally, Leippold and Strømberg (2012) assume a non-homogenous L´ evy process which is governing the forward LIBOR rate with the similar jump structure. The general characteristic triplet is defined in Theorem 2.2, the L´ evy-Khintchine representation.


c 2014

2.4. PRELIMINARY OUTLINE ON THE S&P 500 DYNAMICS where W Tt,ij

Z :=

32

t W Vs−,ji ds,

1 ≤ i, j ≤ k,

(2.26)

0

e defined by Ft := FT W . Even though f Q is a Brownian motion with respect to another filtration F and W t we will work on the positive cone of symmetric positive semi-definite k × k matrices, we denote, without loss of generality, the off-diagonal elements by different indices. Further, we refer to Carr and Wu (2004) for an overview on time-changed Lévy processes. Note as first now that we can write the price of a zero-coupon bond with maturity t and stochastic interest rate r as i h Rt B (0, t) = EQ e− 0 r(Xs− )ds , where X = {Xt : t ∈ [0, T ]} is a Sk+ -valued stochastic process.29 Moreover, we assume Assumption 2.2. The short rate process r = {r (Xt ) : t ∈ [0, T ]} is affine, i.e. r (x) = ρ0 + ρ1 x, where ρ0 ∈ R≥0 , ρ1 ∈ Sk+ . Nevertheless, the previous assumption is by no means exhaustive. One could also think about the quadratic term structure model of Leippold and Wu (2002). Now, if we compare this expression to the Laplace transform of a time change T under Q, R Definition 2.14. We call LX (λ) := R e−λx µ (dx) the Laplace transform of the random variable X −λX under a given probability µ and defined on the interval λ ∈ R : E e 0, Q J+ ηJ (x) = (2.29) |x| η Q (x) = λe− ν− |x|−1 , x < 0. J−

Note, we preserve the analytical tractability because the Lévy exponent of the jump process in (2.29) is known in closed-form. (2.29) also implies a decreasing jump arrival rate with increasing jump size x. As shown in Figure 2.1, the SPX exhibits sharp downward movements and more regular upward movements. We can control, applying a different scaling, for this sharp upward jumps in (2.29). In general, given any dynamics with the Lévy density as in (2.29) and jump arrival rate ηJQ+ (x) dxνtJ dt (resp. ηJQ− (x) dxνtJ dt) at time t, it allows us to capture any asymmetric discontinuous movements in the jump arrival rate. In Figure 2.8 we plot the VG density of (2.29) for different values of the scaling parameters ν+ and ν− , and the other parameter is set to λ = 0.5. The figure describes the expected number of jumps (corresponding to ηJVG (x) in the plot) of a certain height (corresponding to Jump Size x in the plot) in a time interval of length 1. Notice that we exhibit a singularity at the point x = 0, which is due to the blatant reason of diffusion transition. We can deduce from Figure 2.8 that the VG density in (2.29) consists of highly flexible scaling parameters which can control for any asymmetry in the jump arrival rate. A further 32

Madan and Senata (1990) studied as first the VG process for option pricing purposes. An extension is provided by Madan et al. (1998). This particular process is constructed by a time-changed drifted Brownian motion, where the time change is given by a gamma clock Γ = {Γ (t) : t ∈ [0, T ]}, i.e. Γ is a L´ evy process with gamma distributed increments. Also note, the VG process is a pure jump process with infinite activity, finite variation and all moments exist (compare Definition 2.12 for the notion of infinite activity and finite variation).


c 2014

2.5. THE JOINT COVARIANCE DYNAMICS

35

interesting fact is that the jump arrival rate decreases monotonically with increasing jump size x but still can capture rare high upward– and downward jumps (see the period March 2008 - April 2009 in Figure 2.1), respectively. Thus, extreme market movements, a very important modeling issue, can be taken into account. Variance−Gamma Density 300 ν = 0.1, ν = 1.1 +

−

ν = 2.1, ν = 0.01 +

−

ν = 0.01, ν = 0.1 +

250

−

ν+ = 0.04, ν− = 0.03

ηVG (x) J

200

150

100

50

0 −1

−0.8

−0.6

−0.4

−0.2

0 Jump Size x

0.2

0.4

0.6

0.8

1

Figure 2.8 – The VG density function with the specification given by (2.29) and different values for the scaling parameters ν+ and ν− . The other parameter is set to λ = 0.5.

2.5

The Joint Covariance Dynamics

To complete the theoretical pricing setup, one last assumption is needed. We specified in (2.24) the time changes and defined the activity rate of the time change T J in (2.27) by the seminal CIR process. As discussed in Section 2.3.2, we need a mean-reverting covariance dynamics with stochastic correlation, stochastic skewness and different co-volatility structures in time between the SPX and VIX. In particular, to preserve consistency with the S&P 500 index, we have to introduce a jump component with sharp upward (resp. downward) amplitudes in the case of the VIX (resp. SPX). This can be obtained by applying a different scaling to the matrix-variate Lévy density specified in Section 2.5.2. All these prediscussed stylized facts can be incorporated by a jump-extended version of the Wishart process which is presented as next.33,34 33 34

The Wishart process is a multivariate extension of the CIR process. Therefore, the activity rate of the time change T J = TtJ : t ∈ [0, T ] in (2.27) preserves consistency with our modeling framework. The pure diffusion Wishart process, i.e. with a zero-jump intensity, is studied in Gouri´ eroux et al. (2009) and Gouri´ eroux and Sufana (2010).


c 2014


2.5.1

36

Matrix Affine Jump Diffusion Wishart Process

Throughout this section, we follow in particular Leippold and Trojani (2010) who developed the new class of flexible and tractable MAJD. We suppose that V W is an adapted Markov process in the state space Sk+ , the positive cone of symmetric positive semi-definite k × k matrices.35 Assumption 2.3 (Multidimensional joint covariance dynamics under Q). The joint covariance dynamics (or the stochastic activity rate of the absolutely continuous time change under the pricing measure Q for the time change T W defined in (2.24)) of S ∈ Rd and Y ∈ Sn+v under the risk-neutral probability measure Q, for which the dynamics of S = {St : t ∈ [0, T ]} is defined in (2.23) and Y = {Yt : t ∈ [0, T ]} is another Sn+v -valued Wishart process, for which we assume nv ≥ 1, and k := nv + d, can be represented by the Sk+ -valued Markov process V W which solves the SDE:36 q q dVtW = ΩΩ> + M VtW + VtW M > dt + VtW dBt Q + Q> dBt> VtW + dJtV , V0W = v ∈ Sk+ , where Ω, M, Q ∈ Rk×k , B√is a matrix of standard Brownian motions in Rk×k , J V is a pure jump process W taking values in Sk+ , and V W denotes the unique square root on Sk+ . The jump sizes ξ V are i.i.d. and W follow a finite probability distribution Y V on Sk+ \ {0}.37 We impose the restriction ΩΩ> (k − 1) Q> Q which guarantees that V W remains positive semi-definite. Furthermore, to preserve our affine framework, W : t ∈ [0, T ] defined in (2.28).38 we assume that the jumps are realized with the intensity ν J Vt− This particular jump-extended Wishart process represents the matrix analogue of a squared Bessel process defined in Example 2.2 with an additional pure jump process J ρ , i.e. for Y ∈ R≥0 we have Y ∼ BESQδ,ρ x . A moment’s reflection yields the result that in full analogy with the CIR process (see e.g. Remark 2.13 for a comment) the term ΩΩ> is related to the expected long-term variance-covariance W matrix V∞ through the solution of the linear (Lyapunov) equation (cf. Fonseca et al. (2012)) W W M V∞ + V∞ M > = −ΩΩ> .

(2.30)

The matrix M can be compared to the level of mean reversion in the CIR model. A further relationship with the parameters of the CIR process is given by Q which can be identified as the volvol parameter, i.e. the volatility of the volatility matrix. To illustrate the high flexibility of this process to model multivariate q risk structures, we state the analytical solution of the implied correlation process ρij := VijW / ViiW VjjW , for 1 ≤ i, j ≤ k. The result is due to Leippold and Trojani (2010). The proof involves the application of a matrix version of Itˆ o’s lemma and we skip it here for the reader’s convenience. Proposition 2.4. Let ei (resp. ej ), for i = 1, . . . , k, (resp. for j = 1, . . . , k,) be the i-th (resp. j-th) unit vector in Rk . The dynamics of the ij-th correlation coefficient implied by the process in Assumption 2.3

35

36 37 38

Mathematically speaking, a symmetric cone is a self-dual convex cone S, such that for any two points x, y ∈ S a linear automorphism f of S exists and which maps x into y. Moreover, it is called irreducible if it cannot be written as a non-trivial direct sum of two other symmetric cones. Consider (Cuchiero et al., 2011b, Section 2) for a thorough mathematical definition. (Mayerhofer et al., 2011, Section 3) provide the technical conditions for the existence of a unique global strong solution of the following SDE. For the avoidance of a too heavy notation, we write Sk+ \ {0} for the positive cone of symmetric positive definite k × k matrices. To obtain a Wishart transition density of this process, we need to set ΩΩ> = γQ> Q, γ > k − 1.


c 2014


37

is given by: dρij,t

p

p VtW dBt Qj + e> VtW dBt Qi j q = m (ρij,t ) dt + WV W Vii,t jj,t  p  p > W dB Q e> VtW dBt Qj e V t i j t i VW  + ρij,t ζij,t q q − ρij,t  dJt% , + W W Vii,t Vii,t e> i

% where Qi and Q j Jare the i-th and j-th column of the matrix Q, respectively, and J is a pure jump process W with intensity ν Vt− : t ∈ [0, T ] and i.i.d. (percentage) correlation jump size given by: W

1+

W

V ζij,t = s 1+

W

V ξii W Vii,t

V ξij W Vij,t

1+

W

V ξjj W Vjj,t

− 1.

m (ρij,t ) = Aij,t ρ2ij,t + Bij,t ρij,t + Cij,t is a quadratic drift function with stochastic coefficients Aij,t

Bij,t

Cij,t

e> Q> Qej = qi − Mji WV W Vii,t jj,t

s

W Vii,t W Vjj,t

s − Mij

W Vjj,t W Vii,t

,

s s ! W W > > > > > X V V``,t e Q Q − 2ΩΩ e e> Q Q − 2ΩΩ e j i j ``,t i = + − Mi` ρi`,t + Mj` ρj`,t , W W W W 2Vii,t 2Vjj,t Vii,t Vjj,t `6=i,j s s s s ! W W W W > > X V``,t V``,t Vii,t Vjj,t e> i ΩΩ − 2Q Q ej q = + Mji . + Mij + Mi` ρj`,t + Mj` ρi`,t W W W W Vjj,t Vii,t Vii,t Vjj,t WV W Vii,t `6=i,j jj,t

We can immediately deduce the nonlinear persistence properties of the correlation process with a stochastic volatility of volatility. In particular, both the drift and the volatility are non-autonomous processes and depend in general on all components of V W . W

We are left now with the specification of the finite probability distribution Y V .

2.5.2

Gamma-Type L´ evy Density W

W

To define the finite probability distribution Y V on Sk+ \ {0} for the jump sizes ξ V , we follow BarndorffNielsen and Pérez-Abreu (2008) who developed infinitely divisible gamma-type– and simple tempered matrix distributions.39,40 To our best knowledge, we are the first that vigorously examine a jumpextended Wishart process with a matrix-variate Lévy density. We define the matrix-variate Lévy density 39 40

This generalizes the ordinary matrix-valued gamma distribution which is not infinitely divisible. Compare Definition 2.9 for the notion of infinite divisibility. See also Crameri et al. (2010). The authors propose a general family of multivariate time-changed L´ evy processes based on a gamma-type, tempered– and inverse Gaussian matrix subordination approach.


c 2014

2.5. THE JOINT COVARIANCE DYNAMICS W

in terms of ξ V

38

by YV

W

:= hβ ξ V , Σ :=

det (Σ)

(k+1) 2

W exp −tr ξ V Σ−1 , k(k+1) +β W 2 tr ξ V Σ−1

−

W

(2.31)

W

where Σ ∈ Sk+ \ {0} and 0 ≤ β < 1. We write ξ V ∼ Gβ (Σ) to indicate that the random jump matrix W W ξ V has the matrix distribution associated to (2.31). To simulate the random matrix ξ V , one could also use the following approach. For 0 ≤ β < 1, we define, in terms of the random matrix M , the Lévy + density hM β (M ) : Sk \ {0} → R≥0 hM β (M ) :=

exp (−tr (M )) (tr (M ))

k(k+1) +β 2

.

(2.32)

Note that any random matrix M with Lévy density (2.32) has an orthogonal-symmetric distribution, d

i.e. U > M U = M for any U in the orthogonal group O (k).41 This implies that M is distributionally invariant under rotations and reflections, which is also a well-known property of the wide class of spherical W W distributions. Now we can simulate ξ V from the decomposition (or linear transformation) ξ V = 1 1 1 1 1 Σ 2 M Σ 2 , where Σ 2 denotes the unique square root on Sk+ \ {0} such that Sk+ \ {0} 3 Σ = Σ 2 Σ 2 . In Table 2.4 we provide an overview on the parameter specification for the gamma-type distribution in (2.31). This summary is based on proofs and indications in Barndorff-Nielsen and Pérez-Abreu (2008) W

and Crameri et al. (2010). The random variable tr ξ V

W

, where ξ V

∼ Gβ (Σ), follows an univariate

gamma distribution in the case of β = 0 and Σ = Ik , and (2.31) parameterizes the natural extension of the univariate Lévy density of a gamma process with unit mean and scaling coefficient σ. For the case W of β = 0 and Σ ∈ Sk+ \ {0}, the random variable tr Σ ξ V follows a 1-dimensional gamma convolution, introduced by Thorin (1977), which build an important W class of infinitely divisible distributions. If V VW 0 < β < 1 and Σ = Ik , the random variables tr ξ and tr Σ ξ follow obviously the same law. In the general case of 0 < β < 1, we obtain a 1-dimensional tempered β-stable distribution introduced by Rosi´ nski (2007). The special case of β = 1/2 extension of the univariate yields the matrix W

inverse Gaussian (IG) distribution which implies that tr ξ V distribution.

