Applications of semimartingales and Lévy processes in finance: duality and valuation

Transcription

1 Applications of semimartingales and Lévy processes in finance: duality and valuation Dissertation zur Erlangung des Doktorgrades der Fakultät für Mathematik und Physik der Albert-Ludwigs-Universität Freiburg im Breisgau vorgelegt von Antonis Papapantoleon Dezember 26

2 Dekan: eferenten: Prof. Dr. Jörg Flum Prof. Dr. Ernst Eberlein Prof. Dr. Dr. h.c. Albert N. Shiryaev Moscow Datum der Promotion: 2. März 27 Abteilung für Mathematische Stochastik Albert-Ludwigs-Universität Freiburg Eckerstr. 1 D-7914 Freiburg im Breisgau

3 Abstract. The complexity of modern financial derivatives very often leads to valuation problems that require the knowledge of the joint distribution of several random variables. This thesis aims to simplify and solve such valuation problems. Duality is related to the simplification of the valuation problem. We investigate changes of probability measures in an effort to reduce the multivariate problem to a univariate one. The asset price processes are driven either by general semimartingales or by Lévy processes and their dynamics are expressed in terms of their predictable characteristics. Imposing some very natural conditions on the driving processes, a battery of derivative products including Asian, lookback and Margrabe options can be simplified considerably. Valuation is related to the solution of the problem. We provide general valuation formulae for options on single and multi-asset derivatives. These formulae require the knowledge of the characteristic function, while most of the commonly used payoff functions can be treated. Using the Wiener Hopf factorization, we provide expressions for options on the maximum of a Lévy process. Finally, we consider term structure models driven by time-inhomogeneous Lévy processes and provide duality and valuation results.

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5 Contents Acknowledgements vii Introduction 1 Overview synopsis 2 emarks 3 Part 1. Exponential semimartingale and Lévy models 5 Chapter 1. On the duality principle in option pricing: semimartingale setting Introduction Semimartingales and semimartingale characteristics Exponential semimartingale models Martingale measures and dual martingale measures The call-put duality in option pricing 28 Chapter 2. On the duality principle in option pricing II: multidimensional PIIAC and α-homogeneous payoff functions Introduction Time-inhomogeneous Lévy processes Asset price model General description of the method Options with α-homogeneous payoff functions Options on several assets 56 Chapter 3. Valuation of exotic derivatives in Lévy models Introduction Option valuation: general formulae Lévy processes and their fluctuations Examples of payoff functions Applications 82 Part 2. Term structure models 87 Chapter 4. Duality and valuation in Lévy term structure models Introduction Time-inhomogeneous Lévy processes Lévy fixed income models Caplet-floorlet duality Valuation of compositions 111 v

6 vi CONTENTS Appendix A. Transformations 119 Appendix B. An application of Itô s formula 121 Bibliography 123

7 Acknowledgements If you look back, you understand that what you wanted was to return home, and what happened is that you started seeking the route that takes you home. And therein lies the problem. You started looking for a route that did not exist. You should have made that route. Old-Antonio smiles satisfied. Then why do you say that we made the route? You made the route, I simply walked behind you I said feeling rather uncomfortable. Oh, no continues to smile old-antonio. I did not make the route alone. You also made it, since you walked in front for some part. Ah! But that part was useless! I interrupt. On the contrary. It was useful, since we learned it is not useful and thus we did not return to walk that route, since it did not take us where we wanted to go. Hence, we could make another route to take us home says old-antonio. S. I. Marcos, IstorÐec tou gèro-antìnio. I tend to consider this thesis as the end of a journey and I would like to thank all those people who have shared this journey with me. It was long, since I often searched for a route instead of creating one; hence, this personal statement will also be long. I would like to express my deepest gratitude to my advisor Prof. Dr. Ernst Eberlein. He was always present, supervising and helping, yet in a very natural and subtle way. He shared with me his insights on mathematics and finance, provided me with the freedom to explore my own sometimes vague ideas and gave me the opportunity of several educational excursions. Indeed, I will not be overstating his help if I say that he has given me more than I could have asked for. I would like to thank Prof. Dr. Albert N. Shiryaev for several interesting and amazing discussions during his visit at the University of Freiburg. His immense appetite for mathematical research has been a guiding light for me ever since. Many thanks are, of course, due to my doctoral brothers and sister, Jan Bergenhtum, Wolfgang Kluge and Zorana Grbac. We have discussed with Jan several mathematical and non-mathematical ideas and his opinion and advice was always of great help. I had the pleasure to share my afternoon tea with Wolfgang regularly; I would like to thank him for the numerous discussions and his patience for my questions. I am particularly glad these discussions led to some joint work, which is contained in this thesis. Zorana is a true friend, an attribute of both scientific and human dimensions; I am particularly grateful to her for thoroughly reading this thesis and some very helpful discussions on Chapter 2. vii

8 viii ACKNOWLEDGEMENTS I would also like to thank all my other colleagues from the Department of Mathematical Stochastics and especially Monika Hattenbach, for her help with accommodation, paperwork and L A TEX. I would like to thank Andreas Kyprianou for his warm hospitality at Heriot Watt University and for some very helpful discussions. I am grateful to a number people who have discussed several ideas and shared their insights with me. In no particular order, I thank Jan Kallsen, Thorsten Schmidt, Josef Teichmann, Pierre Patie, Evangelia Petrou, Stefan Ankirchner, George Skiadopoulos, Aleš Černý, Ernst August v. Hammerstein, Mikhail Urusov and Kathrin Glau. I am grateful to Maria Siopacha for her friendship throughout the years and her invaluable support during the last few months. I would like to thank my parents for all their love and support, and for what they have sacrificed for the education of their children. I hope this thesis makes them happy, although they will not understand its contents. I thank my sister, Clio, for being my best friend, sincerest advisor and hardest critic more often than not, I wish I had your wisdom! I extend my gratitude to my uncle Notis Polymeropoulos and his family. They have been of immense help throughout my time in Germany; and I have sincerely enjoyed the numerous discussions with Notis about life, politics and everything else. I am very grateful to all my friends from the different places and times. I thank Stelios Chronopoulos for the endless discussions during our Sunday meetings in Paradies, his crisp advice and his great analytical skills. Thanks are due to Manolis Man Of The Year Havakis, Spyros Koutsoumpos and Christos Lekatsas for their friendship; to Michaella Wenzlaff for her friendship and her efforts to teach me the German language. I thank Tino, Steffi, Tino, Uwe, Vasilis, Christina, Marcus, Martha, Jorge, Carmen, Sarah and Jorge. I thank Eugenia for some beautiful moments. I thank Cl.M. for the good times and the hard lessons. Many thanks go to my friends from Athens, especially Alexandros Papadias, Diogenis Brilakis and Antonis Atsaros; and to Costas The Doctor Liatsos, for his subtle but long-standing influence. Last but not least, I acknowledge the financial support provided through the European Community s Human Potential Programme under contract HPN-CT-2-1 DYNSTOCH and the financial support provided from the Deutsche Forschungsgemeinschaft DFG. Now, we will resume the use of the second person in plural and dive into the cold mathematical universe, where there is little room for emotions.

