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.

34 26 1. ON THE DUALITY PINCIPLE: SEMIMATINGALE SETTING The Meixner distribution possesses finite moments of all orders. Moreover, the moment generating function exists, hence, Assumption ES is again satisfied. emark Note that H, in all the parametric examples considered, remains a P -Lévy process from the same class of processes, just with a new parameter ϑ in representation emark Theorem 1.5 cannot be applied, for example, to stable processes, because they do not satisfy Assumption ES. In fact, stable processes may not even have finite first moment, cf. Samorodnitsky and Taqqu 1994, Property This fact makes them particularly unsuitable for option pricing, although these models are applied for risk management purposes, cf. achev 23. Example 1.16 Stochastic volatility Lévy models. This class of models was proposed by Carr, Geman, Madan, and Yor 23 and further investigated in Schoutens 23. Let X = X t t T be a pure jump Lévy process and Y = Y t t T be an increasing process, independent of X. The process Y acts as a stochastic clock measuring activity in business time and has the form Y t = t y s ds where y = y s s T is a positive process. Carr, Geman, Madan, and Yor 23 consider the CI process as a candidate for y, i.e. the solution of the stochastic differential equation dy t = Kη y t dt + λy 1 2 t dw t, where W = W t t T is a standard Brownian motion. For other choices of Y see Schoutens 23. The stochastic volatility Lévy process is defined by time-changing the Lévy process X with the increasing process Y, that is Z t = X Yt, t T. The process Z is a pure jump semimartingale with canonical decomposition Z = Z + B Z + hx µ Z ν Z + x hx µ Z, where the compensator of the random measure of jumps of Z has the form ν Z ds, dx = ysν X dxds, where ν X denotes the Lévy measure of X. Asset prices are modeled as S = e H, where H is a semimartingale such that ν H ds, dx = ν Z ds, dx = ysν X dxds and S M loc P, therefore, TH P = B,, ν H, where B = e x 1 hx ν H. If S MP, cf. Kallsen 26, Proposition 4.1 for a sufficient condition, then applying Theorem 1.5, we get that TH P = B,, ν with and B = e x 1 hx ν. 1 A x ν = 1 A xe x ν H = 1 A xe x ν X dxysds

35 1.4. MATINGALE MEASUES AND DUAL MATINGALE MEASUES 27 Example 1.17 Local volatility models. Local volatility models were introduced by Dupire 1994; we refer to Skiadopoulos 21 for a survey of this literature. The dynamics of the asset price process is given by the stochastic differential equation ds t = S t σt, S t dw t, S = 1, 1.86 where W = W t t T is a standard Brownian motion. If the local volatility function σ : [, T ] + + is Lipschitz, i.e. satisfies the conditions a σt, x σt, y K x y, t [, T ], K constant, b t σt, x is right continuous with left limits, x +, then the SDE 1.86 has a unique strong solution cf. Protter 24, Theorem V.6, for which S t = E = exp = exp σu, S u dw u t t σu, S u dw u 1 2 σu, S u dw u 1 2 t t σu, S u dw u σ 2 u, S u du. Therefore, assuming the canonical setting Jacod and Shiryaev 23, p. 154, these models fit in the general exponential semimartingale framework with driving process H = H t t T and triplet TH P = B, C, ν where B = 1 2 C = ν, σ 2 u, e Hu du σ 2 u, e Hu du and, of course, S = e H M loc P. Now, S = e H MP holds if, for example, Novikov s condition is satisfied, cf. evuz and Yor 1999, Proposition VIII.1.15; then applying Theorem 1.5, we get that TH P = B, C, ν, where B = B C = 1 2 C = ν. σ 2 u, e Hu du σ 2 u, e Hu du t

36 28 1. ON THE DUALITY PINCIPLE: SEMIMATINGALE SETTING 1.5. The call-put duality in option pricing Let S = S t t T be the price process of a financial asset and f T = f T S the payoff of an option on this asset. Here f T S = f T S t, t T is an FT S-measurable functional, where F T S = σs t, t T. In order to simplify the notation we assume that the current interest rate is zero; for detailed formulae in the case of a positive interest rate and dividend yield we refer to sections 2.4 and 2.5 or to Eberlein and Papapantoleon 25b. As is well known, in a complete market, where the martingale measure P is unique, the rational or arbitrage-free price of the option is given by Ef T = E P f T. In incomplete markets one has to choose an equivalent martingale measure. In this work we do not discuss the problem of the choice of a reasonable martingale measure, for example, in the sense of minimization of a distance L 2 -distance, Hellinger distance, entropic distance, etc. from the physical measure or in the sense of constructing the simplest possible measure e.g. Esscher transformation. The practitioners point of view is that the choice of this measure is the result of a calibration to market prices of plain vanilla options. We will assume that the initial measure P is a martingale measure and all our calculations of E P f T will be done with respect to this measure P. In the case of an incomplete market this option price E P f T could be called a quasi rational option price. A. European call and put options. In case of a standard call option the payoff function is whereas for a put option it is f T = S T K +, K >, 1.87 f T = K S T +, K > The corresponding option prices are given by the formulae C T S; K = E[S T K + ] 1.89 and P T K; S = E[K S T + ] 1.9 where E is the expectation operator with respect to the martingale measure P. From 1.89 for S = e H we get [ f ] T C T S; K = E S T = E [ f ] T = E [1 KS T + ] S T S T = KE [ 1 + ] K S T = KE [K S T + ] 1.91 where K = 1 K. Comparing here the right hand side with 1.9 we obtain the following result. Theorem For standard call and put options the option prices satisfy the following duality relations: 1 K C T S; K = P T K ; S and 1 K P T K; S = C T S ; K