W

and tr Σ ξ V

follow an univariate IG

In Figure 2.9 we plot the gamma-type Lévy density function with the specification given by (2.31). The parameter Σ and β are characterized by ! 3 0.5 Σ= and β = 0.5. 0 1 We do not allow for jumping co-volatilities, which means we set the off-diagonal elements of the random W VW VW jump matrix ξ V to zero. Furthermore, we specify different value intervals for ξ11 and ξ22 , respectively. W W V V ∈ [1, 2] and ξ22 ∈ [0, 1]. The figure describes the In particular, the interval constraints are set to ξ11 expected number of jumps (corresponding to hβ (ξ, Σ) in the plot) of a certain contemporaneous height W VW VW for the different jump components of ξ V (corresponding to Jump Size ξ11 and ξ22 in the plot) in a W V time interval of length 1. Note again that we exhibit a singularity at the point ξ = 02×2 , which is due 41

A map U ∈ Rk×k is orthogonal if U U > = U > U = Ik . Geometrically, orthogonal maps describe rotations or reflections.


c 2014

2.6. SPX PRICE– AND JOINT SPX-VIX2 COVARIANCE DYNAMICS

Random Variable W tr ξ V , W

ξV

∼ Gβ (Σ)

tr Σ ξ W

ξV

W

V

,

∼ Gβ (Σ)

β = 0, Σ = Ik

Parameters 0 < β < 1, Σ = Ik

39

β = 1/2, Σ = Ik

1-Dimensional Gamma Distribution

1-Dimensional Tempered β-Stable Distribution

Univariate Inverse Gaussian Distribution

β = 0, Σ ∈ Sk+ \ {0}

0 < β < 1, Σ = Ik

β = 1/2, Σ = Ik

1-Dimensional Gamma Convolution

1-Dimensional Tempered β-Stable Distribution

Univariate Inverse Gaussian Distribution

Table 2.4 – This table provides an overview on the parameter for the gamma-type distribution specification VW VW VW in (2.31). Three particular cases for the random variable tr ξ resp. tr Σ ξ , where ξ ∼ Gβ (Σ), are depicted. This summary is based on proofs and indications in Barndorff-Nielsen and Pérez-Abreu (2008) and Crameri et al. (2010).

to the blatant reason of diffusion transition. We can also see that the density decreases monotonically with increasing simultaneous jump size. Nonetheless, rare joint high upward resp. downward jumps can W be captured due to the smoothing-out behavior of the tails. Notice, due to the support of hβ ξ V , Σ , we are only able to incorporate negative jumps via the off-diagonal elements. So far we have elaborated the general framework on which we are working with. Now we have all ingredients to delve into the SPX price– and joint SPX-VIX2 covariance dynamics.

2.6

SPX Price– and Joint SPX-VIX2 Covariance Dynamics

Before we define the SPX price dynamics and its joint covariance dynamics with the VIX2 , we discuss two particular and distinctive properties of VIX options and a particular implementation issue for the SPX. This discussion is of high importance because it will affect our modeling setup and simplifies, to some extent, the further implementation. Firstly, the underlying of VIX options is not the current VIX spot price itself but rather the corresponding futures contract. This implies that no-arbitrage considerations of VIX options must be examined relative to the VIX futures. Let us provide a short and intuitive explanation why the futures value is 30 the true underlying. First define τ ≡ 365 . Definition 2.4 and Definition 2.5 imply that a VIX call option C (t, T ), i.e. with current time t and maturity T in fractional years, is an option on the volatility on the time interval [T, T + τ ]. But on the contrary, the VIX = {VIXt : t ∈ [0, T ]} is related to the volatility in the time interval [t, t + τ ]. Clearly we have [t, t + τ ] 6= [T, T + τ ] for all t 6= T . To circumvent this asynchronism, we can consider futures on the VIX with maturity T . Obviously, VIX futures are related to the volatility on the time interval [T, T + τ ], which is evidently the underlying of the corresponding VIX option. Secondly, to avoid having to estimate the dividend yield, the S&P 500 index futures are used instead of the underlying index value. The futures incorporate the value of the dividend yield. Let the relationPascal Marco Caversaccio

c 2014


40

Gamma−Type Lévy Density

7 6

hβ(ξ,Σ)

5 4 3 2 1 0 1 0.8

2 0.6

1.8 1.6

0.4 1.4

0.2

1.2 0

Jump Size ξ22

1

Jump Size ξ11

Figure 2.9 – The gamma-type ! Lévy density function with the specification given by (2.31). We set the W 3 0.5 parameters to Σ = and β = 0.5. The random jump size matrix ξ V is characterized by the 0 1 ! VW W ξ11 0 VW resp. . Furthermore, we specify different value intervals for ξ11 diagonal matrix ξ V = VW 0 ξ22 W

W

V V ξ22 . Specifically, we have ξ11

W

V ∈ [1, 2] and ξ22

∈ [0, 1].

ship between the spot price and the futures price, where rt,T is the risk-free interest rate and qt,T the continuous dividend yield for the time period T − t, be given by: Ft (T ) = St e(rt,T −qt,T )(T −t) . We infer the value of the S&P 500 index futures at closing by backing out the value using the ATM forward put-call parity:42 CtMkt (·) + Ke−rt,T (T −t) = PtMkt (·) + Ft (T ) e−rt,T (T −t) ,

(2.33)

with CtMkt (·) := CtMkt (Ft (T ) , K ≈ Ft (T ) , T, rt,T ) , PtMkt (·) := PtMkt (Ft (T ) , K ≈ Ft (T ) , T, rt,T ) , 42

We consider the ATM put-call parity due to the liquidity of the call and put.


c 2014


41

and where Ft (T ) denotes the closing futures price today at time t with maturity T . We denote the ATM (K ≈ Ft (T )) observed market prices with the same maturity T by CtMkt and PtMkt , respectively. The risk-free interest rate rt,T with time-to-maturity T − t can be extracted from e.g. the zero-coupon yields or the implied London Interbank Offered Rate (LIBOR) swap rates for long maturities. Eventually, because the VIX itself is not a tradable asset, there is no cost-of-carry relationship between VIX futures and the spot VIX value (see, e.g., Zhang and Zhu (2006), Zhu and Lian (2012)), i.e. F (t, T, VIXt = x) 6= xert,T (T −t) . This clearly affects the put-call parity for VIX options which can however be modified trivially (see, for instance, Lian and Zhu (2013)) and is given by C (t, T, VIXt = x) − P (t, T, VIXt = x) = e−rt,T (T −t) F (t, T, VIXt = x) − Ke−rt,T (T −t) .

(2.34)

The solely difference to the ordinary put-call parity is the replacement of the underlying price by the discounted forward volatility which can be traded via the VIX future contract. Taking into account the former considerations and properties, we can now frame our modeling setup. We denote the futures price of the SPX by F = {Ft : t ∈ [0, T ]} and its 1-dimensional logarithmic price process by S = {St : t ∈ [0, T ]} = {log (Ft ) : t ∈ [0, T ]}.43 Furthermore, we define the univariate LévyItˆ o-semimartingale, based on Assumption 2.1, of the following form: Assumption 2.4 (S&P 500 index dynamics under Q). Assume Ft := exp (St ) ,

0 ≤ t ≤ T,

where we denote the 1-dimensional logarithmic price process by (St )0≤t≤T starting a.s. at S0 = s = 0. We assume the following representation of the logarithmic price process S under the risk-neutral probability measure Q:44 2 X

dSt = bSt dt +

fQ W + dW ij,T

i,j=1

Z +

∞

x 0

JS+

Z

t,ij

(dt, dx) −

ηJQ+

0

−∞

x JS− (dt, dx) − ηJQ− (x) dxdTtJ

(x) dxdTtJ

(2.35)

,

where T J = TtJ : t ∈ [0, T ] and T W = TtW : t ∈ [0, T ] denote the time changes defined in (2.24) with the stochastic activity rate defined in (2.27) for T J and (2.38) for T W , and • bS denotes a R-valued locally bounded process and is derived specifically in Proposition 2.5, f Q , for 1 ≤ i, j ≤ 2, is a R-valued time-changed Q-Brownian motion defined in (2.25) and the • W ij time change TijW is given by (2.26), • the discontinuous shocks J are split up into separable additive components, i.e. Jt := Jt+ + Jt− , ∀t, • JS− (ds, dx) and JS+ (ds, dx) are the random measures associated with the upward and downward jumps of S, respectively, • the characterization of the jump structure is given by the corresponding Lévy densities ηJQ− (x) and ηJQ+ (x) with the associated specification given by (2.29), • the jump size is x, • the arrival rate of upward (resp. downward) jumps of size x at time t is determined by ηJQ+ (x) TtJ (resp. ηJQ− (x) TtJ ), 43 44

We replace X by F in Assumption 2.1 to better grasp the random variable ”Futures” at first glance. Note that, by definition in (2.24), it holds dTtJ = νtJ dt.


c 2014


42

• ηJQ+ (x) dxdTtJ (resp. ηJQ− (x) dxdTtJ ) defines the Q-compensator for upward (resp. downward) jumps. No arbitrage considerations require Lévy processes employed in mathematical finance to be martingales.45 We provide the explicit martingale drift condition for the representation (2.35) in the following proposition. Proposition 2.5. The futures S&P 500 index price is a local martingale under Q, i.e. the risk-neutral measure, if the following drift condition at time t is satisfied: !! Z 2 X 1 Q VW . (2.36) bSt := − (ex − 1 − x) ηJ (dx) νtJ − 2 i=1 t−,ii R\{0} Proof. Let F , the futures price process, be an exponential Lévy process which admits an affine representation for the Lévy logarithmic price process S. Then, which is shown in Lemma 2.1 below, F is a local martingale. Lemma 2.1. Let the short rate process r = {r (Xt ) : t ∈ [0, T ]} be affine, i.e. r (x) = ρ0 + ρ1 x, where ρ0 , ρ1 ∈ R≥0 , and let X = {Xt : t ∈ [0, T ]} be the underlying price process. hFurther, assume that i r ∈ R Xt − tT r(Xτ − )dτ 0 Q := C (R; R≥0 ). Then the futures price F (t, T ) = B(t,T ) , where B (t, T ) E e Ft is the price of a zero-coupon bond, is a local martingale under Q, i.e. the risk-neutral measure.46 Proof. The result follows straightforward by the seminal result of the first fundamental theorem of asset pricing, i.e. the discounted price process of every tradable asset is a Q-martingale (cf. Delbaen and Schachermayer (1994)). This yields to i h RT EQ [ F (T, T )| Ft ] = EQ [ XT | Ft ] = Xt EQ e t r(Xτ − )dτ Ft =

Xt =: F (t, T ) , B (t, T )

which proves the martingale property. We can now apply the Itˆ o formula for Lévy processes, which is given in differential form by 1 c df (St ) = ∂s f (St− ) dSt + ∂ss f (St ) d [S]t + f (St ) − f (St− ) − ∆St ∂s f (St− ) , 2

(2.37)

to f (s) = es .47 We obtain, by a straightforward and careful application of (2.37) and by taking into

45

46 47

Because L´ evy processes belong to the broader class of semimartingales (see Corollary 2.1), which can be decomposed as the sum of a local martingale and an adapted c` adl` ag process with trajectories of finite variation, we prove everything in terms of local martingales. Nevertheless, it does not change the argument due to the result that every martingale is a local martingale (the reverse however is not necessarily true in general). The affine dependence of the short rate process is not required in general. Nevertheless, we impose this restriction due to our affine framework. Because we do not need, by Lemma 2.1, the discounted version of F to obtain a semimartingale process, there will not be a risk-free rate in the drift component.


c 2014


43

account the Lévy structure given by the Lévy-Itô decomposition in Theorem 2.3,48 ! Z 0 2 2 X 1 X S W fQ W + (ex − 1 − x) ηJQ− (dx) dTtJ dSt = bt dt + dTt,ii + dW ij,Tt,ij 2 i=1 −∞ i,j=1 Z ∞ Z 0 Q x J + (e − 1 − x) ηJ+ (dx) dTt + (ex − 1) JS− (dt, dx) − ηJQ− (x) dxdTtJ 0 −∞ Z ∞ Q + (ex − 1) JS+ (dt, dx) − ηJ+ (x) dxdTtJ . 0

By what we proved in Lemma 2.1, the price process (Ft )0≤t≤T is an (exponential) local martingale which implies that the log-return dynamics (St )0≤t≤T needs to be a local martingale. Therefore, its dynamics has zero drift. Hence, by a reverse application of the time changes in (2.24), we obtain at time t !! Z 2 1 X W Q S x J . bt = − (e − 1 − x) ηJ (dx) νt − V 2 i=1 t−,ii R\{0}

To close our consistent framework in which we want to allow for multifactor volatility, stochastic correlation, stochastic skewness, and jumps in returns and second moments, we are left now with the specification of the stochastic activity rate of the absolutely continuous time change under Q for the time change T W defined in (2.24). Assumption 2.5 (2×2-dimensional joint SPX-VIX2 covariance dynamics under Q). We denote the VIX2 dynamics by Y = {Yt : t ∈ [0, T ]}. The joint covariance dynamics (or the stochastic activity rate continuous time change under the pricing measure Q for the time change T W = Wof the absolutely Tt : t ∈ [0, T ] defined in (2.24)) of S ∈ R and Y ∈ R≥0 under the risk-neutral probability measure Q, for which the dynamics of S is defined in (2.35) and Y is a squared Bessel process defined in Example 2.2 + with an additional pure jump process J ρ , i.e. Y ∼ BESQδ,ρ x , can be represented by the S2 -valued Markov + W W process V starting at V0 = v ∈ S2 and which solves the SDE: q q dVtW = βQ> Q + M VtW + VtW M > dt + VtW dBt Q + Q> dBt> VtW + dJtV , (2.38) 2×2 V where M, Q ∈ R2×2 , B is a matrix √ of standard Brownian motions in R ,+ J is a pure jump proW + cess taking values in S2 , and V W denotes the unique square root on S2 . The jump sizes ξ V + are i.i.d. and follow the gamma-type Lévy density defined in (2.31) with k = 2, Σ ∈ S2 \ {0} and 0 ≤ β < 1. The ”Gindikin” coefficient β is a real number larger than 1 and we impose the restriction ΩΩ> = βQ> Q, compared to Assumption 2.3, which guarantees that V W remains positive semi-definite.49 Furthermore, to preserve our affine framework, we assume that the jumps are realized with the intensity J W ν Vt− : t ∈ [0, T ] defined in (2.28).