9 Introduction Mathematics is a field of science that has a very subtle but far-reaching influence on the life of human beings. Here one could think of number theory and its applications in encoding, that make internet transactions possible; or, of functional analysis and its applications in quantum mechanics. However, what we have in mind are the applications of mathematics in financial markets. Ever since the seminal articles of F. Black, M. Scholes, and. Merton cf. Black and Scholes 1973 and Merton 1973 and their reformulation in terms of martingale theory by M. Harrison, S. Pliska, and D. Kreps cf. Harrison and Kreps 1979, Harrison and Pliska 1981, Kreps 1981, stochastic analysis has become the playground of modern finance. As has already been noted elsewhere, stochastic analysis and martingale theory seem to be tailor-made for their application in mathematical finance; indeed, the proceeds from the investment in an asset can be represented as a stochastic integral, while the rational price of an option on an asset equals its discounted expected payoff under a martingale measure. Initially, the applications relied on the use of Brownian motion as the driving process, but empirical evidence showed that this assumption is too restrictive. One remedy was to consider more general continuous semimartingales as driving processes. Another one, was to consider Lévy processes as the driving force; this line of research was pioneered by E. Eberlein, D. Madan and their co-workers and paved the way for the application of general semimartingales in mathematical finance. Lévy processes are becoming increasingly popular in mathematical finance because they can describe the observed reality of financial markets in a more accurate way than models based on Brownian motion. In the real world, we observe that asset price processes have jumps or spikes and risk-managers have to take them into account. Moreover, the empirical distribution of asset returns exhibits fat tails and skewness, behavior that deviates from normality; hence, models that accurately fit return distributions are essential for the estimation of profit and loss P&L distributions. In the risk-neutral world, we observe that implied volatilities are constant neither across strike nor across maturities as stipulated by the Black and Scholes 1973 actually, Samuelson 1965 model. Therefore, traders need models that can capture the behavior of the implied volatility smiles more accurately, in order to handle the risk of trades. Lévy processes and general semimartingales provide us with the appropriate framework to adequately describe all these observations, both in the real and in the risk-neutral world. 1

10 2 INTODUCTION Now, the everyday life of a mathematician working for a bank or another financial institution consists of four main tasks: 1 design models that accurately fit return distributions and volatility surfaces; 2 develop valuation formulae for derivatives; 3 calibrate the models to market data; 4 derive hedging strategies. In this thesis, we aim at exploiting several aspects of the application of semimartingales and Lévy processes in mathematical finance, especially related to the valuation of exotic derivatives. Our starting point is that the model has been chosen; we assume that it is driven by a Lévy process or a time-inhomogeneous Lévy process or even a general semimartingale. Now, the complexity of some financial products, especially of exotic derivatives, means that the joint density of several random variables must be known in order to price the product. The aim is to simplify this problem; for this we exploit the so-called duality principle. In the simplest case, the duality principle relates a European plain vanilla call option to a European plain vanilla put option. One could think of it in the following simple setting: consider an investor trading options on the Euro/Dollar rate; then, she can intuitively understand that a Euro denominated call option must equal a Dollar denominated put option on the reciprocal exchange rate. Nevertheless, if the investor assumes some dynamics for the exchange rate, it is not immediately clear what the dynamics of the reciprocal rate are. The answer to this question is the central point of the duality principle; the appropriate tool to express this answer turned out to be the triplet of predictable characteristics of a semimartingale cf. Jacod Then, we exploit several other aspects of this idea, with a view towards simplifying valuation problems. A second point of interest is to solve the simplified problem; we provide formulae that allow to price a wide range of products and which can be evaluated fast. This also has important consequences to the speed of the calibration algorithms. Overview synopsis The thesis is divided into two parts. In the first part, we consider models driven by general semimartingales and by Lévy processes, which correspond to models for the dynamics of stocks or short-dated FX products. In the second part, we consider term structure models driven by time-inhomogeneous Lévy processes; such models are applied for interest rate and long-dated FX derivatives. Chapter 1 is based on Eberlein, Papapantoleon, and Shiryaev 26 and Eberlein and Papapantoleon 25a. We develop the appropriate mathematical tools to study the duality principle in a general semimartingale framework, and the central result provides the explicit form of the triplet of predictable characteristics of the dual process under the dual martingale measure. Several examples are provided, which contain discrete time models, Brownian motion and models driven by Lévy processes. Subsequently, we apply these results to option pricing problems. More specifically, we prove