37 1.5. THE CALL-PUT DUALITY IN OPTION PICING 29 where K = 1 K. P T K ; S and C T S, K are the corresponding prices for put and call options with S as underlying price process, computed with respect to the dual measure P. Corollary Call and put prices in markets S, P and S, P which satisfy the duality relation, are connected by the following call-call parity and the following put-put parity C T S; K = KC T S ; K + 1 K P T K; S = KP T K ; S + K 1. Proof. From the identity S T K + = K S T + + S T K we get, taking expectations with respect to the measure P, the well-known call-put parity: C T S; K = P T K; S + 1 K. The result follows from the duality relations in Theorem B. American call and put options. The general theory of pricing of American options see, for example, Shiryaev 1999, Chapters VI and VIII states that, for payoff functions described by the process e λt f t, t T, λ, the price V T S of the American option is given by the formula V T S = sup τ M T E[e λτ f τ ], 1.92 where M T is the class of stopping times τ such that τ T. For a standard call option f τ = S τ K + and for a standard put option f τ = K S τ +, where K > is a constant strike. Denote and Ĉ T S; K = sup τ M T E[e λτ S τ K + ] 1.93 P T K; S = sup τ M T E[e λτ K S τ + ] Similarly to the case of European options, we get for f τ = S τ K + [ Ĉ T S; K = sup E e λτ S ] T f τ = sup E [ e λτ f ] τ τ M T S T τ M T S T = sup E [ ] e λτ f τ S T τ M T = sup E [ ] e λτ f τ S τ τ M T = sup τ M T E [ e λτ 1 KS τ +] = K sup τ M T E [ e λτ K S τ +] = K P T K ; S, = sup τ M T E [ e λτ f τ E S T F τ = sup τ M T E [ e λτ S τ K + S τ ] ]

38 3 1. ON THE DUALITY PINCIPLE: SEMIMATINGALE SETTING Option type\payoff Asian Lookback Fixed Strike call Σ T K + S T K + Fixed Strike put K Σ T + K S T + Floating Strike call S T Σ T + S T S T + Floating Strike put Σ T S T + S T S T + Table 1.1. Types of payoffs for Asian and lookback options where again K = 1 K. Thus, similarly to the statements in Theorem 1.18 we have for American options the following duality relations: and also 1 K ĈT S; K = P T K ; S 1 K P T K; S = Ĉ T S ; K. C. Lookback options. Let S = S t t T and S = S t t T denote the supremum and infimum processes of the asset price process, that is S t = sup S u and S t = inf S u. u t u t There exist two types of lookback options traded in the market: floating and fixed strike options. In the case of the floating strike option, the supremum resp. infimum plays the role of the strike. The different variants of the lookback option are summarized in Table 1.1. A variant of floating strike lookback options are partial lookback options, with payoff S T αs T + and βs T S T + for the call and put option respectively. Here, α [1, and β, 1] denote the degree of partiality. The incentive behind trading partial lookback options is that classical lookback options are too expensive relative to their European plain vanilla counterparts, since S S S a.s., which makes them unattractive for investors. Suppose S MP, then for a floating strike, or partial, lookback call option we get [ + ] C T S; α inf S = E ST α inf t T S t = E [S T 1 α inf t T S + ] t S T = E [ + ] 1 αe inf t T H t H T = E [ + ] 1 αe H T sup t T H t = αe [ 1 α eh T sup t T H t + ]. 1.95

39 1.5. THE CALL-PUT DUALITY IN OPTION PICING 31 In order to further simplify the last expression, let us assume that the process H = H t t T satisfies the following reflection principle: Law H t H T P = Law inf t T H t P sup t T This property holds, of course, if the process H is a Lévy process with respect to P see e.g. Kyprianou 26, Lemma 3.5. Combining 1.95 and 1.96 we get that 1 α C [ T S; α inf S = E 1 + ] α einf t T H t = E [ 1 + ] 1 α inf t T S t = P T α ; inf S Similarly, assuming the following reflection principle Law H T inf t T H t P = Law H t P sup t T which again holds for Lévy processes Kyprianou 26, Lemma 3.5 we get 1 β P T β sup S; S = C T sup S ; β Concluding, we have the following result. Theorem 1.2. Let H be a Lévy process. The calculation of the prices of floating strike lookback call and put options C T S; α inf S and P T β sup S; S, α 1, < β 1, can be reduced via the duality relations and 1 α C T S; α inf S = P 1 T ; inf S α 1 β P T β sup S; S = C T sup S ; 1 β to the calculation of the prices of fixed strike lookback put and call options P T 1 α ; inf S and C T sup S ; 1 β respectively. D. Asian options. Define Σ T to be the arithmetic average of the asset price process S during the time interval [, T ]. In case the price process is continuously monitored, then Σ T = 1 T T S udu, while in the case of discrete monitoring we have Σ T = 1 N N i=1 S T i, where = T < T 1 < < T N = T. Similarly to lookback options, there exist floating and fixed strike Asian options traded in the market. In the floating strike case, the average plays the role of the strike. The different variants of the Asian option are again summarized in Table 1.1. Note that for Asian options there exists a put-call parity relationship, which can be derived by the elementary equality Σ T S T K + K + S T Σ T + = Σ T S T K.