Assumption 2.4 is highly interrelated with Assumption 2.5 via the time change T W . Hence, this allows for the consistent pricing of VIX options while preserving consistency with the SPX dynamics and its options. Furthermore, we can analytically calibrate and price jointly SPX– and VIX options 48 49

W = V W dt, for 1 ≤ i, j ≤ 2. We have, by definition in (2.24), dTt,ij t−,ji According to Fonseca et al. (2012), an increase of the Gindikin coefficient β will shift the distribution of the smallest eigenvalue to positive values. Therefore, this parameter is referred to as the global variance shift factor .


c 2014

2.7. TRANSFORM ANALYSIS

44

due to our affine framework (see Section 2.7). Summarizing, our model provides a high analytical tractability, incorporates the empirical evidence and allows for parsimonious pricing. Recently, Bardgett et al. (2013) provide a thorough analysis of the empirical performance of affine jump diffusion models to jointly represent the values of the SPX– and VIX indices and their options. One main distinction to our setting is their two-factor stochastic volatility model with jumps in opposite to our double jump matrixvariate stochastic volatility model with co-volatilities, i.e. the off-diagonal elements model the volatility of volatility between the SPX and VIX, which can replicate the stochastic skewness and –correlation in the markets.

2.7

Transform Analysis

The entire framework is based on an affine setting which allows us, according to Duffie et al. (2000), to efficiently solve our financial pricing problem by means of transform methods. Therefore, we aim an analytical characterization of the discounted Laplace transform of VtW (resp. St ) at time t under Q:50 " ! # Z T W W ΨV Γ, VtW , t, T := EQ exp − r Vs− ds + tr ΓVTW Ft , (2.39) t " ! # Z T S Q W Ψ (ϑ, St , t, T ) := E exp − r Vs− ds + ϑST Ft , (2.40) t where Γ ∈ S2+ , ϑ ∈ R, and the short rate process r = Assumption 2.2.

2.7.1

W r Vt− : t ∈ [0, T ] is affine in the sense of

Pricing SPX Options

We first target the pricing of SPX options, i.e. the elaboration of the solution of (2.40). Further, in Section 2.7.2 we follow another approach as presented below to price VIX options. In order to capture the well-known leverage effect in our matrix-variate setting, it is necessary to introduce a correlation between the Brownian motion in the dynamics without time change and the Brownian motion component in the activity rate of the corresponding time change. Further on, we assume that the instantaneous correlation between the time change of the jump T J and the continuous component is zero. Translated into our framework, we obtain, in matrix notation, by considering Assumption 2.1 for k = 1, Assumption 2.5 and making use of the orthogonal decomposition of the R2×2 -valued Q-Brownian motion W Q p (2.41) W Q := BP > + Z I2 − P P > , with the standard Brownian motions B ∈ R2×2 and Z ∈ R2×2 , which are independent of each other, and P ∈ R2×2 is a deterministic correlation matrix such that P P > ≤ I2 is a positive semi-definite matrix. 50

We adopt the notation of Leippold and Trojani (2010) for consistency reasons. See also Remark 2.12 for a comment on the different notations of the Laplace transform.


c 2014


45

Under this specification, in the computation of the transform ΨS (Γ, St , t, T ) defined in (2.40) enters W two main sources of stochastic volatility. The first linked the to W time change T of the Brownian motion W Q and the second to the stochastic activity rate ν J Vt− : t ∈ [0, T ] of the absolutely continuous time change T J . Carr and Wu (2004) introduce a change of measure technique in order to first account for a leverage effect in time-changed Lévy models and to remove the dependence structure between S and a general time change T in a second step. We provide a short introduction to this framework which is partly based on (Jeanblanc et al., 2009, Section 11.3). We assume to be under the pricing measure Q. Let X be a Lévy process, and assume that EQ eϑ·Xt < ∞ for some ϑ ∈ Rk . We define a probability Q(ϑ) , locally equivalent to Q, by the formula Q(ϑ) |Ft =

eϑ·Xt Q|Ft . (eϑ·Xt )

EQ

(2.42)

Note that, from the Wald martingale property and Definition 2.11, the process Lt :=

eϑ·Xt = eϑ·Xt −tΨ(ϑ) , (eϑ·Xt )

EQ

(2.43)

with the function Ψ defined in (2.21), is a martingale.51 Hence, this ϑ-Esscher transform (or sometimes called exponential tilting) preserves the Lévy process property as we show in the next proposition. Proposition 2.6. Let X be a Q-Lévy process with parameters (m, A, ν). Let ϑ ∈ Rk be such that Q ϑ·Xt E e < ∞ and suppose that Q(ϑ) is defined by (2.42). Then X is a Q(ϑ) -Lévy process and the Lévy-Khintchine representation of X under Q(ϑ) is ! Z (ϑ) u · Au EQ eiu·Xt = exp iu · m(ϑ) − + eiu·x − 1 − iu · x1|x|≤1 ν (ϑ) (dx) 2 Rk \{0} with (ϑ)

m ν

(ϑ)

1 := m + A + A> ϑ + 2

(dx) := e

ϑ·x

Z

x eϑ·x − 1 ν (dx) ,

|x|≤1

ν (dx) .

The characteristic exponent of X under Q(ϑ) is κ(ϑ) (u) := κ (u − iϑ) − κ (−iϑ) , and the Laplace exponent is Ψ(ϑ) (u) := Ψ (u + ϑ) − Φ (ϑ) for u ≥ −ϑ ∧ 0. The functions κ and Ψ are defined in (2.20) and (2.21). Proof. For the detailed proof, we refer to (Jeanblanc et al., 2009, Section 11.3). We can use now this result in order to eliminate the dependence structure between S and T W . In order to be consistent with the notation, we need to redefine the time changes in (2.24) due to the integration limits t and T . The absolutely continuous time changes are redefined by Z T Z T W TTW := Vs− ds and TTJ := νsJ ds. (2.44) t 51

t

We proved a similar result in Proposition 2.3 with the characteristic exponent κ defined by (2.20).


c 2014


46

Further, as proven in Lemma 2.1, we do not need a discounted version of the futures dynamics to form a semimartingle process. Thus, let us write the ordinary Laplace transform ΨS (Γ, St , t, T ) in (2.40) by ΨS (ϑ, St , t, T ) = EQ exp ϑST − TT> Ψ (ϑ) +TT> Ψ (ϑ) FTT | {z } (2.43)

= LTT

(ϑ)

(2.45) exp TT> Ψ (ϑ) FTT , > > where TT := TTW , TTJ , Ψ (ϑ) := ΨW (ϑ) , ΨJ (ϑ) , and FTT := FTTW ⊗ FTTJ .52 The second line is obtained by applying the Radon-Nikod´ ym derivative in (2.42) and the exponential martingale process in (2.43). Precisely, the new measure Q(ϑ) is defined by the exponential Q-martingale dQ(ϑ) = exp ϑST − TT> Ψ (ϑ) . dQ FT = EQ

T

Under this new measure, calculations are shifted into a world where expectations can be computed as if there were no leverage. Therefore, Q(ϑ) is called the leverage-neutral measure (or sometimes also called correlation-neutral measure). Consequently, the problem of finding the discounted Laplace transform of the time-changed Lévy process in presence of leverage effects is reduced to the computation of the Laplace transform of the random time under the leverage-neutral measure, evaluated at the Laplace exponent of the involved Lévy process. If we let ϑ ∈ D ⊆ C, where D denotes a subset of the complex plane under which the expectation in (2.45) is well defined, Q(ϑ) defines a complex-valued measure which is an extension of the real valued measures usually used in finance. Also notice, in the case ϑ ∈ D the measure Q(ϑ) is no longer a probability measure. From the representation in (2.45) we can deduce that we can decompose our problem into two separative parts. We need to find independently the Laplace exponents of the jump– and diffusion part. If all of these parts are analytically tractable, then the Laplace exponent of the time-changed Lévy process is tractable. We start with the Laplace exponent ΨW (ϑ) of the diffusion part. To be in line with the literature, we first derive the Lévy exponent and switch to the Laplace exponent in the second step. The characteristic function µ bW (u) can be obtained by    ! 2 2 X X 1 W fQ W −  W µ bW (u) = EQ exp iu  TT,ii ij,TT ,ij 2 i,j=1 i=1    ! ! ! 2 2 2 2 X X X X 1 Q W W W f W −  − κW (u)  = EQ exp iu  W TT,ii TT,ii + κW (u) TT,ii ij,TT ,ij 2 i,j=1 i=1 i=1 i=1 " !!# 2 X (u) W = EQ exp −κW (u) TT,ii , i=1

with the exponential Q-martingale   2 (u) X dQW fQ W − 1   = exp iu W ij,TT ,ij dQW 2 i,j=1 T

52

2 X i=1

! W  + κW (u) TT,ii

2 X

! W , TT,ii

i=1

Notice that in our notation we have STT ≡ ST in opposite to the notation in Carr and Wu (2004), where they specify an additional variable for a time-changed L´ evy process, i.e. STT ≡ YT .


c 2014


47

where the general proof can be found in Proposition 2.3, and we follow (2.20) for the definition of the Lévy exponent. Hence, we obtain κW (u) = iu + 2u2

and by (2.19)

ΨW (ϑ) = ϑ (2ϑ − 1) .

We see that κW (u) is the characteristic exponent of the convexity-adjusted continuous Lévy component P2 P2 Q 1 evy exponent κJ (ϑ) of the convexity-adjusted i=1 T . Next, we can calculate the L´ i,j=1 Wij,T − 2 jump component (see e.g. Carr and Wu (2004) for further details) by Z κJ (u) = − eiux − 1 ηJQ (x) dx − iuϕJ (1) R\{0}

= ln ((1 − iuν+ ) (1 + iuν− )) − iu ln ((1 − ν+ ) (1 + ν− )) , where we used the VG density in (2.29), which describes the arrival rate of the jumps, and ϕJ (1) represents the cumulant exponent of the Lévy jump component which is defined, for s ∈ R, by Z ϕJ (s) := (1 − esx ) ηJQ (x) dx. R\{0}

Thus, we get again by (2.19) and a straightforward calculation the Laplace exponent ΨJ (ϑ) = ϑ ln ((1 − ν+ ) (1 + ν− )) − ln ((1 − ϑν+ ) (1 + ϑν− )) . The corresponding exponential Q-martingale is defined by (u) h i dQJ Q = exp iu JT J − ϕJ (1) TTJ + κJ (u) TTJ , T dQJ T

n o where JTQJ = JTQJ : T ≥ 0 represents the time-changed pure jump process. Putting everything together, T we obtain the Radon-Nikod´ ym density of the time-changed Lévy process S given by   ! ! 2 2 2 (u) (u) (u) X X X dQW dQJ dQS 1 Q W W W f    T Wij,T W − + κ (u) TT,ii = = exp iu T ,ij dQS dQW dQJ 2 i=1 T,ii i,j=1 i=1 T T i + iu JTQJ − ϕJ (1) TTJ + κJ (u) TTJ . T

As next, we need to assure that the stochastic activity rates of the time changes T W and T J remain affine under the new measure Q(ϑ) . Proposition 2.7. Let the activity rate under Q of T W be given in (2.38) for the time horizon [t, T ]. Then, the new activity rate of T W under the leverage-neutral measure Q(ϑ) remains affine. In particular, the dynamics is given by the S2+ -valued Markov process V W starting at V0W = v ∈ S2+ and which solves the SDE: q q (ϑ) (ϑ) (ϑ) (ϑ) (ϑ) dVtW = βQ> Q + M Q VtW + VtW M Q > dt + VtW dBtQ Q + Q> dBtQ > VtW + dJtV,Q , (ϑ)

:= M + P > Q with P ∈ R2×2 a deterministic correlation matrix such that where Q ∈ R2×2 , M Q (ϑ) > P P ≤ I2 is a positive semi-definite matrix, B Q is a matrix of Q(ϑ) -standard √ Brownian motions in (ϑ) R2×2 , J V,Q is a pure jump process under Q(ϑ) taking values in S2+ , and V W denotes the unique Pascal Marco Caversaccio

c 2014


48 W

square root on S2+ . The jump sizes ξ V are i.i.d. and follow the gamma-type Lévy density defined in (2.31) with k = 2, Σ ∈ S2+ \ {0} and 0 ≤ β < 1. The ”Gindikin” coefficient β is a real number larger than 1 and we retain the restriction ΩΩ> = βQ> Q (cf. also Assumption 2.3) which guarantees that V W remains positive semi-definite. Furthermore, the jumps are realized with the affine intensity (ϑ) (ϑ) (ϑ) (ϑ) W W dν J,Q Vt− = tr ν1J,Q dVt− , ν0J,Q ∈ R≥0 , ν1J,Q ∈ S2+ . Additionally, the time change T J remains the same under Q(ϑ) with the stochastic activity rate under the leverage-neutral measure Q(ϑ) √ (ϑJ ) , dνtJ = κJ θJ − νtJ dt + γ J ν J dBtQ (ϑJ )

where B Q

ν0J = v ∈ R≥0 ,

κJ , θJ , γ J ∈ R+ ,

(2.46)

is a Q(ϑ) -Brownian motion.