11 EMAKS 3 a call-put duality for European and American options in the general semimartingale framework. We also prove duality results between floating and fixed strike Asian and lookback options, and between forward-start and plain vanilla options. Chapter 2 stems from Eberlein and Papapantoleon 25b; we provide a new proof of a key result, and detailed proofs of some other results; moreover, the published paper contained a review part, which has been omitted from the thesis. Here, we continue our study of the duality principle in two directions: firstly and more importantly, by considering options on several assets; secondly, by considering options with α-homogeneous payoff functions. The driving process is a time-inhomogeneous Lévy process, although most of the results can be proved for general semimartingales as driving processes. We provide a detailed account of multidimensional time-inhomogeneous Lévy processes, provide an alternative view of the duality principle and then prove duality relationships for options with α-homogeneous payoff functions. The final section is central in this chapter; the key result provides the triplet of predictable characteristics under a change of probability measure and projection of a multidimensional time-inhomogeneous Lévy process. This result is then applied to derive duality relationships between options on several assets and plain vanilla call and put options. Chapter 3 deals with valuation problems for vanilla and exotic derivatives on assets driven by general semimartingales and by Lévy processes. We first provide valuation formulae for single and multi-asset options, where the payoff functions can be arbitrary functions and the asset price process is driven by a general semimartingale. Then, we focus on exotic options on assets driven by Lévy processes. Using the Wiener Hopf factorization we derive the characteristic function of the supremum of a Lévy process. As an application of the developed methods, we consider the pricing of lookback and one-touch options on Lévy-driven assets. Chapter 4 is based on Eberlein, Kluge, and Papapantoleon 26 and Kluge and Papapantoleon 26. We present a detailed overview of the three predominant methods for modeling the term structure of interest rates: a forward rate HJM model, a LIBO model and a forward price model, all driven by time-inhomogeneous Lévy processes. Then, we derive duality relationships between caplets and floorlets in each of these models; these results are similar is spirit to the call-put duality of Chapter 1. Finally, we apply the valuation formulae developed in Chapter 3 to price an exotic interest rate derivative, namely an option on a composition of LIBO rates, in the forward rate and forward price frameworks. emarks Each chapter of the thesis is self-contained and has its own introduction. This naturally entails several repetitions, especially regarding notation and conventions. However, in the introductory part of Chapter 1, we state several facts from stochastic analysis that are used throughout the thesis. In general, we follow the notation of Jacod and Shiryaev 23 for stochastic analysis and semimartingale theory; for fluctuation theory of Lévy processes, we follow Kyprianou 26.

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13 Part 1 Exponential semimartingale and Lévy models

14 The real research trip does not consist of seeking new land, but of observing with new eyes. Marcel Proust, In Search of Lost Time.

15 CHAPTE 1 On the duality principle in option pricing: semimartingale setting 1.1. Introduction The purpose of this section is to develop the appropriate mathematical tools for the study of the call-put duality in option pricing. The dynamics of asset prices are modeled as general exponential semimartingales, hence we work in the widest possible framework, as far as arbitrage theory is concerned. The duality principle states that the calculation of the price of a call option for a model with price process S = e H, with respect to the measure P, is equivalent to the calculation of the price of a put option for a suitable dual model S = e H with respect to a dual measure P. From the analysis it becomes clear that appealing to general exponential semimartingale models leads to a deeper insight into the essence of the duality principle. The most standard application of the duality principle relates the value of a European call option to the value of a European put option. Carr 1994 derived a put-call duality for the Black and Scholes 1973 model and more general diffusion models. Chesney and Gibson 1995 considered a two-factor diffusion model and Bates 1997 considered diffusion and jump-diffusion models. Schroder 1999 worked in a general semimartingale framework, but calculated the dynamics under the dual measure only in specific examples diffusion and jump-diffusion models. Fajardo and Mordecki 26b considered Lévy processes. These results where used to derive static hedging strategies for some exotic derivatives, using standard European options as hedging instruments; see e.g. Carr, Ellis, and Gupta They were also used by Bates 1997, and more recently by Fajardo and Mordecki 26a, to calculate the socalled skewness premium from observed market prices. Naturally, once the duality for European options was derived, researchers looked into analogous results for American options. The duality between American call and put options is even more interesting than its European counterpart, since for American options the put-call parity holds only as an inequality. Carr and Chesney 1996 proved the put-call duality for American options for general diffusion models, Detemple 21 studied dualities for American options with general payoffs in diffusion models, while Fajardo and Mordecki 26b proved analogous results in Lévy models. The duality principle demonstrates its full strength when considering exotic derivatives. In certain cases it allows to reduce a problem involving two random variables for example, the asset price and its supremum to a problem involving just one random variable in this example, the 7

16 8 1. ON THE DUALITY PINCIPLE: SEMIMATINGALE SETTING supremum under a dual measure. This is the case for Asian, lookback and forward-start options in Lévy models. Sometimes, one can solve the problem involving a single random variable, see e.g. Borovkov and Novikov 22 and Benhamou 22, while the problem involving both random variables remains very hard to tackle. Henderson and Wojakowski 22 showed an equivalence between floating and fixed strike Asian options in the Black Scholes model. Vanmaele et al. 26 extended those results to forward-start Asian options in the Black Scholes model. Večeř 22 and Večeř and Xu 24 used this change of measure to derive a one-dimensional partial integro-differential equation for floating and fixed strike Asian options in the Black Scholes and a general semimartingale model respectively. Andreasen 1998 also used this change of measure to derive a one-dimensional partial integro-differential equation for floating and fixed strike lookback options in the Black Scholes and in a jump-diffusion model. The connection between the choice of an appropriate numeraire and a subsequent change of measure has been beautifully described in Geman, El Karoui, and ochet Nevertheless, the change of measure method has also been used in earlier work, see e.g. Shepp and Shiryaev 1994 and Shiryaev et al This chapter is organized as follows: in section 1.2 we collect some facts from stochastic analysis, describe the general semimartingale process and introduce the characteristics of a semimartingale. In section 1.3 we present the exponential semimartingale model for the dynamics of a financial asset and in section 1.4 we discuss the structure of the dual martingale measure. The main result describes the dynamics of the price process under the dual martingale measure; several examples are also discussed. Finally, in section 1.5 the call-put duality is proved for European, American, lookback, Asian and forward-start options Semimartingales and semimartingale characteristics In this section we gather some results from stochastic analysis and semimartingale theory that will be used throughout the thesis. The presentation follows Jacod and Shiryaev 23 closely; any unexplained notation is also used as in this monograph. Other standard references on these topics are Jacod 1979, 198 and Shiryaev Peskir and Shiryaev 26, Chapter II present a comprehensive overview on stochastic analysis; Kallsen 26 provides a motivated introduction to the notion of semimartingale characteristics. 1. We assume that B = Ω, F, F, P is a stochastic basis, that is a probability space Ω, F, P equipped with a filtration F = F t t T ; T is a finite time horizon. A filtration is an increasing and right-continuous family of sub-σ-algebras of F = F T, i.e. F s F t for all s t T and F t = s>t F s for all t < T. A filtration is interpreted as the flow of information. Furthermore, we assume that the stochastic basis Ω, F, F, P satisfies the usual conditions, i.e. the σ-algebra F is P -complete and each F t contains all P -null sets of F.