40 32 1. ON THE DUALITY PINCIPLE: SEMIMATINGALE SETTING Suppose that S MP and consider the price of the floating strike Asian call option; then C T S; 1 [ S = E S T 1 T + ] S t dt = E [S T 1 1 T + ] S t dt T T T S T = E [ = E [ 1 1 T 1 1 T T T S T S t dt + ] = E [ e H T H T u du + ] 1 1 T T e H T H tdt + ] If H is a P -Lévy process, then the following duality property holds see e.g. Kyprianou 26, Lemma 3.4 Law H T H T t ; t < T P = Law H t; t < T P. 1.1 From 1.99 and 1.1 we conclude C T S; 1 [ S = E 1 1 T + ] e H udu T T = E [ 1 1 T T S udu + ] Similarly, 1 1 P T S; S = C T T T Therefore, we have the following result. = P T 1; 1 T S S ; Theorem Let H be a Lévy process. Then, calculating the prices of floating strike Asian call and put options C T S; 1 T S and PT 1 T S; S, can be reduced via the duality relations C T S; 1 S = P T 1; 1 S T T and P T 1 T 1 S; S = C T T S ; 1 to the calculation of the prices of fixed strike Asian put and call options P T 1; 1 T S and C T 1 T S ; 1 respectively. emark The duality relationships of Theorem 1.21 remain true if we replace the arithmetic average Σ T by the geometric or harmonic average, i.e. when the averaging is of the form N Γ T = i=1 S Ti 1 N or A T = N N i=1 1 S Ti. 1.13

41 1.5. THE CALL-PUT DUALITY IN OPTION PICING 33 Proof. Indeed, using 1.1 we have that N 1 + ] [ N C T S; Γ = E [S T S Ti = E 1 i=1 N 1 + ] = E [1 e H T H N T i i=1 N = E [1 k=1 N = E [1 k=1 1 + ] e H T H N T T k 1 + ] e H N T k = P T 1; Γ, N i=1 1 + ] S N Ti S T where Γ denotes the geometric average corresponding to S. Similarly, we get for the harmonic average: + ] [ + ] N C T S; A = E [S T N i=1 1 = E N 1 S Ti S N T i=1 1 S Ti + ] [ + ] = E [1 N = E N 1 = E [1 N S T i i=1 S T N N 1 k=1 e H T k + ] = P T 1; A, N 1 i=1 e H T H T i where now A is the harmonic average corresponding to S. The proofs of the dualities between floating strike put and fixed strike call options follow along the same lines. E. Forward-start options. The payoff of a forward-start call option is S T S t +, where t [, T is some prespecified future date; similarly, the payoff of the forward-start put option is S t S T +. Note that both the forward start call and the put option are at-the-money at time t, when the strike is activated. Suppose S MP. Therefore, for the price of the forward-start call option we get that C t,t S; S = E [ + ] [ S T S t + ] S T S t = E S T S T = E [ 1 S t + ] = E [ 1 S + ] T S T S t = E [ 1 e H T H t + ] = E [ 1 e H T H T u + ], where u = T t. Appealing once again to the duality property 1.1, we obtain the following relation between a forward start call option and a

42 34 1. ON THE DUALITY PINCIPLE: SEMIMATINGALE SETTING European plain vanilla put option C t,t S; S = E [ 1 e H + ] T t = P T t1; S Similarly, we get a relationship between a forward-start put option and a European plain vanilla call option Therefore, we have the following result. P t,t S; S = C T ts ; Theorem Let H be a Lévy process. Then, calculating the prices of forward-start call and put options C t,t S; S and P t,t S; S, can be reduced via the duality relations and C t,t S; S = P T t1; S P t,t S; S = C T ts ; 1. to the calculation of the prices of European put and call options P T t 1; S and C T t S ; 1 respectively.

43 CHAPTE 2 On the duality principle in option pricing II: multidimensional PIIAC and α-homogeneous payoff functions 2.1. Introduction The main aim of this chapter is to continue our study of the duality principle and its applications in option pricing. In Chapter 1 we concentrated on options depending on a single asset and on exotic options on this asset. Here, the focal point are options depending on several assets. Moreover, we will consider options on a single asset with an α-homogeneous payoff function, see Definition 2.21, like power options. The driving process is chosen to be a time-inhomogeneous Lévy process, although most of the results can equally well be proved for general semimartingales. Nevertheless, the class of time-inhomogeneous Lévy processes constitutes a very convenient class for financial modeling, since it provides enough flexibility to model the empirically observed behavior in financial markets, while allowing the fast pricing of the most liquidly traded assets. It is well known that the efforts to calibrate standard Gaussian models to the empirically observed volatility surfaces very often do not produce satisfactory results. This phenomenon is not restricted to data from equity markets, but it is observed in interest rate and foreign exchange markets as well. There are two basic aspects to which the classical models cannot respond appropriately: the underlying distribution is not flexible enough to capture the implied volatilities either across different strikes or across different maturities. The first phenomenon is the so-called volatility smile and the second one the term structure of smiles; together they lead to the non-flat implied volatility surface, a typical example of which can be seen in Figure 2.1. One way to improve the calibration results is to use stochastic volatility models; we refer to Psychoyios, Skiadopoulos, and Alexakis 23 for a thorough review of the various stochastic volatility approaches. A fundamentally different approach is to replace the driving process. Lévy processes offer a large variety of distributions that are capable of fitting the return distributions in the real world and the volatility smiles in the risk-neutral world. Nevertheless, they cannot capture the term structure of smiles adequately. In order to take care of the change of the smile across maturities, one has to go a step further and consider time-inhomogeneous Lévy processes also called additive processes as the driving processes. For term structure models this approach was introduced in Eberlein et al. 25 and further investigated in Eberlein and Kluge 26a, where cap and swaption volatilities were calibrated quite successfully. 35