Proof. Due to ϑ ∈ R, we can apply the ordinary Radon-Nikod´ ym density which is given in our case by dQ(ϑ) f QW , =E W T T dQ F W T

T

where we denote by E (·) the Doléans-Dade exponential (or sometimes also called the stochastic exponential ). Note that whenever we have ϑ ∈ D ⊆ C, there is a complex-valued extension available in e.g. Carr et al. (2003). By a straightforward application of Girsanov’s theorem, we obtain q (ϑ) W d B, W Q dBtQ = dBt − Vt− t q (2.41) W P > dt, = dBt − Vt− where P ∈ R2×2 is a deterministicqcorrelation matrix such that P P > ≤ I2 is a positive semi-definite (ϑ) W P > dt into (2.38) and incorporating also the change of measure matrix. Plugging dBt = dBtQ + Vt− for the jump process J V , yields the expression q q (ϑ) (ϑ) (ϑ) (ϑ) (ϑ) dVtW = βQ> Q + M Q VtW + VtW M Q > dt + VtW dBtQ Q + Q> dBtQ > VtW + dJtV,Q , (ϑ)

(ϑ)

where B Q is a matrix of Q(ϑ) -standard Brownian motions in R2×2 , J V,Q is a pure jump process (ϑ) under Q(ϑ) taking values in S2+ , and M Q := M + P > Q. Further, because the jump intensity is by W definition in (2.28) an affine n o function of V , we can preserve the structure with a new affine intensity (ϑ) W ν J,Q Vt− : t ∈ [0, T ] . Finally, because the driving Brownian motion of the time change T J is by definition independent of the MAJD process driving the SPX dynamics directly, the assertion is proven. As the affine structure is preserved, the ordinary Laplace transform under the leverage-neutral measure Q(ϑ) is given by (ϑ)

exp TTW ΨW (ϑ) + TTJ ΨJ (ϑ) FTT = exp φW (τ ) + tr ψW (τ ) VtW + φJ (τ ) + ψJ (τ ) νtJ ,

ΨS (ϑ, St , t, T ) = EQ


c 2014


49

where τ := T −t. Thus, the transform is exponential affine in the state variables (cf. Carr and Wu (2004)). The coefficients of the Laplace transform are determined by the following (matrix) ODEs:53 "Z # (ϑ) W dφW (τ ) J,Q = βtr ψW (τ ) Q> Q + ν0 exp tr ψW (τ ) v W Y V dv W − 1 , (2.47) dτ S2+ \{0} (ϑ) (ϑ) dψW (τ ) = ΨW (ϑ) + M Q > ψW (τ ) + ψW (τ ) M Q + 2ψW (τ ) Q> QψW (τ ) dτ "Z # VW J,Q(ϑ) W W + ν1 Y −1 , exp tr ψW (τ ) v dv

(2.48)

S2+ \{0}

dφJ (τ ) = ψJ (τ ) κJ θJ , dτ 2 2 ψJ (τ ) dψJ (τ ) = ΨJ (ϑ) − ψJ (τ ) κJ − γJ , dτ 2

(2.49) (2.50)

for the functions φW (τ ), φJ (τ ), ψJ (τ ) ∈ R, and ψW (τ ) ∈ S2+ , subject to the terminal conditions φW (0) = 0, φJ (0) = 0, ψJ (0) = 0, and ψW (0) = 0. We assume that the notion of regularity is satisfied (see Definition 2.3). The ODEs governing the coefficients φJ (τ ) and ψJ (τ ) in (2.49) & (2.50) can be solved analytically, κJ θJ ζ − κJ J (1 − exp (−ζτ )) + ζ − κ τ , φJ (τ ) = 2 ln 1 − 2 2ζ (γ J ) q 2ΨJ (ϑ) (1 − exp (−ζτ )) 2 2 := ψJ (τ ) = , ζ (κJ ) + 2 (γ J ) ΨJ (ϑ). 2ζ − (ζ − κJ ) (1 − exp (−ζτ )) To retain our analytical tractability, we impose an additional restriction on the jump intensity driving the jumps in the jump-extended Wishart process in Proposition 2.7. Nevertheless, one could continue also with the given setting and solve the Riccati equations governing the coefficients φW (τ ) and ψW (τ ) with the standard Runge-Kutta 4th order method. The following result is due to Crameri et al. (2010) and can be replicated by an application of Radon’s lemma to the ODE of ψW (τ ). (ϑ)

Proposition 2.8. Assume ν1J,Q = 0 and let (2.47), (2.48), Definition 2.3 and some additional regu−1 larity conditions in Cuchiero et al. (2011b) be satisfied. Then, ψW (τ ) = C22 (τ ) C21 (τ ), where C21 (τ ) and C22 (τ ) are 2 × 2 blocks of the following matrix exponential: ! " !# (ϑ) C11 (τ ) C12 (τ ) MQ −2Q> Q := exp τ . (ϑ) C21 (τ ) C22 (τ ) ΨW (ϑ) −M Q > Given the solution for ψW (τ ), the coefficient φW (τ ) follows by direct integration: "Z Z # τ i (ϑ) (ϑ) W β h φW (τ ) = − tr ln C22 (τ ) + τ M Q > + ν0J,Q exp tr ψW (s) v W Y V dv W ds − τ , 2 0 S2+ \{0} where ln (·) denotes the matrix logarithm. Proof. See Appendix B. 53

See Proposition 2.9 for a mathematical proof, without using the leverage-neutral measure change technique, concerning the matrix ODEs in (2.47) and (2.48). The ODEs (2.49) and (2.50) can be obtained along the same lines as in Appendix A.


c 2014


50

Last but not least, we can use the so-called fast Fourier transform (FFT), developed in Carr and Madan (1999), which relates the characteristic function of the S&P 500 price to the SPX option price and can therefore be applied to derive the derivative price. We denote by C (Ft , K, t, T ) the SPX option price at time t with maturity T and perform the following rescaling. Let c (k) be given by Z c (k) := C (Ft , K, t, T ) /Ft = EQ eST − ek 1{ST ≥k} Ft = es − ek 1{s≥k} dFS (s) , (2.51) R

where k := ln K/Ft and c (k) represents the option forward price in percentage of the underlying forward as a function of moneyness defined as the log strike over forward. Subsequently, we can derive the Fourier transform which is characterized by the following expression Z µ b (u − i) , (2.52) χ (u) := eiuk c (k) dk = (iu) (iu + 1) R with u := z − iα, z ∈ R, α ∈ D ⊆ R+ for the option transform to be well defined. Generally, the range of α depends on the payoff structure and the model, and Carr and Madan (1999) refer to α as the dampening coefficient. Further, the exact value choice of α is a numerical issue. Equation (2.52) is obtained by substituting the definition of c (k) into (2.52), calculating the integral by simple integration and checking carefully the boundaries. The exact calculation steps are provided in Appendix C. The conditional characteristic function µ b (u) of the non-homogeneous Lévy process S is given, for u ∈ D ⊆ C, by µ b (u) = EQ [ exp (iuST )| Ft ] = exp −φW (τ ) − tr ψW (τ ) VtW − φJ (τ ) − ψJ (τ ) νtJ , where τ := T − t and the coefficients φW (τ ), ψW (τ ), φJ (τ ) and ψJ (τ ) are given by ζ − κJ κJ θJ J 2 ln 1 − (1 − exp (−ζτ )) + ζ − κ τ , φJ (τ ) = 2 2ζ (γ J ) q 2κJ (u) (1 − exp (−ζτ )) 2 2 := ψJ (τ ) = , ζ (κJ ) + 2 (γ J ) κJ (u), 2ζ − (ζ − κJ ) (1 − exp (−ζτ )) where κJ (u) := ln ((1 − iuν+ ) (1 + iuν− )) − iu ln ((1 − ν+ ) (1 + ν− )) , and the solutions of φW (τ ), ψW (τ ) are stated in Proposition 2.8. Eventually, we get to the option price by inversion of (2.52). This can be done via the inversion formula for PDFs Z Z 1 1 e−iux µ b (u) du = e−iux µ b (u) du, fX (x) = 2π R π R≥0 i.e. 1 c (k) = 2π

Z

−iα+∞

e

−iuk

−iα−∞

e−αk χ (u) du = π

Z

∞

e−izk χ (z − iα) dz.

0

Thus, the exact solution is given by c (k) =

e−αk π

Z 0

∞

e−izk

µ b (z − iα − i) dz. (iz + α) (iz + α + 1)

Let us remark that the price of a put option can be calculated analogously. Another approach is to back out the put option price via the ordinary put-call parity since its application is not violated due to the European-style contract. Pascal Marco Caversaccio

c 2014


51

Remark 2.14. In the approach of Carr and Madan (1999) the Fourier transform is taken with respect to the log-strike price. Obviously a call is not L1 -integrable with respect to the logarithm of the strike price, where we write L1 for the set of all integrable random variables. To circumvent this technical difficulty we need to dampen the payoff, which explains the rationale behind the dampening coefficient α. J

2.7.2

Pricing VIX Options

In an analogous manner as above, we first derive the discounted Laplace transform of the VIX2 . For simplicity’s sake we set the risk-free rate constant, i.e. ρ1 = 0. One may argue that the error of a constant interest rate can become substantially large, but this only holds for long dated financial derivatives, e.g. 20, 30 years. Since this is not the case for VIX options, where the options are dated much more shortterm, we do not encounter this issue in our setting. The following proposition is due to the results in Leippold and Trojani (2010). Proposition 2.9. Let Assumption 2.5, Definition 2.3 and some additional regularity conditions in Cuchiero et al. (2011b) be satisfied. Furthermore, assume a constant risk-free rate, i.e. ρ1 = 0. Then, the discounted Laplace transform in (2.39) is exponentially affine: W

ΨV

Γ, VtW , t, T = exp φV (τ ) + tr ψV (τ ) VtW

with functions φV (τ ) ∈ R and ψV (τ ) ∈ S2+ that solve the system of matrix Riccati equations: "Z # VW dφV (τ ) > J W W = − ρ0 + βtr ψV Q Q + ν0 exp tr ψV (τ ) v Y dv −1 , dτ S2+ \{0} dψV (τ ) = M > ψV (τ ) + ψV (τ ) M + 2ψV (τ ) Q> QψV (τ ) dτ "Z # VV J W W + ν1 exp tr ψW (τ ) v Y dv −1 ,

(2.53)

(2.54)

S2+ \{0}

subject to the terminal conditions φV (0) = 0 and ψV (0) = Γ. Proof. Suppose that V W is an adapted Markov process in the state space X ⊂ S2+ . Under some W regularity conditions, the Lévy infinitesimal generator AV of the matrix Markov process V W is defined for bounded f v W ∈ C 2 (X ; R) functions by W AV f v W := tr βQ> Q + M v W + v W M > D + 2v W DQ> QD f v W Z W + ν J vW f v W + z − f v W Y V (dz) , S2+ \{0}

where D is a 2 × 2 matrix of differential operators with the ij-component given by

∂ ∂vW

. Since the

ij

generator exhibits an affine dependence on the state space v W ∈ S2+ , we obtain, by separation of the variables, the exponential affine property with the matrix Riccati equations given in the proposition.


c 2014


52

W Again, to preserve our analytical tractability, we choose ν J Vt− : t ∈ [0, T ] to be constant. The numerical alternative is given by the standard Runge-Kutta 4th order method. Another interesting approach is pointed out in Leippold and Trojani (2010), where they propose to use asymptotic approximations by starting from the closed-form solution for the case ν1J = 0. Our choice is based on the rationale that it simplifies the numerical implementation and exhibits a high robustness and accuracy due to the lack of discretization errors. Corollary 2.2. Assume ν1J = 0 and let (2.53), (2.54), Definition 2.3 and some additional regularity con−1 ditions in Cuchiero et al. (2011b) be satisfied. Then, ψV (τ ) = (ΓC12 (τ ) + C22 (τ )) (ΓC11 (τ ) + C21 (τ )), where C11 (τ ), C12 (τ ), C21 (τ ) and C22 (τ ) are 2 × 2 blocks of the following matrix exponential: ! " !# C11 (τ ) C12 (τ ) M −2Q> Q := exp τ . C21 (τ ) C22 (τ ) 0 −M > Given the solution for ψV (τ ), the coefficient φV (τ ) follows by direct integration: β φV (τ ) = − ρ0 τ − tr ln (ΓC12 (τ ) + C22 (τ )) + τ M > "Z 2 Z # τ VW W W J Y dv ds − τ , exp tr ψV (s) v + ν0 0

S2+ \{0}

where ln (·) denotes the matrix logarithm. Proof. See Appendix D. Now the particular solution for the VIX2 is readily obtained. We remark that the VIX2 dynamics is W modeled via the second diagonal component V22 . The following corollary is a special case of Corollary 2.2 and therefore we skip the proof. Corollary 2.3. Assume ν1J = 0 and let (2.53), (2.54), Definition 2.3 and some additional regularity 2 conditions inh Cuchiero et al. (2011b) be satisfied. Then, i the coefficient of the VIX is obtained by −1

[ψV (τ )]22 = (ΓC12 (τ ) + C22 (τ )) (ΓC11 (τ ) + C21 (τ )) are 2 × 2 blocks of the following matrix exponential: ! " C11 (τ ) C12 (τ ) := exp τ C21 (τ ) C22 (τ )

22

, where C11 (τ ), C12 (τ ), C21 (τ ) and C22 (τ )

M 0

!# −2Q> Q , −M >

and the subscript of [·]22 denotes the second diagonal element of a matrix. Given the solution for ψV (τ ), the coefficient φV (τ ) follows by direct integration: β φV (τ ) = − ρ0 τ − tr ln (ΓC12 (τ ) + C22 (τ )) + τ M > # "Z 2 Z τ VW W J W + ν0 exp tr ψV (s) v Y dv ds − τ , 0

S2+ \{0}

where ln (·) denotes the matrix logarithm.


c 2014


53

For the pricing of options on the VIX, we cannot follow the previous technique applied in Section 2.7.1 of Carr and Madan (1999). To illustrate why, we follow the arguments in Bardgett et al. (2013), who have pointed out the technical difficulties in this framework. We can write the price of a call option with strike K and maturity T at time t as C (VIXt , K, t, T ) = e

−r(T −t)