17 1.2. SEMIMATINGALES AND SEMIMATINGALE CHAACTEISTICS 9 All stochastic processes H = H t t T considered throughout this work have càdlàg trajectories, i.e. they are right continuous for t < T with left hand limits for < t T. As usual, we assume that the process H is adapted to the filtration F = F t t T. Consider the space Ω [, T ] = {ω, t : ω Ω, t [, T ]} and a process Y with left continuous càg trajectories. The predictable σ-algebra P is the σ-algebra on Ω [, T ] generated by all càg adapted processes Y, considered as mappings ω, t Y t ω on Ω [, T ]. An adapted process H = H t ω t T, ω Ω, that is P-measurable is called a predictable process. The optional σ-algebra O is the σ-algebra generated by all càdlàg adapted processes Y, considered as mappings ω, t Y t ω. A process H that is O-measurable is called an optional process. Consider the space Ω [, T ] = {ω; t, x : ω Ω, t [, T ], x }. Then, P = P B denotes the σ-algebra of predictable sets in Ω = Ω [, T ] and Õ = O B denotes the σ-algebra of optional sets in Ω. A function W : Ω [, T ] is called predictable, resp. optional, if it is P-measurable, resp. Õ-measurable. 2. A process H = H t t T defined on the stochastic basis Ω, F, F, P is a semimartingale if it admits a representation H = H + M + A 1.1 where X is a finite-valued, F -measurable random variable, M is a local martingale with M = M M loc and A is a bounded variation process with A = A V. The representation 1.1 is, in general, not unique. A semimartingale H is a special semimartingale if the process A in the representation 1.1 is, in addition, predictable A P V. As a consequence of the Doob Meyer decomposition, we conclude that the canonical representation 1.1 for a special semimartingale is unique. The class of semimartingales remains invariant under several transformations such as stopping, localization, change of time, change of filtration, absolutely continuous change of measure, etc. More importantly, it is the widest class of stochastic processes for which a stochastic integral can be defined for reasonable integrands i.e. bounded predictable processes. 3. Let LH denote the set of predictable processes that are integrable with respect to the semimartingale H. Let K LH, then K H denotes the stochastic integral K sdh s. If H = A+M, then K H = K A+K M, where K A is the stochastic integral with respect to the bounded variation part of H and K M the stochastic integral with respect to the local martingale part of H. A random measure µ on [, T ] is a family µω ω Ω of measures on [, T ], B[, T ] B with µω; {} = for all ω Ω. Let µ = µω; dt, dx, be an integer-valued random measure on [, T ]. Let W = W ω; t, x be an optional function on Ω [, T ] ; then W µ denotes the integral process W ω; t, xµω; dt, dx,

18 1 1. ON THE DUALITY PINCIPLE: SEMIMATINGALE SETTING often written as W dµ. Let ν denote the predictable compensator of the random measure µ. For a predictable function W : Ω [, T ] in G loc µ, W µ ν denotes the stochastic integral W ω; t, xµ νω; dt, dx, again abbreviated as W dµ ν. 4. Every semimartingale H = H t t T admits a canonical representation H = H + B + H c + hx µ ν + x hx µ 1.2 or, in detail t H t = H + B t + Ht c + t hxdµ ν + x hxdµ, 1.2 where a h = hx is a truncation function, i.e. a bounded function with compact support that behaves as hx = x in a neighborhood of zero; a canonical choice of h is hx = x1 { x a} where 1 A x, or 1A, denotes the indicator of the set A; b B = B t t T is a predictable process of bounded variation; c H c = Ht c t T is the continuous martingale part of H; d ν = νω; dt, dx is the predictable compensator of the random measure of jumps µ = µω; dt, dx of H; for clarity we write also ν H and µ H instead of ν and µ. The continuous martingale part H c of any semimartingale H is uniquely defined up to indistinguishability. The predictable quadratic variation H c of the continuous martingale H c will be denoted by C = C t t T. An application of the Doob Meyer decomposition yields that H c 2 H c M loc actually M c loc, the space of continuous local martingales. The random measure of jumps µ = µω; dt, dx of the semimartingale H is an integer-valued random measure of the form µω; dt, dx = 1 { Hsω }ε s, Hsωdt, dx, s where H s = H s H s and ε a denotes the Dirac measure at point a. The compensator of the random measure µ can be characterized as the unique up to indistinguishability predictable random measure ν such that for every non-negative P-measurable function W = W ω; t, x on Ω [ T E ] [ T W ω; t, xµω; dt, dx = E ] W ω; t, xνω; dt, dx. 1.3 Equivalently, we have that the process W ω; t, xµω; dt, dx W ω; t, xνω; dt, dx M loc. 1.4

19 1.2. SEMIMATINGALES AND SEMIMATINGALE CHAACTEISTICS 11 In addition, we have that the process x 2 1νω; dt, dx A + loc. 1.5 The processes B, C, and the measure ν are called the triplet of predictable characteristics of the semimartingale H with respect to the probability measure P, and will be denoted by TH P = B, C, ν. The characteristics are uniquely defined, up to indistinguishability of course. 5. It is important to underline that the canonical representation 1.2 of a semimartingale H depends on the selected truncation function h = hx. However, the characteristics C and ν do not depend on the choice of h while B = Bh does. If h and h are two truncation functions, then Bh Bh = h h ν. In the sequel, we assume that the truncation function h = hx satisfies the following antisymmetry property: h x = hx. We will see that this property simplifies many formulae. Note that, for example, the canonical choice hx = x1 { x a} satisfies this property. 6. An equivalent way to define the characteristics of a semimartingale H, which reveals some additional properties, is the following cf. Theorem II.2.42 and Corollary II.2.48 in Jacod and Shiryaev 23. Let B be a realvalued, predictable process in V, C a non-negative-valued predictable process in V and ν the predictable compensator of the random measure of jumps of H. Then, B, C, ν is called the triplet of predictable characteristics of H if and only if e iuh e iuh Kiu 1.6 is a complex-valued local martingale for all u, where K is the cumulant process of H Ku = ub + u2 2 C + eux 1 uhx ν. 1.7 Equivalently, we have that TH P = B, C, ν if and only if for all u e iuh Giu M locp, 1.8 where Gu = EKu, assuming it never vanishes. Here, E denotes the stochastic exponential, cf and In addition, there exist an increasing predictable process A, predictable processes b and c and a transition kernel F from Ω [, T ], P into, B such that t t t B t = b s da s, C t = c s da s, ν[, t] E = F s dxda s, 1.9 E