44 36 2. ON THE DUALITY PINCIPLE II: MULTIDIMENSIONAL PIIAC implied vol % delta % or strike maturity Figure 2.1. Implied volatility surface of vanilla options on the Euro/Dollar rate; date: 5 November 21. Data available at We should also point out two more aspects of our modeling approach. Firstly, since we are interested in options on several assets, we consider a multi-dimensional time-inhomogeneous Lévy process as the driving process. Nevertheless, each asset is driven by one coordinate of the driving process. This happens because the rich structure of the time-inhomogeneous Lévy process allows us to capture all the phenomena we are interested in, hence we do not need to employ a multi-factor model. Of course, given that a Lévy process can have an infinite number of jumps on each finite time interval, hence an infinite number of assets should be employed to hedge a contingent claim perfectly, it is rather simplistic to call a Lévy-driven model a one-factor model. On the contrary, a non-trivial i.e. not simply Gaussian Lévy process is already a high dimensional object. Secondly, the theory is developed for general time-inhomogeneous Lévy processes. Nevertheless, when we calibrate the model to empirical data, it is normally sufficient to split the trading horizon in two or three pieces e.g. short, middle and long term and employ a Lévy process for each of these pieces. Hence, the driving motion is a piecewise Lévy process. As a result, the number of parameters used in the calibration is an input to the calibration routine and not an output, depending on the data. The chapter is organized as follows: in section 2.2 we present a detailed account of time-inhomogeneous Lévy processes, and in section 2.3 we describe the asset price model. In section 2.4 we describe a method for exploring the duality principle in option pricing. The next section 2.5 contains duality relationships between options with α-homogeneous payoff functions and in the final section 2.6 we derive duality results for options depending on several assets.

45 2.2. TIME-INHOMOGENEOUS LÉVY POCESSES Time-inhomogeneous Lévy processes 1. Let d denote the d-dimensional Euclidean space. The Euclidean scalar product between two vectors u, v d is denoted by u, v or u v, where u denotes the transpose of the vector or matrix u. The Euclidean norm is denoted by, e i denotes the unit vector where the i-th entry is 1 and all others zero, i.e. e i =,..., 1,..., and 1 denotes the vector with all entries equal to 1, i.e. 1 = 1,..., 1. The inner product is extended from real to complex numbers as follows: for u = u k 1 k d and v = v k 1 k d in C d, set u, v := d k=1 u kv k ; therefore we do not use the Hermitian inner product d k=1 u kv k. Moreover, we denote iv := iv k 1 k d and v := v k 1 k d d. Let M d denote the space of real d d matrices and let denote the norm on M d induced by the Euclidean norm on d. In addition, let M nd denote the space of real n d matrices and similarly denotes the induced norm on this space. Note that we could equally well work with any vector norm on d and the norms induced by, or consistent with, it on M d and M nd. Define the set D := {x d : x > 1}, hence D c is the unit ball in d. The function h = hx denotes a truncation function, where the canonical choice is hx = x1 { x 1} = x1 D cx. We assume that h satisfies the antisymmetry property h x = hx. 2. Let B = Ω, F, F t t T, P be a complete stochastic basis and denote by MP, resp. M loc P the class of martingales, resp. local martingales, on this stochastic basis. Throughout this chapter, we will work with the following definition of a time-inhomogeneous Lévy process. Definition 2.1. A time-inhomogeneous Lévy process, is an adapted, càdlàg d -valued stochastic process L = L t t T with L = a.s., such that the following conditions hold: D1: L has independent increments, i.e. L t L s is independent of F s, s < t T, D2: the law of L t, for all t [, T ], is described by the characteristic function [ E e i u,lt ] t = exp i u, b s 1 2 u, c su + e i u,x 1 i u, hx λ s dx ds, 2.1 d where b t d, c t is a symmetric non-negative definite d d matrix and λ t is a Lévy measure on d, i.e. it satisfies λ t {} = and d x 2 1λ t dx < for all t [, T ]. Moreover, Assumption AC holds. Assumption AC. The triplets b t, c t, λ t satisfy T b t + c t + 1 x 2 λ t dx dt <. d