Z

∞

q + W W W VIX2t = v0W dv22 , v22 − K f v22

0

W W W given the VIX2t = v0W is the PDF of v22 where v22 denotes the value of the VIX2 at time T , f v22 2 value today, i.e. VIXt , and r is the (constant) risk-neutral interest rate. Comparing this expression with Equation (2.51), we see that to apply the Fourier method of Carr and Madan (1999) we would need to have an affine dependence on the log of the VIX, which is however incompatible with affine models. To circumvent this technical issue, we henceforth follow Fang and Oosterlee (2008). Fang and Oosterlee (2008) develop an option pricing method for European-style options, called the COS method, based on Fourier-cosine series expansions. The main idea is to decompose a density function into a linear combination of cosine functions since the series coefficients of many density functions can be accurately obtained from their characteristic functions. This particular decomposition allows for easy and highly efficient numerical computations which attain, in most cases, an exponential convergence rate and a linear computational complexity. Remark 2.15. The COS method is solely applied to the pricing of VIX options. The FFT is used instead for SPX options, whose payoff does not necessarily require the COS method. Nonetheless, one can also easily adopt the technique of Fang and Oosterlee (2008) to SPX options (see in particular (Fang and Oosterlee, 2008, Section 3)). Furthermore, there exist other highly efficient transform techniques in the literature for the pricing of plain vanilla options with different merits and demerits. Worth mentioning are the fast Gauss transform of Broadie and Yamamoto (2003), the double-exponential transformation elaborated in Mori and Sugihara (2001) and Yamamoto (2005), the fractional FFT detailed in Chourdakis (2005), the convolution method (CONV) developed by Lord et al. (2008), and the saddlepoint method worked out by Carr and Madan (2009). J Proposition 2.10. Given the truncation interval [a, b] ⊂ R≥0 for the (compact) support of the PDF W f v22 VIX2t = v0W , the price Υ (VIXt , K, t, T ) of a European contingent claim on the VIX with payoff + √ function g VIX2 = VIX2 − K and time t ∈ [0, T ] is

Υ (VIXt , K, t, T ) = e

−r(T −t)

N −1 X0

Ak (a, b) ϕk (a, b) ,

k=0

with the coefficients h −iakπ i W Ak (a, b) := Re exp φV (T − t) + [ψV (T − t)]22 v22 e b−a ,  p + v W −a W W −K ω := R b , k > 0, v22 cos kπ 22 dv22 b−a a h 2 i ϕk (a, b) :=  2 2 b3/2 + K K − b , k = 0, b−a


3

3

c 2014


54

where ω can be obtained in closed-form:   √ b −ikπ b−a a 2 π 1  −ikπ b−a √ e ω= + Re e  b ikπ 3/2 b−a 2 −ikπ b−a

r erfz

−ikπb b−a

!

r − erfz K

 !! −ikπ   , b−a

b−a

Rz 2 with erfz (z) := √2π 0 e−t dt and z ∈ C the complex Gauss error function. The prime superscript in the P0 sum indicates that the first term in the summation is divided by 2 and the coefficients φV (T − t) and [ψV (T − t)]22 are given in Corollary 2.3 with ! Γ :=

ikπ b−a ikπ b−a

ikπ b−a ikπ b−a

.

Proof. The general COS formula for European contingent claims is taken from (Fang and Oosterlee, 2008, Section 3). The corresponding coefficients are provided in Bardgett et al. (2013) for which we merely have to substitute our Laplace transform into Ak (a, b) while ϕk (a, b) remains the same. Hence, the result follows. Let us remark that the parameter Γ still belongs to S2+ since, for k ∈ R≥0 and b > a, all of the eigenvalues remain at least non-negative. Our highly analytical and tractable model yields a semi-closed solution and we can therefore circumvent the need of supplementary numerical solutions which induces additional approximation errors. Moreover, the price of a put option can be calculated analogously. Another approach is to back out the put option price via the modified put-call parity in (2.34).


c 2014

Chapter

3

Numerical Investigation Failure is not an option. — Apollo XIII Mission to the Moon Rescue Motto

Every theoretical model needs to search its justification in the data. This is the aim of this chapter. Since a full implementation of the simultaneous pricing of SPX– and VIX options is beyond the scope of this thesis and far from being trivial because of the need of a numerical evaluation of the matrix integrals of the jump components in Corollary 2.3 and Proposition 2.8, we set the jump part to zero in the following. Furthermore, we only consider the pricing of VIX options because this is the ultimate goal of this thesis and leave the numerical pricing implementation of SPX options for future research. We first provide in Section 3.1 the detailed description of the estimation procedure conducted in this thesis. Further, Section 3.2 examines the pricing performance of our model by means of implied volatilities.

3.1

Parameter Estimation

For the implementation of the COS method for VIX options specified in Proposition 2.10, we need the following 9 parameter estimates for the full model specification: Θ = {β, M11 , M12 , M21 , M22 , Q11 , Q12 , Q21 , Q22 } , 2 which govern W the joint SPX-VIX covariance dynamics under the pricing measure Q, and the latent state variable v22 . As already indicated above, we set for simplicity’s sake the jump component to zero, i.e. ν0J = 0. Moreover, we use the so-called market implied approach for the calibration which relies on the existence of semi-closed– or closed-form expressions for the prices of benchmark derivatives. For most cases the calibration is based on the minimization of a loss function which has to be defined according to the purpose of the model. Different loss functions need to be taken into consideration for hedging, speculating or market making purposes for instance. For an in-depth discussion of the choice of the loss functions we refer to Christoffersen and Jacobs (2004) which recommend and show empirically that the alignment of the estimation– and evaluation loss function is a crucial issue.

55

3.1. PARAMETER ESTIMATION

56

We fix the date 27.02.2009 to demonstrate the flexibility of our model design. The motivation of this particular choice is based on the following reasoning. As already described in Section 2.3.2, on this day the investors experienced two main events which caused uncertainty and therefore high fluctuations in terms of SPX– and VIX option trading. Hence, the accurate replication of the IVS slice shapes would show the high flexibility of the option pricing model even in times of financial turmoils and also would justify an estimation of the model across a time series of options using a particle filter or generalized method of moments (GMM) in future research. Let I ⊂ N denote the set of (cleaned) option data available on this particular date. For the detailed data treatment we refer to Section 2.3 where we describe which exclusionary criteria are applied and how the implied volatilities are calculated. Additionally to these exclusionary criteria, we delete all observed options with price levels less than 0.5$. We focus on European plain vanilla call options on the VIX and end up with a total of 42 (noiseless) option data points for four different maturities. Following Christoffersen and Jacobs (2004) and Bardgett et al. (2013), we define 2 different loss functions. The first is often referred to as root mean squared pricing error (or sometimes also called root mean squared dollar error ) and is defined by s 1 X W W 2, := (3.1) Ci − Ci Θ, v22 $RMSE Θ, v22 #I i∈I

W

where C and C Θ, v22 are the data– and model option prices, respectively, and #I denotes the cardinality of the set I. The $RMSE exhibits the virtue that the errors are easily interpreted as $-errors, but raises the problem of heteroskedasticity for the $RMSE-based parameter estimation (cf. Christoffersen and Jacobs (2004)). We can also deduce that (3.1) gives more weight to expensive options, i.e. (3.1) emphasis ITM options and long time-to-maturity contracts rather than the more liquid short-term OTM options. This particular drawback brings us to our second loss function which we refer to as implied volatility root mean squared error and is defined by s 1 X W W 2, IVRMSE Θ, v22 := σi − σi Θ, v22 (3.2) #I i∈I

where the implied volatilities are defined, for all i ∈ I, by W W := BS−1 Ci Θ, v22 σi := BS−1 (Ci , τi , Ki , Fi , ri ) and σi Θ, v22 , τi , Ki , Fi , ri ,

(3.3)

where BS−1 is the inverse Black-Scholes formula, τ stands for the time-to-maturity, K represents the strike price, F is the ATM futures price, and r denotes the (varying) risk-free interest rate. Due to the different time-to-maturities in our data set, the risk-free rate changes and therefore explains the subscript in (3.3). Since the implied volatilities are the highest for OTM VIX options, (3.2) emphasizes the most liquid options. We face a so-called inverse problem formulation which means that we need to find a parameter set W Θ and a latent state v22 such that almost all options are matched. Unfortunately, this problem is highly ill-posed which entails a non-unique solution, or it may even not exist at all. Furthermore, (3.1) and (3.2) imply a high-dimensional non-convex optimization problem. Using a gradient-based algorithm (local optimizer) will not necessarily lead to the global solution. Instead, due to this non-convexity and the high sensitivity of these optimizers with respect to small changes in the data and starting values, we will end up with many different local minima. Further issues are the possible remaining noise in the data or the mid-price as approximation for the market price. To mitigate the ill-posedness problem and to find the global minimum we use the following strategy: Pascal Marco Caversaccio

c 2014

3.1. PARAMETER ESTIMATION

57

Algorithm Algorithm for finding a global minimum 1:

Minimize the root mean squared pricing error in (3.1) using a differential evolution (DE) algorithm W to obtain a parameter set Θ and a latent state v22 which produce reasonable call option prices.

2:

Minimize the implied volatility root mean squared error in (3.2) using again a DE algorithm with W the parameter set Θ and the latent state v22 obtained in step 1 as initial values.

3:

Using the optimal parameters obtained in step 2 as initial values, apply a gradient-based optimizer to (3.2) to make sure the result from step 2 is indeed a global solution.

Remark 3.1. A DE algorithm is an efficient heuristic for the global optimization over continuous spaces. The inception of such evolutionary-based optimization strategies can be traced back to Storn and Price (1997) and some earlier papers cited therein. In brief, a DE algorithm is a population-based optimizer that attacks the starting point problem by sampling the objective function at multiple, randomly chosen initial points and generates new points that are perturbations of existing points. We refer to Price et al. (2005) and Chakraborty (2008) for a review of differential evolution algorithms. Another worth mentioning approach for the calibration is based on regularization of the objective function in order to make it convex. We refer to Cont and Tankov (2004b) for a comprehensive treatment of this particular method. J Remark 3.2. Like in the case of the FFT method, the COS method can break down for deep OTM options where it often gives rise to negative prices. With regard to our optimization strategy, we incorporate this fact by first using a loss function which does not break down if a set of parameters is examined by the DE algorithm which leads to negative prices. In a second step the loss function based on implied volatilities is used, which would collapse with negative prices, by taking the reasonable parameters of the first step. Accordingly, we can circumvent the deterioration of the calibration in most cases. J J Following the algorithm above and using the closed-form expression in Proposition W 2.10 with ν0 = 0, we obtain the following estimates for the parameter set Θ and the latent state v22 .

M -8.0616 8.9894 8.9707 -10

Q 10 7.8412

27/02/2009 W v22 8.2678 0.125 10

β 1.01

$RMSE 0.31

IVRMSE 0.0538

Table 3.1 – The fitted parameter values of the Wishart option pricing model without jumps, i.e. ν0J = 0, W for the parameter set Θ and the latent state v22 on February 27, 2009. We use the closed-form expression in Proposition 2.10 for the model implied price for VIX options. The detailed data treatment is outlined in Section 2.3 and thereunto we additionally delete all observed options with price levels less than 0.5$. To mitigate the ill-posedness problem and to find the global minimum we use a DE algorithm. We report the $RMSE (see (3.1) for the definition) of step 1 and the IVRMSE (see (3.2) for the definition) of step 2 from the optimization algorithm above.

Let us remark that the boundaries for the components of M and Q in the DE algorithm were set to -10 and 10, respectively, which are attained in 3 cases, namely M22 , Q11 and Q22 . We inspected this peculiarity and a closer look on Corollary 2.3 reveals that the sensitivity of the solution with respect to the matrix Q is less high than for the matrix M and is therefore neglected, to some degree, in the optimization of the DE algorithm which could explain these figures. Further, if we compare our Pascal Marco Caversaccio

c 2014

3.2. PRICING PERFORMANCE

58

parameter estimates of M and Q with the calibration results of Leippold and Trojani (2010), Buraschi et al. (2010) or Gruber et al. (2010), among others, they substantially differ in the size of the numbers. One possible explanation of this difference is due to the used data which is in their case SPX option market data, where we are, to our best knowledge, the first using only VIX option data for the calibration of a MAD model. Moreover, since we treat a high turbulent day during the financial crisis, this implies higher figures accordingly. Despite this contrast to previous research, we will show in Section 3.2 that our estimates can, to some extend, accurately replicate the VIX implied volatility skews. Also observe that the estimates satisfy the theoretical requirements, in particular Q is invertible, β > 1 and the linear Lyapunov equation in (2.30) does not explode. Nonetheless, in future research special attention needs to be given on the calibration, in particular for a particle filter based estimation, due to the high sensitivity of the optimization with respect to the data points. To compare our zero-jump model with another stochastic volatility model without jumps, we choose the seminal Heston (1993) model and obtain the following estimates for the parameters.54

√

ν0 0.1546

ρ 0.579

κ 1.8656

27/02/2009 θ γ 0.1866 0.4787

$RMSE 0.8627

IVRMSE 0.2058

Table 3.2 – The fitted parameter values of the Heston (1993) model, using VIX option data on February 27, 2009, in which the instantaneous volatility is modeled as a square root process (see, for instance, (2.27)). √ The parameter ν0 is the estimated value of stochastic volatility at t = 0, ρ is the correlation coefficient between the two driving Brownian motions, κ is referred to as the speed of mean reversion, θ represents the level of mean reversion and the parameter γ denotes the volatility of volatility. The detailed data treatment is outlined in Section 2.3 and thereunto we additionally delete all observed options with price levels less than 0.5$. To mitigate the ill-posedness problem and to find the global minimum we use a DE algorithm. We force the parameters to satisfy the Feller condition, i.e. γ 2 < 2κθ (see also Remark 2.13). We report the $RMSE (see (3.1) for the definition) of step 1 and the IVRMSE (see (3.2) for the definition) of step 2 from the optimization algorithm above.

Using the obtained fitted parameter values in Table 3.1 and Table 3.2, we analysis the model pricing performance by means of implied volatilities in the next section.