20 12 1. ON THE DUALITY PINCIPLE: SEMIMATINGALE SETTING where E B. Moreover, we have that Ku = κu A, where κu = ub + u2 2 c + e ux 1 uhxf dx. 1.1 In several applications, the characteristics B, C, ν are absolutely continuous, in which case we can choose the process A t = t. Then, we call the triplet b, c, F the differential characteristics of H Exponential semimartingale models 1. Let B = Ω, F, F, P be a stochastic basis and S = S t t T be an exponential semimartingale i.e. a stochastic process with representation S t = e Ht, t T 1.11 shortly: S = e H, where H = H t t T is a semimartingale, H =. The process S is interpreted as the price process of a financial asset, e.g. a stock or an FX rate. Together with the compound interest representation 1.11 for positive prices S, which is appropriate for the statistical analysis of S, the following simple interest representation S t = E H t, t T 1.12 shortly: S = E H with some suitable semimartingale H = H t t T, is convenient for the study of the process S by martingale methods; see details in Shiryaev In 1.12 we used the standard notation EX = EX t t T for the stochastic exponential of a semimartingale, defined as the unique strong solution of the stochastic differential equation dex t = EX t dx t, X =, 1.13 that has the following explicit solution EX t = e Xt 1 2 Xc t 1 + X s e Xs, 1.14 <s t where X c is the predictable quadratic characteristic of the continuous martingale part X c of X and X s = X s X s. From 1.11 and 1.12 it follows that the process H should satisfy the equation e Ht = E H t, t T 1.15 shortly: e H = E H which implies H > 1. In other words, and vice versa H t = log E H t, t T 1.16 H t = Loge Ht, t T 1.17 shortly: H = Loge H where Log X denotes the stochastic logarithm of a positive process X = X t t T : Log X t = t dx s X s. 1.18

21 1.3. EXPONENTIAL SEMIMATINGALE MODELS 13 Note that for a positive process X with X = 1 we have for Log X Log X = log X + 1 2X 2 X c log 1 + X s X s ; 1.19 X s X s <s for details see Kallsen and Shiryaev 22a or Jacod and Shiryaev 23, Chapter II. From one may get the following useful formulae: H = H Hc + e Hs 1 H s 1.2 and <s H = H 1 2 H c + log1 + H s H s <s If µ H = µ H ω; ds, dx and µ eh = µ eh ω; ds, dx are the random measures of jumps of H and H, then the formulae 1.2 and 1.21 may be written in the form H = H Hc + e x 1 x µ H 1.2 and H = H 1 2 H c + log1 + x x µ eh It is useful to note that discrete time sequences H = H n n with H = and F n -measurable random variables H n can be considered as a semimartingale H = H t t in continuous time, where H t = H n for t [n, n+1, given on the stochastic basis B = Ω, F, F t t, P with F t = F n for t [n, n + 1. In the discrete time setting S = S n n has a compound interest representation S n = e Hn, H n = h h n, n 1, 1.22 S = 1, where h n are random variables with h = ; the analogue of the simple interest representation has the form S n = E H n = 1 + h k 1.23 k n with h k = e h k 1, Hk = h h k, k 1, H =. We see that S n = S n 1 H n where S n = S n S n 1, H n = H n H n 1 = h n compare with From formulae 1.2 and 1.21 it is not difficult to find the relationships between the triplets TH P = B, C, ν and T H P = B, C, ν with respect to the same truncation function h: B = B + C 2 + hex 1 hx ν C = C A x ν = 1 A e x 1 ν, A B\{}

22 14 1. ON THE DUALITY PINCIPLE: SEMIMATINGALE SETTING and B = B C 2 + hlog1 + x hx ν C = C A x ν = 1 A log1 + x ν, A B\{}; for more details we refer to Kallsen and Shiryaev 22a and Jacod and Shiryaev 23. In particular, if H is a Lévy process with triplet of local characteristics b, c, F, then H will also be a Lévy process with the triplet b, c, F for which c b = b he x 1 hxf dx Correspondingly, c = c 1.24 F A = 1 A e x 1F dx, A B\{}. b = b c 2 + hlog1 + x hx F dx c = c 1.25 F A = 1 A log1 + x F dx, A B\{} Martingale measures and dual martingale measures 1. Let M loc P be the class of all local martingales on a given stochastic basis B = Ω, F, F, P. It is known and easily follows from the canonical representation 1.2 that if TH P = B, C, ν then H M loc P B + x hx ν = Similarly, for the process H = Loge H we have H M loc P B + x hx ν =. In the sequel, we will assume that the following condition is in force. Assumption ES. The process 1 {x>1} e x ν has bounded variation Under Assumption ES the property 1.26 can be rewritten, taking into account 1.24, in the following form: H M loc P B + C 2 + ex 1 hx ν = emark 1.1. The assumption that the process 1 {x>1} e x ν has bounded variation is equivalent, by Kallsen and Shiryaev 22a, Lemma 2.13, to the assumption that the semimartingale H is exponentially special, i.e. the

23 1.4. MATINGALE MEASUES AND DUAL MATINGALE MEASUES 15 price process S = e H is a special semimartingale. This justifies to call it Assumption ES. Since H M loc P if and only if E H M loc P, we get from 1.27 E H M loc P B + C 2 + ex 1 hx ν =, 1.28 and, therefore, using 1.15 S = e H M loc P B + C 2 + ex 1 hx ν = In the sequel we shall assume that S is not only a local martingale but also a martingale S MP on [, T ]. Thus ES T = 1, which allows us to define on Ω, F, F t t T a new probability measure P with Since S is a martingale dp dp = S T. 1.3 dp F t dp F t = S t, t T 1.31 and since S > P -a.s., we have P P and Let us introduce the process dp dp = S T S = 1 S Then, denoting by H the dual of the semimartingale H, i.e. H = H, we have S = e H The following simple but, as we shall see, useful lemma plays a crucial role in the problem of duality between call and put options. It also explains the name of dual martingale measure for the measure P. Lemma 1.2. Suppose S = e H MP i.e. S is a P -martingale. Then the process S MP i.e. S is a P -martingale. Proof. The proof follows directly from Proposition III.3.8 in Jacod and Shiryaev 23, which states that if Z = dp dp then S MP iff S Z is a P -martingale. In our case Z = S and S S 1. Thus S MP. 3. The next theorem is crucial for all calculations of option prices on the basis of the duality principle see Section 1.5. We first prove an auxiliary proposition of independent interest, about the characteristics of the stochastic integral process fdh.