46 38 2. ON THE DUALITY PINCIPLE II: MULTIDIMENSIONAL PIIAC Applications in mathematical finance postulate that the asset price process is a martingale with respect to a risk-neutral measure; this naturally leads to the existence of exponential moments for the driving process. Therefore we will often use the following assumption. Assumption EM. There exists a constant M > 1, such that the Lévy measures λ t satisfy T D exp u, x λ t dxdt <, u [ M, M] d. Moreover, without loss of generality, we assume D exp u, x λ tdx < for all t [, T ] and u [ M, M] d. emark 2.2. A time-inhomogeneous Lévy process L = L t t T is an additive process, i.e. a càdlàg, stochastically continuous process with independent increments and L = a.s. Sato 1999, Definition A time-inhomogeneous Lévy process that satisfies Assumption AC is a semimartingale on the stochastic basis Ω, F, F, P. Indeed, it is a process with independent increments and absolutely continuous characteristics in the sequel abbreviated PIIAC; cf. Lemmata 1.4 and 1.5 in Kluge 25. The canonical representation of L = L t t T cf. Jacod and Shiryaev 23, II.2.34 and Eberlein et al. 25 is L t = t t b s ds + t cs 1/2 dw s + d t hxdµ L ν + d x hxdµ where c 1/2 is a measurable version of the square root of c, W = W t t T is a P -standard Brownian motion on d, µ L is the random measure of jumps of the process L and νdt, dx = λ t dxdt is the P -compensator of the jump measure µ L. The triplet of predictable or semimartingale characteristics of L with respect to P, TL P = B, C, ν, is B t = t b s ds, C t = t c s ds, ν[, t] A = t A L, λ s dxds, 2.2 where A B d. The triplet b, c, λ is called the triplet of differentiable or local characteristics of L. A semimartingale with absolutely continuous characteristics admits no fixed times of discontinuity, therefore it is a quasi-left-continuous process. Consequently, the cumulant process of the time-inhomogeneous Lévy process L = L t t T, defined as Ku = u, B u, Cu + e u,x 1 u, hx ν, 2.3 is continuous, hence EKu = e Ku and never vanishes cf. Proposition II.2.9 and Theorem III.7.4 in Jacod and Shiryaev 23. In addition, we have

47 2.2. TIME-INHOMOGENEOUS LÉVY POCESSES 39 that, for all t [, T ] and all u d [ E e i u,lt ] = e Kiut and e i u,l e Kiu M locp. 2.4 Note that since the increments of the process L are independent, the cumulant and the predictable characteristics are deterministic processes. emark 2.3. Assumption EM renders the process L = L t t T a special semimartingale. Therefore, the canonical representation resumes the form Jacod and Shiryaev 23, II.2.38 L t = t t bs ds + t c 1/2 s dw s + d xµ L νds, dx, 2.5 where B B = x hx ν. Of course, the characteristic function 2.1 and the cumulant 2.3 are modified accordingly, omitting the use of a truncation function and replacing b by b. In the sequel, whenever we work with a time-inhomogeneous Lévy process that is a special semimartingale, we will suppress the notation B and b and write B and b instead. emark 2.4. In addition, subject to Assumption EM every component L i = L i t t T, i {1..., d}, of the time-inhomogeneous Lévy process L = L 1,..., L d becomes an exponentially special semimartingale cf. Kallsen and Shiryaev 22a, 2.12, An important consequence of the independent increments property is that the triplet of semimartingale characteristics TL P determines the law of the random variables generating the process L. This is the subject of the next result. Lemma 2.5. The distribution of L t, for a fixed t [, T ], is infinitely divisible with Lévy triplet b, c, λ, where b := t b s ds, c := t c s ds, λdx := The integrals should be understood componentwise. t λ s dxds. 2.6 Proof. Firstly, we immediately have that b d and c is a symmetric non-negative definite d d matrix. Secondly, an application of the monotone convergence theorem yields that λ is a Borel measure on d and for any integrable function f we have d fxλdx = t d fxλ s dxds. 2.7 Therefore, using Assumption AC we get that d 1 x 2 λdx < and λ{} =, i.e. λ is a Lévy measure. Now, it suffices to notice that we can

48 4 2. ON THE DUALITY PINCIPLE II: MULTIDIMENSIONAL PIIAC rewrite 2.1 as [ E e i u,lt ] = exp i u, b 1 2 u, cu + e i u,x 1 i u, hx λdx, d and the result follows from the Lévy Khintchine formula The finiteness of the g-moment of the random variable L t, for a PI- IAC L = L t t T and a submultiplicative function g, is related to an integrability property of its compensator measure ν. For the notions of the g-moment and submultiplicative function, we refer to Definitions 25.1 and 25.2 in Sato Lemma 2.6 g-moment. Let g be a submultiplicative, locally bounded, measurable function on d. Then, the following statements are equivalent t a gxλ s dxds <, for any t [, T ] D b E [ gl t ] <, for any t [, T ]. Proof. We argue along the lines of Kluge 25, Lemma 1.6. Assume that condition a holds and consider the Lévy process L = L t t T such that L 1 d = Lt, for some fixed t [, T ]. Then, the Lévy triplet of L, b, c, λ, is given by Lemma 2.5 and we have that D gxλdx = t D gxλ s dxds <. 2.9 Applying Theorem 25.3 in Sato 1999, we get that E [ g L t ] <, or equivalently E [ g L 1 ] <, which immediately yields E [ gl t ] <. Conversely, assume that condition b holds and consider again the Lévy process L = L t t T as above. By definition, E [ g L 1 ] = E [ gl t ] <, which yields that E [ g L t ] <. Applying Theorem 25.3 in Sato 1999, we conclude that t The result is proved. D gxλ s dxds = D gxλdx <. 2.1 Consequently, since gx = exp u, x is a submultiplicative function, we immediately get the following result concerning Assumption EM. Corollary 2.7. The following statements are equivalent T 1 exp u, x νdt, dx <, u [ M, M] d D 2 E [ exp u, L T ] <, u [ M, M] d.