3.2

Pricing Performance

At first glance we see that, based on Table 3.1 and Table 3.2, our MAD model outperforms the Heston model in terms of $RMSE and IVRMSE. The difference is substantial and consistent with the findings in Gatheral (2008) who show that, using parameters based on Monte Carlo simulations, the Heston model is inappropriate for the replication of the VIX implied volatility skew and should not be used W therefore for the pricing– or hedging of VIX options. Further, the latent state v22 exhibits a reasonable high value which is needed for the replication of the IVS shape on this day. In Figure 3.1 we depict the obtained implied volatility skews implied by our MAD– and the Heston model for 4 different time-tomaturities on February 27, 2009. The moneyness m is defined in (2.11) and τ := T − t is the option’s 54

We also used another evolutionary-based algorithm, a so-called genetic algorithm (GA), which is nevertheless outperformed by the DE algorithm in terms of $RMSE, IVRMSE and calculation time. Therefore, we only report the results of the DE algorithm.


c 2014


59

time-to-maturity in fractional years. The first, even though non-surprising, observation which can be deduced from Figure 3.1 is that the Heston model fails to fit the IVS slice for each time-to-maturity. In three out of four maturities it does not even attain any market data point. The only exception is given for τ = 0.1306, where it approximately touches the data point around the moneyness level of 0.5. Second, our zero-jump model however performs well, due to the high flexibility even without jumps, in the replication of the IVS slice shapes and their levels. The best fits are obtained for τ = 0.1306 and τ = 0.3056. In these cases we can perfectly reproduce a couple of data points and the others are very close to the predicted MAD implied volatilities. For the maturities τ = 0.0528 and τ = 0.2278 the performance is not striking and we can only match two (resp. one) market data point(s) for τ = 0.0528 (resp. τ = 0.2278). VIX: 27/02/2009, τ = 0.0528

VIX: 27/02/2009, τ = 0.1306 1.2

3 MAD Heston Market

2 1.5 1 0.5 0 −1

MAD Heston Market


Implied Volatility

2.5

1 0.9 0.8 0.7 0.6

0 1 Moneyness, m

0.5 −0.5

2

VIX: 27/02/2009, τ = 0.2278

1.5

2

2 MAD Heston Market

MAD Heston Market

Implied Volatility


0.5 1 Moneyness, m

VIX: 27/02/2009, τ = 0.3056

1

0.8 0.7 0.6 0.5 −0.5

0

0 0.5 Moneyness, m

1

1.5

1

0.5

0

0

0.5 1 Moneyness, m

1.5

Figure 3.1 – The fitted VIX implied volatility skews implied by the MAD– and Heston model, using the parameter estimates in Table 3.1 and Table 3.2, for 4 different time-to-maturities on February 27, 2009, ln(K/FtV (T )) √ as a function of the standardized log-moneyness m := ATMIV(t,T , where K is the strike level of the ) T −t European-style option, FtV (T ) denotes the closing VIX futures price today at time t with maturity T , ATMIV (t, T ) stands for the at-the-money implied volatility quote, and τ := T − t is the option’s timeto-maturity which is depicted in fractional years in the plots. The detailed data treatment is outlined in Section 2.3 and thereunto we additionally delete all observed options with price levels less than 0.5$. To back out the implied volatilities, a hybrid algorithm, consisting of the Newton-Raphson algorithm and the bisection method, is used.


c 2014


60

There is space for model improvements for very short-term maturities, as for τ = 0.0528 in Figure 3.1. The introduction of jumps and e.g. applying a different scaling to the matrix-variate Lévy density in Section 2.5.2 can help to overcome this shortcoming. Moreover, it is possible that we obtain different results for different data providers since some of them preprocess the data.55 Among other things (see Chapter 4), these issues will be incorporated into the future research on this topic.56 As emphasized in Bardgett et al. (2013), daily calibration is a multiple curve-fitting exercise where one checks whether the models which come under scrutiny can fit the risk-neutral distributions inferred by option prices for different time-to-maturities. This approach is obviously extremely myopic, namely only holds on a daily basis, and cannot show whether the model dynamics can replicate the time evolution of the entire IVS shapes and their levels. Furthermore, as indicated in Broadie et al. (2007) and Lindström et al. (2008), the daily calibrated parameters are unstable and can vary a lot over time and therefore should not be used to infer the model performance over a time series– and cross section of options. Instead, to fit both the cross-sectional– and time-series properties we could consider a particle filter based estimation strategy. This topic also belongs to the extensive body of future research.

55 56

Special thanks go to Prof. Dr. Leippold for pointing this out. In Figure 3.1 we also see less data points depicted than available on this day (42), since some of them did not allow for a solution in the implied volatility calculations of the optimization procedure even though we used a hybrid algorithm consisting of the Newton-Raphson algorithm and the bisection method. This numerical issue will be also part of future research.


c 2014

Chapter

4

Conclusion If there is something very slightly wrong in our definition of the theories, then the full mathematical rigor may convert these errors into ridiculous conclusions. — Richard Phillips Feynman (1918 – 1988), Nobel Prize in Physics 1965

In this thesis we address the issue of pricing VIX options while preserving consistency with the SPX options. Our model design traces back to the preliminary data analysis conducted at the beginning of this thesis. We show, using descriptive statistical measures, that the SPX and VIX exhibit a negative linkage with a time-varying correlation coefficient and moreover that the VIX has a mean-reverting behavior. Further, the log-returns of the SPX exhibit a lower skewness than the VIX but on the contrary a higher excess kurtosis compared to the VIX, which implies a strong leptokurtic behavior indicating rare but sharp movements in the tails. Finally we take a look at the implied volatility skews of the SPX– and VIX options. The results show that the backed out implied volatilities of the SPX options are negatively skewed and the reverse is true for VIX options. The main driver of these shapes is the seminal leverage effect arising from the negative correlation between the SPX log-returns and its volatility. For the specification of the theoretical model, we assume, due to its generality, a time-changed Lévy process for the SPX dynamics. The motivation of this approach is twofold. First, Lévy processes cannot capture stochastic volatility, stochastic risk reversal (skewness) and stochastic correlation. These drawbacks can be resolved, to some extent, by considering time-changed Lévy processes for which it is possible to generate distributions which vary over time. Second, by specifying a suitable time change for the Brownian motion part, we can integrate the VIX dynamics via a stochastic activity rate and can therefore preserve consistency between the markets and their options. In particular, we specify the 2 × 2-dimensional joint SPX-VIX2 covariance dynamics under the risk-neutral measure Q, which can be seen equivalently as the stochastic activity rate for the Brownian motion component, by a jump-extended Wishart process. Hence, we obtain a double jump matrix-variate stochastic volatility model in which we allow for multifactor volatility, stochastic correlation, stochastic skewness, and jumps in returns and second moments. Summarizing, our model design accounts for the different empirical evidence such as the volatility smile, fat tails of the return distribution, the power-law scaling property of return moments and the seminal leverage effect of returns.

61

4. CONCLUSION

62

Since we preserve the affine structure, we can solve our financial pricing problem by means of transform methods. In order to capture the well-known leverage effect we introduce a correlation between the Brownian motion in the SPX dynamics without time change and the Brownian motion component in the activity rate of the corresponding time change. By using the leverage-neutral measure change method and the FFT technique we obtain a semi-closed expression for SPX options. For VIX options the COS method similarly yields a semi-closed expression. We estimate a simplified version of the VIX option pricing formulae, namely with a zero-jump component, on February 27, 2009, and demonstrate that the Heston (1993) model fails to fit the IVS slice for each time-to-maturity where our model however performs well, due to the high flexibility even without jumps, in the replication of the IVS slice shapes and their levels. Finally, future research on this topic will include daily calibration exercises for the simultaneous pricing of SPX– and VIX options with jumps on different dates. To evaluate the performance a comparison to SVJ– and SVJJ models need to be done. Further, to fit the model to the cross-sectional– and time-series properties one needs to consider, e.g., a particle filter based estimation strategy.


c 2014

Appendices

Appendix

A

The Ft-Conditional Characteristic Function of the CIR Process First, fix the closed state space X ⊂ R≥0 with non-empty interior. According to Definition 2.2, Theorem 2.1 and Example 2.1, we need to find the C-valued functions φ(T − t, z) and ψ(T − t, z) with φ(0, z) = 0 and ψ(0, z) = z, such that J Mt := f t, νtJ = eφ(T −t,z)+ψ(T −t,z)νt is a martingale. This implies h J i J EQ ezνT Ft = EQ [ MT | Ft ] = Mt = eφ(T −t,z)+ψ(T −t,z)νt . Assuming the functions φ and ψ satisfy Definition 2.3, we apply the Itô-formula to J f t, νtJ = eφ(T −t,z)+ψ(T −t,z)νt . The relevant derivatives are57 ft t, νtJ = − φ˙ (T − t, z) + ψ˙ (T − t, z) νtJ f t, νtJ , fv t, νtJ = ψ (T − t, z) f t, νtJ , 2 fvv t, νtJ = ψ (T − t, z) f t, νtJ . We obtain, by plugging in the CIR dynamics of (2.27), 2 df t, νtJ 1 2 = − φ˙ (T − t, z) + ψ˙ (T − t, z) νtJ dt + ψ (T − t, z) dνtJ + ψ (T − t, z) γ J νtJ dt J 2 f t, νt 1 2 2 = − φ˙ (T − t, z) + ψ˙ (T − t, z) νtJ − ψ (T − t, z) κJ θJ − νtJ + ψ (T − t, z) γ J νtJ dt 2 q + ψ (T − t, z) γ J νtJ dBtQ . 57

We denote by φ˙ (·) and ψ˙ (·), respectively, the corresponding time derivatives.

64

A. THE Ft -CONDITIONAL CHARACTERISTIC FUNCTION OF THE CIR PROCESS

65

Hence, f t, νtJ is a local martingale iff 1 2 2 φ˙ (T − t, z) + ψ˙ (T − t, z) νtJ = ψ (T − t, z) κJ θJ − νtJ + ψ (T − t, z) γ J νtJ . 2

(A.1)

We can now separate the variables, since (A.1) must hold for all νtJ ∈ X with t ∈ [0, T ], and obtain the system, ∀s ∈ R≥0 , φ˙ (s, z) = κJ θJ ψ (s, z) , (A.2) 2 1 2 ψ˙ (s, z) = −κJ ψ (s, z) + ψ (s, z) γ J , 2 with boundary conditions φ (0, z) = 0, ψ (0, z) = z. The second equation in (A.2) is a so-called Riccati equation. We refer to Remark 2.3 for a historical anecdote on Riccati equations. Solving the system in (A.2) by some involved calculations, we get ! 2 J γJ 2κJ θJ ze−κ (T −t) −κJ (T −t) z 1−e . ψ (T − t, z) = , φ (T − t, z) = − J 2 log 1 − J 2 2κJ (γ ) 1 − (γ )J z 1 − e−κJ (T −t) 2κ

This result allows us to write the Ft -conditional characteristic function of the CIR process under the risk-neutral measure Q as h

EQ e

J zνT

i Ft =

! 2κJ θJ 2 − (γ J )2 γJ J 1− z 1 − e−κ (T −t) exp 2κJ


J

νtJ ze−κ 1−

(γ J )2 2κJ z

!

(T −t)

1 − e−κJ (T −t)

.

c 2014

Appendix

B

Proof of Proposition 2.8 According to Radon’s lemma, we can represent the function ψW (τ ) as ψW (τ ) = J (τ )

−1

K (τ ) ,

(B.1)

where K (τ ) and J (τ ) are square matrices with J (τ ) invertible. Multiplying (2.48), where we assume (ϑ)

ν1J,Q

= 0, by J (τ ) yields (ϑ)

J (τ ) ∂τ ψW (τ ) = ΨW (ϑ) J (τ ) + J (τ ) M Q

>

(ϑ)

ψW (τ ) + J (τ ) ψW (τ ) M Q

+ 2J (τ ) ψW (τ ) Q> QψW (τ ) .

(B.2)

Now, we differentiate J (τ ) ψW (τ ) = K (τ )

(B.3)

J (τ ) ∂τ ψW (τ ) = ∂τ (J (τ ) ψW (τ )) − ∂τ J (τ ) ψW (τ )

(B.4)

∂τ (J (τ ) ψW (τ )) = ∂τ K (τ ) .

(B.5)

in light of (B.1), and obtain

and

Plugging (B.3), (B.4) and (B.5) into (B.2), we get (ϑ)

∂τ K (τ ) − ∂τ J (τ ) ψW (τ ) = ΨW (ϑ) J (τ ) + J (τ ) M Q (ϑ)

+ J (τ ) ψW (τ ) M Q

>

ψW (τ )

+ 2J (τ ) ψW (τ ) Q> QψW (τ ) .

By collecting the coefficients of ψW (τ ), the following matrix ODEs are induced: (ϑ)

∂τ K (τ ) = ΨW (ϑ) J (τ ) + K (τ ) M Q >

∂τ J (τ ) = −2K (τ ) Q Q − J (t) M 66

,

Q(ϑ) >

,

B. PROOF OF PROPOSITION 2.8

67

or ! M Q(ϑ) −2K (τ ) Q> Q d . K (τ ) J (τ ) = K (τ ) J (τ ) (ϑ) dτ ΨW (ϑ) −M Q >

(B.6)

We can solve (B.6) by exponentiation:

K (τ ) J (τ ) = K (0)

" J (0) exp τ "

= ψW (0) I2 exp τ

!# (ϑ) MQ −2K (τ ) Q> Q (ϑ) ΨW (ϑ) −M Q > !# (ϑ) MQ −2K (τ ) Q> Q (ϑ) ΨW (ϑ) −M Q >

= ψW (0) C11 (τ ) + C21 (τ ) ψW (0) C12 (τ ) + C22 (τ ) = C21 (τ ) C22 (τ ) . Finally from (B.1), we can conclude that the solution is given by −1

ψW (τ ) = C22 (τ )

C21 (τ ) .

A moment’s reflection reveals that φW (τ ) is obtained by simple integration, "Z Z # Z τ τ VW J,Q(ϑ) > W W φW (τ ) = tr βQ Q ψW (s) ds + ν0 exp tr ψW (s) v Y dv ds − τ 0

0

i (ϑ) (ϑ) β h = − tr ln C22 (τ ) + τ M Q > + ν0J,Q 2

S2+ \{0}

"Z 0

τ

#

Z S2+ \{0}

exp tr ψW (s) v

W

W

YV

dv

W

ds − τ .