24 16 1. ON THE DUALITY PINCIPLE: SEMIMATINGALE SETTING Proposition 1.3. Let f be a predictable, bounded process. The triplet of predictable characteristics of the stochastic integral process J = fdh, denoted by TJ P = B J, C J, ν J, is B J = f B + [hfx fhx] ν 1.35 C J = f 2 C A x ν J = 1 A fx ν, A B\{} Proof. The last two statements follow directly from the properties of the stochastic integral J = f H: and J c = f H c 1.38 J = f H Indeed, 1.36 follows directly from 1.38 and Jacod and Shiryaev 23, I.4.41: C J = J c = f 2 H c = f 2 C. From 1.39 we deduce 1 A x µ J = 1 A fx µ H, A B\{} 1.4 which gives for ν J, the compensator of the random measure of jumps µ J of J, the relation For the proof of relation 1.35 we recall the canonical representation of the semimartingale H: H = H + B + M + x hx µ H 1.41 where M is a local martingale in fact M = H c + hx µ H ν and the canonical representation of the semimartingale J: J = J + B J + J c + hy µ J ν J + y hy µ J From the definition J = f H and the representation 1.41 we get J = f B + f M + fx fhx µ H 1.43 which gives, together with 1.4, the following formula: J y hy µ J = = f B + f M + fx fhx µ H fx hfx µ H = f B + f M + hfx fhx µ H The process J y hy µ J has bounded jumps. Hence this process is a special semimartingale Jacod and Shiryaev 23, Lemma 4.24, p. 44 and by Proposition 4.23iii, again from Jacod and Shiryaev 23, p. 44, we conclude that the process f B + hfx fhx µ H A loc, i.e. it is a process with locally integrable variation. Note now that the process f B belongs also to the class A loc since it is a predictable process of locally bounded variation Jacod and Shiryaev 23, Lemma 3.1, p. 29. Hence the process hfx fhx µ H A loc and using Jacod and Shiryaev 23, Theorem 3.18, p. 33 there exists a compensator of this process given by

25 1.4. MATINGALE MEASUES AND DUAL MATINGALE MEASUES 17 the formula hfx fhx ν Jacod and Shiryaev 23, Theorem 1.8, pp As a result we get from 1.44 that J y hy µ J = [f B + hfx fhx ν] + [f M + hfx fhx µ H ν] = f B + hfx fhx ν + f H c + hfx µ H ν Comparing the decomposition 1.45 of the special semimartingale J y hy µ J with the representation of J y hy µ J from the canonical representation 1.42 we conclude, by the uniqueness of the representation of a special semimartingale Jacod and Shiryaev 23, I.4.22, that the processes B J and f B + hfx fhx ν are indistinguishable; cf. Jacod and Shiryaev 23, p. 3. Therefore, formula 1.35 is proved. emark 1.4. Variants of Proposition 1.3 are stated in Jacod and Shiryaev 23, IX.5.3 and Kallsen and Shiryaev 22b, Lemma 3. Theorem 1.5. The triplet TH P = B, C, ν can be expressed via the triplet TH P = B, C, ν by the following formulae: B = B C hxe x 1 ν C = C A x ν = 1 A xe x ν, A B\{}. Proof. We give two proofs which are of interest here, since these proofs contain some additional useful relationships between different triplets. The structure of these proofs can be represented by the following diagram: TH P TH P G a c b d G TH P TH P 1.47 where G means that we use Girsanov s theorem for calculating the right side triplet from the left side one and the dual of the semimartingale on the left side. means that we consider G a TH P TH P. For the calculation of the triplet TH P = B +, C +, ν + from the triplet TH P = B, C, ν, we use Girsanov s theorem for semimartingales Jacod and Shiryaev 23, pp which states that B + = B + β + C + hxy + 1 ν 1.48 C + = C 1.49 ν + = Y + ν. 1.5

26 18 1. ON THE DUALITY PINCIPLE: SEMIMATINGALE SETTING Here β + = β + t ω and Y + = Y + ω; t, x are defined by the following formulae see Jacod and Shiryaev 23, p. 173: S c, H c = S β + C 1.51 and S Y + = M P µ P H S In equation 1.52 M P = µ H ω; dt, dxp dω is the positive measure µ H on Ω [, T ], F B[, T ] B defined by M P µ W = EW µ H H T 1.53 for measurable non-negative functions W = W ω; t, x on Ω [, T ]. The conditional expectation M P SS µ P is, by definition, the M P -a.s. H µ H unique P-measurable function Y + with the property S M P µ U = M P H S µ Y + U 1.54 H for all non-negative P-measurable functions U = Uω; t, x. We show that in our special case S = e H, where evidently one may take the following versions of β + and Y + : S S = e H, β + 1 and Y + = e x Indeed, for S = e H, we get applying Itô s formula to e H, see Appendix B, that and, therefore, and e H c = S c, H c = e H c, H c = = e H d H c = S β + C = e H s dh c s e H dh c, H c e H dc = S C 1.56 e H β + dc From this formula and the equality 1.51 we see that one may take β + 1. For the proof that one may choose Y + = e x we need to verify 1.54 with this version of Y +.