49 2.2. TIME-INHOMOGENEOUS LÉVY POCESSES 41 The following result, which is a straightforward generalization of Theorem in Sato 1999, provides the existence of the characteristic function of the random variable L t on some strip in the complex plane. Lemma 2.8. Let L = L t t T be an d -valued time-inhomogeneous Lévy process with triplet of predictable characteristics B, C, ν, that satisfies Assumption EM. For a fixed t [, T ] and z C d with z [ M, M] d, we have that θ s z = z, b s z, c sz + e z,x 1 z, x λ s dx 2.11 d is well defined for all s [, t], E e z,l t < and [ E e z,lt ] t = exp θ s zds Proof. Firstly, as in the proof of Lemma 2.6, consider the Lévy process L = L t t T such that L d 1 = Lt, for some fixed t [, T ]. Then we have that t D e z,x λs dxds = D e z,x λdx = D e z,x λdx <, 2.13 therefore E e z,lt <. Secondly, using Assumption EM and Theorem 25.17iii in Sato 1999, we deduce that θ s in 2.11 is well defined and finite for all s [, t]. Thirdly, using 2.13, Theorem in Sato 1999 and Lemma 2.5, we can define for the Lévy triplet b, c, λ of L Kz = z, b z, cz + e z,x 1 z, x λdx d and [ ] E e z,b L 1 = exp Kz. Finally, using Lemma 2.5 once again, we conclude that Kz = t θ szds, therefore E [ e z,lt ] = exp t θ szds and the result is proved. emark 2.9. The function θ s in 2.11 is called the cumulant associated with the infinitely divisible distribution characterized by the Lévy triplet b s, c s, λ s, s [, T ]. Moreover, the cumulant K of the process L, defined by 2.3, and the cumulant associated with the differential characteristics of L, defined by 2.11, are related via Ku = θ s uds. 2.14

50 42 2. ON THE DUALITY PINCIPLE II: MULTIDIMENSIONAL PIIAC 6. We will later consider linear transformations and projections of timeinhomogeneous Lévy processes. The following result describes the law and the predictable characteristics under such a transformation. Analogous results for Lévy processes can be found in Sato 1999, Proposition Proposition 2.1. Let L = L t t T be a time-inhomogeneous Lévy process on d with predictable characteristics B, C, ν. Let U be a realvalued n d matrix U M nd. Then, UL = UL t t T is an n -valued time-inhomogeneous Lévy process with characteristics B U, C U, ν U, where b U s = Ub s + hux Uhxλ s dx d c U s = Uc s U 2.15 λ U s E = λ s {x d : Ux E}, E B n \{}. Here hx is a truncation function on n. Proof. We clearly have that UL is an n -valued adapted, càdlàg process, UL = a.s. and the linearity of the transformation preserves the independence of the increments. egarding the law, we have for any z n [ E e i z,ult ] = E [e ] i U z,l t t = exp t = exp i U z, b s 1 2 U z, c s U z i U + e z,x 1 i U z, hx λ s dx ds d i z, b U s 1 2 z, Uc su z + n e i z,y 1 i z, hy λ U s dy ds, where b U is given by Now, for ease of notation we consider h and h as the canonical truncation functions. Since b s and λ s satisfy Assumption AC, and from the definition of the induced norm we get that Ux U x for all U M nd and x d, we can conclude that T b U s ds T T U Ub s + b s ds + d hux Uhxλ s dx T d ds hux Uhx λ s dxds <

51 2.2. TIME-INHOMOGENEOUS LÉVY POCESSES 43 because T d hux Uhx λ s dxds = T d T d T Ux 1 { Ux 1} 1 { x 1} λ s dxds Ux 1 { Ux 1< x } 1 { x 1< Ux } λ s dxds { Ux 1< x } T { x >1} T { x >1} T Ux λ s dxds + T λ s dxds + U T λ s dxds + U 2 { x 1< Ux } { x 1< U x } { 1 U < x 1} x Ux λ s dxds x λ s dxds 2 λ s dxds < Similarly, we have that T c U s ds = T T Uc s U ds U U c s ds < and T T y 2 1λ U s dyds = Ux 2 1λ s dxds n d U 2 1 x 2 1λ s dxds <. d T Therefore, the triplets b U t, c U t, λ U t satisfy Assumption AC and the statement is proved. emark Assume that the process L = L t t T is a special semimartingale and consider a matrix U M nd. Then the process UL =

52 44 2. ON THE DUALITY PINCIPLE II: MULTIDIMENSIONAL PIIAC UL t t T is also a special semimartingale, since T { y >1} y λ U s dyds = T { Ux >1} T U T U T Ux λ s dxds U { U x >1} { x >1} T + U 2 { Ux >1} x λ s dxds x λ s dxds 2.16 x λ s dxds { 1 U < x 1} x 2 λ s dxds <, 2.17 where we have implicitly assumed that U 1; otherwise, we can conclude already from Therefore, we can omit the use of truncation functions in 2.15 and we have that B U = U B; compare with emark 2.3. emark Assume that the process L = L t t T satisfies Assumption EM and consider a matrix U M nd, a vector v n, and the process UL = UL t t T. If U v [ M, M] d, then E[e v,ul T ] = E[e U v,l T ] < which follows directly from Assumption EM and Corollary 2.7. Therefore, UL has finite exponential moment if U 1 [ M, M] d. emark In case we want to express condition U 1 [ M, M] d in terms of some norm on the set of n d matrices, then the natural norm to consider is the max-column-sum norm U 1, defined by U 1 = max j=1,...,d Then, UL has finite exponential moment if n U ij. i=1 U 1 M. 7. We can describe the triplet of semimartingale characteristics of the dual of a 1-dimensional time-inhomogeneous Lévy process in terms of the triplet of the original process. In Chapter 1 we proved an analogous relationship for general semimartingales; cf. Theorem 1.5. However, since the triplet of a time-inhomogeneous Lévy process determines the law, the relationship between the triplets implies a relationship between the laws of the original and the dual process. Lemma Let L = L t t T be a time-inhomogeneous Lévy process with characteristic triplet B, C, ν. Then L := L is again a timeinhomogeneous Lévy process with triplet T L P = TL P = B, C, ν,