This concludes the proof.


c 2014

Appendix

C

The Fourier Transform of the Call Option c (k) The Fourier transform of the option price c (k) is characterized by the following expression: Z χ (u) := eiuk c (k) dk,

(C.1)

R

with u ∈ D ⊆ C. The following identity holds due to the payoff structure in log-price of a European Call option: Z c (k) := es − ek 1{s≥k} dFS (s) . (C.2) R

Plugging (C.2) into (C.1), applying Fubini and simple integration yields Z χ (u) = eiuk c (k) dk ZR Z s k = e − e 1{s≥k} dFS (s) ds eiuk dk R R Z Z = es − ek 1{s≥k} eiuk dk dFS (s) R R Z Z s = eiuk+s − e(iu+1)k dk dFS (s) R −∞ s # Z " iuk e(iu+1)k se = e − dFS (s) . iu iu + 1 −∞ R We need to consider the boundary conditions at k = −∞. We have limk→−∞ eiuk = 0 iff the real component of iu is greater than 0. Concerning the other case, we have limk→−∞ e(iu+1)k = 0 iff the real component of iu is greater than −1. Thus, summarizing we need the real component of iu to be greater 68

C. THE FOURIER TRANSFORM OF THE CALL OPTION c (k)

69

than 0. We can write u := z − iα, z, α ∈ R and obtain iu = iz + α. Hence, due to the convergence we need α ∈ R+ . Assuming that iz ∈ R+ , we obtain Z s(iu+1)s e(iu+1)s e χ (u) = − dFS (s) iu iu + 1 R Z e(iu+1)s = dFS (s) R iu (iu + 1) µ b (u − i) . = iu (iu + 1)


c 2014

Appendix

D

Proof of Corollary 2.2 We follow the same rationale as in the proof of Proposition 2.8. We repeat it here for the reader’s convenience. The linearization of the flow of the differential equation is obtained by Radon’s lemma. We can express ψV (τ ) by −1

ψV (τ ) = J (τ )

K (τ ) ,

(D.1)

where K (τ ) and J (τ ) are square matrices with J (τ ) invertible. Multiplying (2.54), where we assume ν1J = 0, by J (τ ) yields J (τ ) ∂τ ψV (τ ) = J (τ ) M > ψV (τ ) + J (τ ) ψV (τ ) M + 2J (τ ) ψV (τ ) Q> QψV (τ ) .

(D.2)

Now, we differentiate J (τ ) ψV (τ ) = K (τ )

(D.3)

J (τ ) ∂τ ψV (τ ) = ∂τ (J (τ ) ψV (τ )) − ∂τ J (τ ) ψV (τ )

(D.4)

∂τ (J (τ ) ψV (τ )) = ∂τ K (τ ) .

(D.5)

in light of (D.1), and obtain

and

Plugging (D.3), (D.4) and (D.5) into (D.2), we get ∂τ K (τ ) − ∂τ J (τ ) ψV (τ ) = J (τ ) M > ψV (τ ) + J (τ ) ψV (τ ) M + 2J (τ ) ψV (τ ) Q> QψV (τ ) . By collecting the coefficients of ψV (τ ), the following matrix ODEs are induced: ∂τ K (τ ) = K (τ ) M, ∂τ J (τ ) = −2K (τ ) Q> Q − J (t) M > ,

70

D. PROOF OF COROLLARY 2.2

71

or M d K (τ ) J (τ ) = K (τ ) J (τ ) dτ 0

! −2K (τ ) Q> Q . −M >

(D.6)

We can solve (D.6) by exponentiation:

" K (τ ) J (τ ) = K (0) J (0) exp τ

!# −2K (τ ) Q> Q −M > " !# M −2K (τ ) Q> Q = ψV (0) I2 exp τ 0 −M > = ΓC11 (τ ) + C21 (τ ) ΓC12 (τ ) + C22 (τ ) . M 0

Notice, the difference to the solution of Proposition 2.8 is solely due to the terminal conditions and the constant interest rate. Finally from (D.1), we can deduce the solution which is given by ψV (τ ) = (ΓC12 (τ ) + C22 (τ ))

−1

(ΓC11 (τ ) + C21 (τ )) .

A moment’s reflection reveals that φV (τ ) is obtained by simple integration, # "Z Z Z τ Z τ τ VW W W > J exp tr ψV (s) v Y dv ds − τ φV (τ ) = −ρ0 ds + tr βQ Q ψV (s) ds + ν0 0

0

0

S2+ \{0}

β = −ρ0 τ − tr ln (ΓC12 (τ ) + C22 (τ )) + τ M > 2 # "Z Z τ VW W W J exp tr ψV (s) v Y dv ds − τ . + ν0 0

S2+ \{0}

This concludes the proof.


c 2014

Bibliography

Y. A¨ıt-Sahalia and J. Jacod. Analyzing the spectrum of asset returns: Jump and volatility components in high frequency data. Journal of Economic Literature, 50(4):1007–1050, 2012. Y. A¨ıt-Sahalia and A. W. Lo. Nonparametric estimation of state-price densities implicit in financial asset prices. Journal of Finance, 53(2):499–547, 1998. C. Albanese, H. Lo, and A. Majatović. Spectral methods for volatility derivatives. Quantitative Finance, 9(6):663–692, 2009. T. Ané and H. Geman. Order flow, transaction clock, and normality of asset returns. Journal of Finance, 55(5):2259–2284, 2000. D. Applebaum. Lévy processes and stochastic calculus, volume 116 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, second edition, 2009. Aristotle. Politics. Book I. C. Bardgett, E. Gourier, and M. Leippold. Inferring volatility dynamics and risk premia from the S&P 500 and VIX markets. Swiss Finance Institute, Research Paper Series No 13-40, University of Zurich, 2013. O. E. Barndorff-Nielsen. Processes of normal inverse Gaussian type. Finance and Stochastics, 2(1):41–68, 1997. O. E. Barndorff-Nielsen and V. Pérez-Abreu. Matrix subordinators and related upsilon transformations. Theory of Probability & Its Applications, 52(1):1–23, 2008. O. E. Barndorff-Nielsen and N. Shephard. Non-Gaussian Ornstein-Uhlenbeck-based models and some of their uses in financial economics. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 63(2):167–241, 2001. O. E. Barndorff-Nielsen and R. Stelzer. Positive-definite matrix processes of finite variation. Probability and Mathematical Statistics, 27(1):3–43, 2007. D. S. Bates. Jumps and stochastic volatility: Exchange rate processes implicit in Deutsche Mark options. Review of Financial Studies, 9(1):69–107, 1996. S. Beckers. The constant elasticity of variance model and its implications for option pricing. Journal of Finance, 35(3):661–673, 1980. 72

BIBLIOGRAPHY

73

L. Bergomi. Smile dynamics. Risk, September:117–123, 2004. L. Bergomi. Smile dynamics II. Risk, October:67–73, 2005. L. Bergomi. Smile dynamics III. Risk, October:90–96, 2008. J. Bertoin. Lévy processes. Cambridge University Press, New York, 1996. F. Black and M. Scholes. The pricing of options and corporate liabilities. Journal of Political Economy, 81(3):637–654, 1973. S. Bochner. Diffusion equation and stochastic processes. Proceedings of the National Academy of Science of the United States of America, 35(7):368–379, 1949. M. Brenner and D. Galai. New financial instruments for hedging changes in volatility. Financial Analysts Journal, 45(4):61–65, 1989. M. Britten-Jones and A. Neuberger. Option prices, implied price processes, and stochastic volatility. Journal of Finance, 55(2):839–866, 2000. M. Broadie and Y. Yamamoto. Application of the fast Gauss transform to option pricing. Management Science, 49(8):1071–1088, 2003. M. Broadie, M. Chernov, and M. Johannes. Model specification and risk premia: Evidence from futures options. Journal of Finance, 62(3):1453–1490, 2007. M.-F. Bru. Wishart processes. Journal of Theoretical Probability, 4(4):725–751, 1991. A. Buraschi, A. Cieslak, and F. Trojani. Correlation risk and the term structure of interest rates. Working Paper, Imperial College London, University of St. Gallen, 2008. A. Buraschi, P. Porchia, and F. Trojani. Correlation risk and optimal portfolio choice. Journal of Finance, 65(1):393–420, 2010. P. Carr and R. Lee. Volatility derivatives. Annual Review of Financial Economics, 1(1):319–339, 2009. P. Carr and D. Madan. Saddlepoint methods for option pricing. Journal of Computational Finance, 13 (1):49–61, 2009. P. Carr and D. B. Madan. Option valuation using the fast Fourier transform. Journal of Computational Finance, 2(4):61–73, 1999. P. Carr and L. Wu. The finite moment log stable process and option pricing. Journal of Finance, 58(2): 753–778, 2003. P. Carr and L. Wu. Time-changed Lévy processes and option pricing. Journal of Financial Economics, 71(1):113–141, 2004. P. Carr and L. Wu. Leverage effect, volatility feedback, and self-exciting market disruptions. Working Paper, New York University, City University of New York, 2011. P. Carr, H. Geman, D. B. Madan, and M. Yor. The fine structure of asset returns: An empirical investigation. Journal of Business, 75(2):305–333, 2002. P. Carr, H. Geman, D. B. Madan, and M. Yor. Stochastic volatility for Lévy processes. Mathematical Finance, 13(3):345–382, 2003. Pascal Marco Caversaccio

c 2014

BIBLIOGRAPHY

74

U. K. Chakraborty, editor. Advances in differential evolution, volume 143 of Studies in Computational Intelligence. Springer Verlag, Berlin Heidelberg, 2008. J. Cheng, M. Ibraimi, M. Leippold, and J. E. Zhang. A remark on Lin and Chang’s paper ’Consistent modeling of S&P 500 and VIX derivatives’. Journal of Economic Dynamics & Control, 36(5):708–715, 2012. K. Chourdakis. Option pricing using the fractional FFT. Journal of Computational Finance, 8(2):1–18, 2005. P. Christoffersen and K. Jacobs. The importance of the loss function in option valuation. Journal of Financial Economics, 72(2):291–318, 2004. S.-L. Chung, W.-C. Tsai, Y.-H. Wang, and P.-S. Weng. The information content of the S&P 500 index and VIX options on the dynamics of the S&P 500 index. Journal of Futures Markets, 31(12):1170–1201, 2011. P. K. Clark. A subordinated stochastic process model with finite variance for speculative prices. Econometrica, 41(1):135–155, 1973. R. Cont and T. Kokholm. A consistent pricing model for index options and volatility derivatives. Mathematical Finance, 23(2):248–274, 2011. R. Cont and P. Tankov. Financial modelling with jump processes. Financial Mathematics Series. Chapman & Hall/CRC, Boca Raton, first edition, 2004a. R. Cont and P. Tankov. Non-parametric calibration of jump-diffusion option pricing models. Journal of Computational Finance, 7(3):1–50, 2004b. J. C. Cox and S. A. Ross. The valuation of options for alternative stochastic processes. Journal of Financial Economics, 3(1-2):145–166, 1976. J. C. Cox, J. E. Ingersoll, and S. A. Ross. A theory of the term structure of interest rates. Econometrica, 53(2):385–407, 1985. R. Crameri, M. Leippold, and F. Trojani. Portfolio business activities, Lévy returns and stochastic multivariate risk. Working Paper, University of Zurich, University of Lugano, 2010. C. Cuchiero, D. Filipović, E. Mayerhofer, and J. Teichmann. Affine processes on positive semidefinite matrices. The Annals of Applied Probability, 21(2):397–463, 2011a. C. Cuchiero, M. Keller-Ressel, E. Mayerhofer, and J. Teichmann. Affine processes on symmetric cones. Working Paper, Vienna University of Technology, Berlin University of Technology, Dublin City University, Swiss Federal Institute of Technology, http://arxiv.org/pdf/1112.1233v1.pdf, 2011b. H. Daouk and J. Q. Guo. Switching asymmetric GARCH and options on a volatility index. Journal of Futures Markets, 24(3):251–282, 2004. D. Davydov and V. Linetsky. Pricing and hedging path-dependent options under the CEV process. Management Science, 47(7):949–965, 2001. F. Delbaen and W. Schachermayer. A general version of the fundamental theorem of asset pricing. Mathematische Annalen, 300(1):463–520, 1994. E. Derman and I. Kani. Stochastic implied trees: Arbitrage pricing with stochastic term and strike structure of volatility. International Journal of Theoretical and Applied Finance, 1(1):61–110, 1998. Pascal Marco Caversaccio

c 2014

BIBLIOGRAPHY

75

J. Detemple and C. Osakwe. The valuation of volatility options. European Finance Review, 4(1):21–50, 2000. G. Drimus and W. Farkas. Local volatility of volatility for the VIX market. Review of Derivatives Research, 16(3):267–293, 2013. L. E. Dubins and G. Schwarz. On continuous martingales. Proceedings of the National Academy of Sciences of the United States of America, 53(5):913–916, 1965. D. Duffie, J. Pan, and K. Singleton. Transform analysis and asset pricing for affine jump-diffusions. Econometrica, 68(6):1343–1376, 2000. D. Duffie, D. Filipović, and W. Schachermayer. Affine processes and applications in finance. The Annals of Applied Probability, 13(3):984–1053, 2003. B. Dumas, J. Fleming, and R. E. Whaley. Implied volatility functions: Empirical tests. Journal of Finance, 53(6):2059–2106, 1998. E. Eberlein, U. Keller, and K. Prause. New insights into smile, mispricing, and value at risk: The hyperbolic model. Journal of Business, 71(3):371–405, 1998. F. Fang and C. W. Oosterlee. A novel pricing method for European options based on Fourier-cosine series expansions. SIAM Journal on Scientific Computing, 31(2):826–848, 2008. W. Feller. Two singular diffusion problems. Annals of Mathematics, 54(1):173–182, 1951. D. Filipović and E. Mayerhofer. Affine diffusion processes: Theory and applications. Radon Series on Computational and Applied Mathematics, 8:1–40, 2009. J. Fleming, B. Ostdiek, and R. E. Whaley. Predicting stock market volatility: A new measure. Journal of Futures Markets, 15(3):265–302, 1995. J. D. Fonseca, M. Grasselli, and F. Ielpo. Estimating the Wishart affine stochastic correlation model using the empirical characteristic function. Working Paper, Auckland University of Technology, University of Padua, Université Paris I Panthéon-Sorbonne, 2012. J.-P. Fouque, G. Papanicolaou, and K. R. Sircar. Derivatives in financial markets with stochastic volatility. Cambridge University Press, Cambridge, 2000. J.-P. Fouque, G. Papanicolaou, R. Sircar, and K. Solna. Multiscale stochastic volatility asymptotics. Multiscale Modeling and Simulation, 2(1):22–42, 2003. P. Friz and J. Gatheral. Valuation of volatility derivatives as an inverse problem. Quantitative Finance, 5(6):531–542, 2005. J. Gatheral. Consistent modeling of SPX and VIX options. In The Fifth World Congress of the Bachelier Finance Society, London, July 2008. C. Gouriéroux and R. Sufana. Derivative pricing with Wishart multivariate stochastic volatility. Journal of Business & Economic Statistics, 28(3):438–451, 2010. C. Gouriéroux, J. Jasiak, and R. Sufana. The Wishart Autoregressive process of multivariate stochastic volatility. Journal of Econometrics, 150(2):167–181, 2009. P. Gruber, C. Tebaldi, and F. Trojani. Three make a smile – Dynamic volatility, skewness and term structure components in option valuation. Working Paper, Università Bocconi, 2010. Pascal Marco Caversaccio