27 1.4. MATINGALE MEASUES AND DUAL MATINGALE MEASUES 19 We have, using that µ H is the random measure of jumps of H: [ T M P µ e x U = E H [ = E [ T = E t T = M P µ H S S U ] e x Uω; t, xµ H ω; dt, dx e Htω Uω; t, H t ω1 { Htω } ] S t ω S t ω Uω; t, xµh ω; dt, dx Consequently in one may put β + 1 and Y + = e x which gives the following result: B + = B + C + hxe x 1 ν C + = C 1.59 ν + = e x ν. emark 1.6. It is useful to note that for the discrete time case the relation dν + = e x dν can be proved with the obvious notation in the following simple way. Let h n = H n and µ n = µ n ω; be the random measure of jumps of H at time n, i.e. µ n ω; A = 1h n ω A for A B\{}. The compensator ν n = ν n ω; of µ n ω; has here the simple form ν n ω; A = P h n A F n 1 ω see Jacod and Shiryaev 23, p. 92 for the definition of the compensator in the discrete time case. If ν n + ω; A = P h n A F n 1 ω then, applying the already used Proposition III.3.8 in Jacod and Shiryaev 23 or, equivalently, applying Bayes formula also called the conversion formula; see Shiryaev 1999, p. 438 we find that ] ν n + ω; A = E [1 A h n F n 1 ]ω = E[1 A h n e hn F n 1 ]ω = A e x ν n ω; dx. Therefore, ν + n ν n and dν+ n dν n ω; x = e x ν n -a.e.. b TH P TH P. Because H = H the triplet TH P = T H P. Now, applying Proposition 1.3 to the function f 1 i.e. J = H and assuming that

28 2 1. ON THE DUALITY PINCIPLE: SEMIMATINGALE SETTING hx = h x, we get B = B C = C A x ν = 1 A x ν, A B\{}. c TH P TH P. The triplet TH P = B +, C +, ν + is given by the formulae Then from 1.6 with the necessary adaptation of the notation we get B = B + = B C hxe x 1 ν C = C + = C A x ν = 1 A x ν + = 1 A xe x ν, so, the proof using steps a and c leads to the formulae d TH G P TH P. Here TH P = T H P = B, C, ν and TH P = T H P = B, C, ν. Similarly to the case a we have the following formulae compare with 1.48: B = B + β C + hxy 1 ν 1.62 C = C 1.63 ν = Y ν 1.64 where β = β t ω and Y = Y ω; t, x are given by the formulae compare with 1.51 and 1.52 S c, H c = S β C 1.65 and S Y = M P µ P H S Since S c, H c = e H c, H c = = e H dh c, H c = e H d H c = e H d H c e H dc = S C, 1.67 comparing 1.65 and 1.67 we see that one may take β 1.

29 1.4. MATINGALE MEASUES AND DUAL MATINGALE MEASUES 21 Similarly to the calculations in a we deduce [ T M P µ e x U = E H [ = E t T [ = E [ T = E t T = M P µ H S S U ] e x Uω; t, xµ H ω; dt, dx e Htω Uω; t, H t ω1 { Htω } e Htω Uω; t, H t ω1 { Htω } ] S t ω S t ω Uω; t, xµ H ω; dt, dx Therefore one may take Y = e x in 1.66 and from and 1.6 we find that B = B C + hx e x 1 ν 1.69 C = C 1.7 ν = e x ν 1.71 where ν is such that 1 A x ν = 1 A x ν, A B\{}. Hence, as one easily sees 1 A x ν = 1 A x e x ν = 1 A xe x ν In addition, if h x = hx hx e x 1 ν = h xe x 1 ν = hxe x 1 ν From we conclude that the triplet TH P = B, C, ν, obtained using steps b and d is given by formulae Hence, Theorem 1.5 is proved. emark 1.7. Note that under Assumption ES we can conclude from formulae 1.46 that x 2 1 ν A loc, because x 2 1 ν K x 2 1 { x 1} ν + 1 {x< 1} ν + e x 1 {x>1} ν. Here K is a constant and the processes on the right-hand side are predictable processes of bounded variation, hence belong to A loc cf. Jacod and Shiryaev 23, Lemma I.3.1. Similarly, we get that ν satisfies Assumption ES, because and 1 {x< 1} ν A loc. 1 {x>1} e x ν = 1 {x< 1} e x e x ν = 1 {x< 1} ν ] ]

30 22 1. ON THE DUALITY PINCIPLE: SEMIMATINGALE SETTING Corollary 1.8. Suppose that H is a P -Lévy process with local characteristics b, c, F. Then the process H is a P -Lévy process with local characteristics b, c, F given by the formulae we take h x = hx: b = b c hxe x 1F dx c = c F A = 1 A xe x F dx, A B\{}. Proof. The proof follows from Theorem 1.5 and Jacod and Shiryaev 23, Corollary 4.19, p Lemma 1.2 states that S is a P -martingale. Now, there exists an alternative path to verify this result. emark 1.9. The formulae 1.46 provide a simple way to confirm that the process S = e H M loc P. Indeed, by 1.29 it is sufficient to check that B + C 2 + ex 1 hx ν = From 1.46 with h x = hx we get B + C 2 + ex 1 hx ν = B C hxe x 1 ν + C 2 + e x 1 h x e x ν = B + C 2 + ex 1 hx ν = where the last equality follows from the assumption S = e H M loc P and criterion Now we consider some examples that show how to calculate the triplet TH P from the triplet TH P and for which particular models in finance Assumption ES is satisfied. Example 1.1 Brownian case. Suppose ν. From 1.29 S = e H M loc P iff B + C 2 =. If S MP then by Theorem 1.5 the triplet TH P = B, C, with B = B + C and C = C. So, B + C 2 = B + C 2 = which implies S M loc P. In particular, if σ2 σwt S t = e 2 t, i.e. H t = σw t σ2 2 t where W = W t t T is a standard Brownian motion Wiener process then B t = σ2 2 t, C t = σ 2 t. Evidently B + C 2 = which implies by 1.29 that S M loc P. In fact, S MP. Note also that ds t = σs t dw t. The process S = e H = e H has stochastic differential ds t = σs tdw t σdt. 1.75