53 where 2.2. TIME-INHOMOGENEOUS LÉVY POCESSES 45 B = B C = C A x ν = 1 A x ν, A B\{}. Proof. From the Lévy Khintchine representation we have that ϕ Lt u = E [ e iult] t = exp We get immediately ϕ Lt u = ϕ Lt u t = exp = exp t [ ib s u c s 2 u2 + [ ib s u c s 2 u2 + [ i b s u c s 2 u2 + ] e iux 1 iuhxλ s dx ds. ] e i ux 1 i uhxλ s dx ds ] e iu x 1 iuh xλ s dx ds. Then, b t = b t, c t = c t, and 1 A xλ sdx = 1 A xλ s dx, and they clearly satisfy Assumption AC. Since L is a process with independent increments and L = a.s., we can conclude it is a time-inhomogeneous Lévy process and has characteristics Bt = t b sds = B t, Ct = t c sds = C t and 1A xν dt, dx = 1 A xλ t dxdt = 1 A xνdt, dx. 8. emark The PIIACs L 1,..., L d are independent if and only if the matrices c t are diagonal and the Lévy measures λ t are supported on the union of the coordinate axes. This follows similarly to Exercise 12.1 in Sato Describing the dependence is a more difficult task; the book of Müller and Stoyan 22 consists of a comprehensive exposition of various dependence concepts and their applications. Tankov 23 and Kallsen and Tankov 26 introduced the notion of a Lévy copula to describe the dependence of the components of multidimensional Lévy processes. emark If the triplet of local characteristics b t, c t, λ t is not timedependent, then the time-inhomogeneous Lévy process L PIIAC becomes a homogeneous Lévy process, i.e. a process with independent and stationary increments PIIS. In that case, the distribution of L is described by the Lévy triplet b, c, λ, where λ is the Lévy measure and the compensator of µ L becomes a product measure of the form ν = λ λ\, where λ\ denotes the Lebesgue measure. In that case, equation 2.1 takes the form [ E e i u,lt ] = e tψu 2.19 where ψu = i u, b 1 2 u, cu + e i u,x 1 i u, hx λdx, 2.2 d which is called the characteristic exponent of the Lévy process L.

54 46 2. ON THE DUALITY PINCIPLE II: MULTIDIMENSIONAL PIIAC 2.3. Asset price model 1. The asset price process is modeled as follows: each component S i of the vector of asset price processes S = S 1,..., S d is an exponential timeinhomogeneous Lévy process, i.e. a stochastic process with representation S i t = S i exp L i t, t T ; 2.21 here the superscript i refers to the i-th coordinate, i {1,..., d}. The driving process L = L t t T is an d -valued time-inhomogeneous Lévy process that satisfies Assumption EM, with canonical decomposition L t = t t b s ds + t cs 1/2 dw s + d xµ L νds, dx We assume that P is a risk-neutral or, martingale measure, i.e. the asset prices have mean rate of return µ i r δ i, where r is the risk-free interest rate and δ i is the dividend yield of the i-th asset. Therefore, the auxiliary price processes S i = S i t t T, where S i t := eδi t St i, i {1,..., d}, once discounted at the rate r, become P -martingales. Consequently, the drift characteristic B of the driving process L is completely determined by the other two characteristics C, ν, the rate of return of the asset and the risk-free interest rate. 2. Assume for the moment that r = δ i and S i 1 for all i {1,..., d}. Then, S i = S i = e Li and we wish to determine the drift characteristic B i in this special case, such that e Li M loc P. Using Theorem II.2.42, Corollary II.2.48 and section III.7a in Jacod and Shiryaev 23, we know that for all predictable, integrable with respect to L processes ϑ i.e. ϑ LL, such that ϑ L is exponentially special, the following hold: and e ϑ L e ϑ L Ku M loc P 2.23 e ϑ L Gϑ M locp, 2.24 where K is the cumulant process of L, cf. 2.3 and emarks 2.3 and 2.9, i.e. Kϑ = θ sϑds, with θ s ϑ = ϑ, b s ϑ, c sϑ + e ϑ,x 1 ϑ, x λ s dx, 2.25 d and Gϑ = EKϑ = e Kϑ As was already mentioned, L is a quasi-left-continuous process and the cumulant process K is continuous; moreover, Kϑ is deterministic if ϑ is deterministic see also emarks III.7.15 in Jacod and Shiryaev 23.