c 2014

BIBLIOGRAPHY

76

A. Gr¨ unbichler and F. A. Longstaff. Valuing futures and options on volatility. Journal of Banking & Finance, 20(6):985–1001, 1996. S. L. Heston. A closed-form solution for options with stochastic volatility with applications to bond and currency options. Review of Financial Studies, 6(2):327–343, 1993. S. L. Heston and S. Nandi. Derivatives on volatility: Some simple solutions based on observables. Working Paper, Federal Reserve Bank of Atlanta, 2000. N. Hilber, O. Reichmann, C. Schwab, and C. Winter. Computational methods for quantitative finance. Springer Verlag, Berlin Heidelberg, 2013. C. Homescu. Implied volatility surface: Construction methodologies and characteristics. Working Paper, Wells Fargo Financial, 2011. S. Howison, A. Rafailidis, and H. Rasmussen. On the pricing and hedging of volatility derivatives. Applied Mathematical Finance, 11(4):317–346, 2004. Y. L. Hsu, T. I. Lin, and C. F. Lee. Constant elasticity of variance (CEV) option pricing model: Integration and detailed derivation. Mathematics and Computers in Simulation, 79(1):60–71, 2008. J. Hull and A. White. The pricing of options on assets with stochastic volatilities. Journal of Finance, 42:281–300, 1987. K. Itˆ o and H. P. McKean. Diffusion processes and their sample paths, volume 125 of Classics in Mathematics. Springer Verlag, Berlin Heidelberg, 1 edition, 1965. J. Jacod and P. Protter. Probability essentials. Springer Verlag, Berlin Heidelberg, second edition, 2004. M. Jeanblanc, M. Yor, and M. Chesney. Mathematical methods for financial markets. Springer Verlag, Berlin Heidelberg, 2009. J. Kallsen, J. Muhle-Karbe, and M. Voss. Pricing options on variance in affine stochastic volatility models. Mathematical Finance, 21(4):627–641, 2011. M. Keller-Ressel. Affine processes - Theory and applications in finance. PhD thesis, Berlin University of Technology, 2008. M. Keller-Ressel, W. Schachermayer, and J. Teichmann. Affine processes are regular. Probability Theory and Related Fields, 151(3-4):591–611, 2011. M. Keller-Ressel, W. Schachermayer, and J. Teichmann. Regularity of affine processes on general state spaces. Electronic Journal of Probability, 18(43):1–17, 2013. S. G. Kou. A jump-diffusion model for option pricing. Management Science, 48(8):1086–1101, 2002. S. G. Kou. Option pricing under a double exponential jump diffusion model. Management Science, 50 (9):1178–1192, 2004. S. Kummer and C. Pauletto. The history of derivatives: A few milestones. EFTA Seminar on Regulation of Derivatives Markets, 2012. M. Leippold and J. Strømberg. Time-changed Lévy LIBOR market model for the joint estimation and pricing of caps and swaptions. Swiss Finance Institute, Research Paper Series No 12-23, University of Zurich, (forthcoming, Journal of Financial Economics), 2012. Pascal Marco Caversaccio

c 2014

BIBLIOGRAPHY

77

M. Leippold and F. Trojani. Asset pricing with matrix jump diffusions. Working Paper, University of Zurich, University of Lugano, 2010. M. Leippold and L. Wu. Asset pricing under the quadratic class. Journal of Financial and Quantitative Analysis, 37(2):271–295, 2002. P. Lévy. Processus stochastiques et mouvement brownien, volume 1 of Monographies des probabilités,. Gauthier-Villars, Paris, 1948. G.-H. Lian and S.-P. Zhu. Pricing VIX options with stochastic volatility and random jumps. Decisions in Economics and Finance, 36(1):71–88, 2013. Y.-N. Lin. Pricing VIX futures: Evidence from integrated physical and risk-neutral probability measures. Journal of Futures Markets, 27(12):1175–1217, 2007. Y.-N. Lin and C.-H. Chang. Consistent modeling of S&P 500 and VIX derivatives. Journal of Economic Dynamics and Control, 34(11):2302–2319, 2010. E. Lindstr¨ om, J. Str¨ ojby, M. Brodén, M. Wiktorsson, and J. Holst. Sequential calibration of options. Computational Statistics & Data Analysis, 52(6):2877–2891, 2008. R. Lord, F. Fang, F. Bervoets, and C. W. Oosterlee. A fast and accurate FFT-based method for pricing early-exercise options under Lévy processes. SIAM Journal on Scientific Computing, 30(4):1678–1705, 2008. D. B. Madan and E. Senata. The variance gamma (V.G.) model for share market returns. Journal of Business, 63(4):511–524, 1990. D. B. Madan, P. P. Carr, and E. C. Chang. The variance gamma process and option pricing. European Finance Review, 2(1):79–105, 1998. E. Mayerhofer, O. Pfaffel, and R. Stelzer. On strong solutions for positive definite jump diffusions. Stochastic Processes and their Applications, 121(9):2072–2086, 2011. J. Menc´ıa and E. Sentana. Valuation of VIX derivatives. Journal of Financial Economics, 108(2): 367–391, 2013. R. C. Merton. Theory of rational option pricing. The Bell Journal of Economics and Management Science, 4(1):141–183, 1973. R. C. Merton. Option pricing when underlying stock returns are discontinuous. Journal of Financial Economics, 3(1-2):125–144, 1976. M. Mori and M. Sugihara. The double-exponential transformation in numerical analysis. Journal of Computational and Applied Mathematics, 127(1-2):287–296, 2001. B. K. Øksendal. Stochastic differential equations: An introduction with applications. Universitext. Springer Verlag, Berlin Heidelberg, 6 edition, 2003. K. V. Price, R. M. Storn, and J. A. Lampinen. Differential evolution - A practical approach to global optimization. Natural Computing Series. Springer Verlag, Berlin Heidelberg, 2005. J. Rosi´ nski. Tempering stable processes. Stochastic Processes and their Applications, 117(6):677–707, 2007.


c 2014

BIBLIOGRAPHY

78

G. Samoradnitsky and M. S. Taqqu. Stable non-Gaussian random processes: Stochastic models with infinite variance. Stochastic Modeling Series. Chapman & Hall/CRC, New York, 1994. K. Sato. Lévy processes and infinitely divisible distributions, volume 68 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 1999. W. Schoutens. The Meixner process: Theory and applications in finance. Technical Report, Katholieke Universiteit Leuven, February 2002. M. Schroder. Computing the constant elasticity of variance option pricing formula. Journal of Finance, 44(1):211–219, 1989. L. O. Scott. Option pricing when the variance changes randomly: Theory, estimation, and an application. Journal of Financial and Quantitative Analysis, 22(4):419–438, 1987. A. Sepp. Pricing options on realized variance in the Heston model with jumps in returns and volatility. Journal of Computational Finance, 11(4):33–70, 2008a. A. Sepp. VIX option pricing in a jump-diffusion model. Risk Magazine, pages 84–89, 2008b. E. M. Stein and J. C. Stein. Stock price distributions with stochastic volatility: An analytic approach. Review of Financial Studies, 4(4):727–752, 1991. R. M. Storn and K. Price. Differential evolution - A simple and efficient heuristic for global optimization over continuous spaces. Journal of Global Optimization, 11(4):341–359, 1997. O. Thorin. On the infinite divisibility of the lognormal distribution. Scandinavian Actuarial Journal, 1977(3):121–148, 1977. G. E. Uhlenbeck and L. S. Ornstein. On the theory of the Brownian motion. Physical Review, 36(5): 823–841, 1930. V. A. Volkonskii. Random substitution of time in strong Markov processes. Theory of Probability & Its Applications, 3(3):310–326, 1958. G. N. Watson. The theory of Bessel functions. Cambridge University Press, New York, 1958. R. E. Whaley. Derivatives on market volatility: Hedging tools long overdue. Journal of Derivatives, 1 (1):71–84, 1993. Y. Yamamoto. Double-exponential fast Gauss transform algorithms for pricing discrete lookback options. Publications RIMS, Kyoto University, 41(4):989–1006, 2005. J. E. Zhang and Y. Zhu. VIX futures. Journal of Futures Markets, 26(6):521–531, 2006. S.-P. Zhu and G.-H. Lian. An analytical formula for VIX futures and its applications. Journal of Futures Markets, 32(2):166–190, 2012.


c 2014

Index

Fourier-cosine series expansion . . . . . . . 53

0-9 3-month volatility index . . . . . . . . . . . . . . 19

G Gamma-type Lévy density. . . . . . . . . . . .37 Gindikin coefficient . . . . . . . . . . . . . . . . . . . 43 Global variance shift factor . . . . . . . . . . . 43

A Absolutely continuous time change . . . 31 Affine dependence . . . . . . . . . . . . . . . . . . . . 10 Affine process . . . . . . . . . . . . . . . . . . . . . . . . . 8 α-stable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ATM forward put-call parity . . . . . . . . . 40

I Implied volatility RMSE . . . . . . . . . . . . . 56 Infinite activity. . . . . . . . . . . . . . . . . . . . . . .27 Infinite divisibility . . . . . . . . . . . . . . . . . . . . 24 Infinite variation . . . . . . . . . . . . . . . . . . . . . 27 Inverse problem formulation . . . . . . . . . . 56

C Central limit theorem . . . . . . . . . . . . . . . . 34 Characteristic exponent . . . . . . . . . . . . . . 26 Characteristic function . . . . . . . . . . . . . . . . 7 CIR Q-dynamics . . . . . . . . . . . . . . . . . . . . . 33 CIR Q(ϑ) -dynamics. . . . . . . . . . . . . . . . . . .48 Compound Poisson process . . . . . . . . . . . 27 Correlation-neutral measure . . . . . . . . . . 46 Cumulant function . . . . . . . . . . . . . . . . . . . 26

J Jump measure . . . . . . . . . . . . . . . . . . . . . . . 23 L Lévy exponent . . . . . . . . . . . . . . . . . . . . . . . 26 Lévy measure . . . . . . . . . . . . . . . . . . . . . . . . 24 Lévy process . . . . . . . . . . . . . . . . . . . . . . . . . 23 Lévy-Itô decomposition. . . . . . . . . . . . . . .25 Lévy-Khintchine representation . . . . . . . 25 Laplace exponent. . . . . . . . . . . . . . . . . . . . .26 Laplace transform . . . . . . . . . . . . . . . . . . . . 32 Level of mean reversion . . . . . . . . . . . . . . 33 Leverage effect . . . . . . . . . . . . . . . . . . . . . . . . 3 Leverage-neutral measure . . . . . . . . . . . . 46 Local volatility model . . . . . . . . . . . . . . . . 21 log-contract . . . . . . . . . . . . . . . . . . . . . . . . . . 13

D Differential evolution algorithm . . . . . . . 57 Discounted Laplace transform . . . . . . . . 44 Doléans-Dade exponential . . . . . . . . . . . . 48 Drifted Brownian motion . . . . . . . . . . . . . 27 E Exponential tilting . . . . . . . . . . . . . . . . . . . 45 F Fast Fourier transform . . . . . . . . . . . . . . . 50 Feller condition . . . . . . . . . . . . . . . . . . . . . . 33 Finite activity . . . . . . . . . . . . . . . . . . . . . . . . 27 Finite variation . . . . . . . . . . . . . . . . . . . . . . 27 Fourier transform . . . . . . . . . . . . . . . . . 50, 68

M Market implied approach . . . . . . . . . . . . . 55 N Non-homogenous Lévy process . . . . . . . 31 79

INDEX P Poisson process. . . . . . . . . . . . . . . . . . . . . . .27 R Radon’s lemma . . . . . . . . . . . . . . . . . . . 66, 70 Random measure . . . . . . . . . . . . . . . . . . . . . 23 Regularity. . . . . . . . . . . . . . . . . . . . . . . . . . . .10 Riccati equation . . . . . . . . . . . . . . . . . . . . . . 11 RMS pricing error . . . . . . . . . . . . . . . . . . . . 56 S Short-term volatility index . . . . . . . . . . . 18 Speed of mean reversion . . . . . . . . . . . . . . 33 SPX Q-dynamics . . . . . . . . . . . . . . . . . . . . . 41 SPX call option price . . . . . . . . . . . . . . . . . 50 SPX-VIX2 Q-covariance dynamics . . . . 43 SPX-VIX2 Q(ϑ) -covariance dynamics . 47 Squared Bessel process . . . . . . . . . . . . . . . 11 Stable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 Standardized log-moneyness . . . . . . . . . . 12


80 Stochastic exponential . . . . . . . . . . . . . . . . 48 Stochastic volatility model . . . . . . . . . . . 21 Strictly stable . . . . . . . . . . . . . . . . . . . . . . . . 28 Subordinator . . . . . . . . . . . . . . . . . . . . . . . . . 23 Symmetric stable law. . . . . . . . . . . . . . . . .29 T ϑ-Esscher transform . . . . . . . . . . . . . . . . . . 45 V Variance swap . . . . . . . . . . . . . . . . . . . . . . . . 13 Variance-gamma density. . . . . . . . . . . . . .34 VIX call option price . . . . . . . . . . . . . . . . . 53 VIX put-call parity . . . . . . . . . . . . . . . . . . . 41 Volatility index. . . . . . . . . . . . . . . . . . . . . . .13 Volatility of volatility. . . . . . . . . . . . . . . . .33 Volatility swap . . . . . . . . . . . . . . . . . . . . . . . 14 W Wishart process . . . . . . . . . . . . . . . . . . . . . . 36

c 2014