31 1.4. MATINGALE MEASUES AND DUAL MATINGALE MEASUES 23 Since S M loc P from formula 1.75 one can deduce that the process W t = W t σt, t T is a P -local martingale. This is a particular case of the classical Girsanov theorem which can be easily checked directly using the fact already mentioned before, in Lemma 1.2 that W M loc P iff W S M loc P. The last property follows from calculating dw S by Itô s formula. The fact that W is a P -Brownian motion follows also from Lévy s characterization of a Brownian motion evuz and Yor 1999, Theorem IV.3.6. So, ds t = σs tdw t. Example 1.11 Poissonian case. Consider S = e H with H t = απ t λe α 1t, α 1.76 where π = π t t T is a Poisson process with parameter λ > Eπ t = λt. Take hx. Then the corresponding triplet B, C, ν has the following form: B t = λe α 1t C t = 1.77 νdt, dx = λ1 {α} dxdt. By 1.29 S M loc P B + e x 1 ν =. With the process given in 1.76 B t + e x 1 ν t = λe α 1t + λe α 1t =. Therefore, S M loc P and even S MP ; moreover, P is the unique martingale measure for the Poisson model cf. e.g. Corcuera et al. 25, pp In addition, with respect to the measure P the process S is a local martingale; this follows directly from criterion 1.29 B + e x 1 ν = By Theorem 1.5 B t = λe α 1t and e x 1 ν t = e x 1e x ν t = λ1 e α t. Hence, the property 1.78 does hold and S M loc P. Example 1.12 Discrete time, C-model. In the binomial model of Cox, oss and ubinstein C-model, asset prices are modeled by S n = e Hn, with H n = h h n, n 1, H =, where h n n 1 is a P -i.i.d. sequence of random variables which have only two values. If h n = e hn 1 then S n = k n 1 + h k and S n = 1 + h n S n 1, n 1, with S =. For simplicity let us assume that the random variables h n take the values ln λ and ln 1 λ with λ > 1. So hn = { a = λ 1 1, b = λ 1. If the probability measure P is such that P h n = ln 1 λ and = P h n = a = b b a = P h n = ln λ = P h n = b = λ 1 + λ a b a = λ

32 24 1. ON THE DUALITY PINCIPLE: SEMIMATINGALE SETTING then we find that Ee hn = E1 + h n = 1. This means that the measure P is a martingale measure for the sequence S = S n n. Indeed it is the unique martingale measure for the C-model; see Shiryaev 1999, Example V.3.2, pp With the truncation function hx = x and the martingale measure P we easily find that the triplet TH P = B,, ν where with B n = B n B n 1 B n = Eh n = 1 λ ln λ λ and with ν n A = ν{n} A ν n {ln λ} = P h n = ln λ = λ { ν n ln 1 } = P h n = ln 1 = λ 1.82 λ λ 1 + λ. Note that from 1.81 and 1.82 we get B n +e x 1 x ν n =, which is another derivation of the martingale property for S under the measure P given by 1.79 and 1.8. Based on formulae 1.46 we find directly that B n = B n, ν n = ν n 1.83 and from the previous note and 1.29 it follows that S M loc P in fact S MP. Example 1.13 Purely discontinuous Lévy models. In this class of models, asset prices are modeled as S = e H, where H = H t t T is a purely discontinuous Lévy process with triplet TH P = B,, ν. We can also work with the triplet of differential characteristics, denoted by b,, F, which using Jacod and Shiryaev 23, II.4.2, is related for our case to the triplet of semimartingale characteristics via B t ω = bt, νω; dt, dx = dtf dx. Since S = e H M loc P, the characteristic b resumes the form b = e x 1 hxf dx. and criterion 1.29 is satisfied. Now, we have in addition that S MP, cf. Lemma 4.4 in Kallsen 2. Then, we can apply Theorem 1.5 and the triplet TH P = B,, ν is given by 1 A x ν = 1 A xe x ν 1.84 and B = B hxe x 1 ν = e x 1 hx ν. Therefore, using 1.29 again, or alternatively emark 1.9, we have that S = e H M loc P. When considering parametric models it is very convenient to represent the Lévy measure F = F dx in the form F dx = e ϑx fxdx 1.85

33 1.4. MATINGALE MEASUES AND DUAL MATINGALE MEASUES 25 where ϑ and f is an even function, i.e. fx = f x. In that case, the triplet of local characteristics of the dual process H is b,, F where 1 A xf dx = 1 A xe 1+ϑx fxdx and, of course, b = ex 1 hxf dx. Examples of parametric models are: Example Generalized hyperbolic model. Let H = H t t T be a generalized hyperbolic process with LawH 1 P = GHλ, α, β, δ, µ, cf. Eberlein 21, p. 321 or Eberlein and Prause 22. Then the Lévy measure of H admits the representation 1.85 with parameters ϑ = β, β < α and fx = 1 exp 2y + α 2 x x π 2 yj λ 2 δ 2y + Y λ 2 δ 2y dy + λe α x 1 {λ>}, where α >, δ >, λ, µ, cf. Eberlein 21, p Here J λ and Y λ are the modified Bessel functions of first and second kind respectively. The moment generating function exists for u α β, α β, hence, Assumption ES is satisfied. The class of generalized hyperbolic distributions contains several other distributions as subclasses, for example hyperbolic distributions Eberlein and Keller 1995, normal inverse Gaussian distributions Barndorff-Nielsen 1998 or limiting classes e.g. variance gamma. We refer to Eberlein and v. Hammerstein 24 for an extensive survey. Example CGMY model. Let H = H t t T be a CGMY Lévy process, cf. Carr, Geman, Madan, and Yor 22; another name for this process is generalized tempered stable process. The Lévy measure of this process admits the representation 1.85 with the following parameters { G, x < ϑ = and fx = C M, x > x 1+Y, where C >, G >, M >, and Y < 2. The CGMY processes are closely related to stable processes; in fact, the function f coincides with the Lévy measure of the stable process with index α, 2, cf. Samorodnitsky and Taqqu 1994, Def Due to the exponential tempering of the Lévy measure, the CGMY distribution has finite moments of all orders. Moreover, the moment generating function exists, hence Assumption ES is satisfied. Again, the class of CGMY distributions contains several other distributions as subclasses, for example the variance gamma distribution Madan and Seneta 199 and the bilateral gamma distribution Küchler and Tappe 26. Example Meixner model. Let H = H t t T be a Meixner process with LawH 1 P = Meixnerα, β, δ, α >, π < β < π, δ >, cf. Schoutens and Teugels 1998 and Schoutens 22. The Lévy measure of the Meixner process admits the representation 1.85 with ϑ = β α and fx = δ x sinh πx α.