55 2.3. ASSET PICE MODEL 47 Since L i = e i, L, e i M, M d d and subject to Assumption EM each coordinate process L i of L is an exponentially special semimartingale, we get from 2.24 that e e i,l Ge i = Now, from 2.27 we immediately get that eli e Ke i M locp e Li M loc P Ke i = ; 2.28 the if part being obvious, the only if part follows from the uniqueness of the multiplicative decomposition of a special semimartingale cf. Jacod 1979, VI.2a and Theoreme Therefore, from 2.27 and 2.28 we conclude that S i M loc P e i, B e i, Ce i + e e i,x 1 e i, x ν = Finally, using that every exponential PIIAC that is a local martingale is indeed a martingale cf. Eberlein, Jacod, and aible 25, pp. 79-8, we conclude that, for all i {1,..., d} S i = e Li MP B i Cii + e xi 1 x i ν = ; 2.3 compare with emark Another way to derive the condition above, is to extract the i-th coordinate process from L and then apply the results of Chapter 1. Indeed, using Proposition 2.1 for U = e i, we get that L i has local characteristics b i, c ii, λ i, where λ i E = λ{x d : x i E}, E B. Then, applying 1.29 we immediately get 2.3. emark Note that since each asset is driven by one component of the multidimensional time-inhomogeneous Lévy process, the off-diagonal elements of the matrix C and the contribution of the jump measure not concentrated on the coordinate axes, do not participate in the martingale condition 2.3. Nevertheless, they describe the dependence between the asset price processes S i and S j, j i. See also Example eturning to the general case, where r, δ i and S i, i {1,..., d}, it is immediately clear from the previous considerations that the i-th component of the drift vector B must have the form B i t = t r δ i ds 1 2 t t c ii x sds e i 1 x i νds, dx 2.31 d and then S i discounted is a P -martingale, for all i {1,..., d}. 4. Markets modeled by exponential time-inhomogeneous Lévy processes are incomplete and there exists a large class of equivalent martingale or, risk neutral measures. Eberlein and Jacod 1997 provide a complete characterization of the class of equivalent martingale measures for exponential Lévy

56 48 2. ON THE DUALITY PINCIPLE II: MULTIDIMENSIONAL PIIAC models; this was later extended to general semimartingales in Gushchin and Mordecki 22. In this work, we do not dive into the theory of choosing a martingale measure, we rather assume that the choice has already taken place. We refer to Eberlein and Keller 1995 and Kallsen and Shiryaev 22a for the Esscher transform, Frittelli 2 and Fujiwara and Miyahara 23 for the minimal entropy martingale measure and Bellini and Frittelli 22 for minimax martingale measures, to mention just a small part of the literature on this subject. A unifying exposition in terms of f-divergences of the different methods for selecting an equivalent martingale measure can be found in Goll and üschendorf 21. Alternatively, one can consider the choice of the martingale measure as the result of a calibration to the smile of the vanilla options market. Hakala and Wystup 22 describe the calibration procedure in detail; we refer to Cont and Tankov 24, 26 and Belomestny and eiß 26 for numerically stable calibration methods for Lévy driven models. emark In the above setting, we can easily incorporate dynamic time-dependent interest rates and dividend yields; the drift term 2.31 would have a similar form, taking r s and δ s into account. emark 2.2. Assumption EM is sufficient for all our considerations, but in general stronger than required. In the sequel we will replace EM, on occasion, by the minimal sufficient assumptions. From a practical point of view though, it is not too restrictive to assume EM, since all examples of Lévy models we are interested in, e.g. the Generalized Hyperbolic model cf. Eberlein and Prause 22, the CGMY model cf. Carr et al. 22 or the Meixner model cf. Schoutens 22, possess moments of all order and moment generating functions General description of the method In Chapter 1 we developed the mathematical tools required to study the duality principle in an abstract semimartingale framework. Here, we aim at exploring the duality principle in the more practically-oriented framework described in sections 2.2 and 2.3: when interest rates and dividend yields are present and the driving process is a time-inhomogeneous Lévy process. In addition, we want to derive put-call dualities for options with payoff functions homogeneous of degree higher than one and for options on several underlying assets. We begin by defining an α-homogeneous payoff function and then describe the method we will use to explore dualities; it is based on the choice of a suitable numéraire and the subsequent change of the probability measure, pioneered by Geman et al Definition A payoff function f : is called α- homogeneous if f is a homogeneous function of degree α 1, that is for c, x, y +, holds fcx, cy = c α fx, y. In order to derive duality relationships in this framework, the discounted asset price process corrected for dividends serves as the numéraire, in case

57 2.4. GENEAL DESCIPTION OF THE METHOD 49 the option payoff is homogeneous of degree one. Using this numéraire, evaluated at the time of maturity, as the adon Nikodym derivative, we form a new measure. Under this new measure, the numéraire asset is riskless while all other assets, including the savings account, are now risky. As a result, the new riskless rate is the dividend yield of the numéraire asset. In case the payoff function is α-homogeneous for α > 1, we will have to modify the asset price process appropriately, so that it serves as the numéraire. Consequently, the asset price dynamics under the new measure will depend on α as well. For the sake of simplicity, we assume here that α = 1 and later in the case of power options we will treat the case α > 1. We consider two cases for the driving process L and the asset price processes: P1: L = L 1 is a 1-dimensional PIIAC, L 2 = k is constant and S 1 = S 1 exp L1, S 2 = exp L 2 = K; P2: L = L 1, L 2 is a 2-dimensional PIIAC and S i = S i exp Li, i = 1, 2. According to the general arbitrage pricing theory, the value of an option on assets S 1, S 2 with payoff function f is equal to its discounted expected payoff under an equivalent martingale measure. Throughout this chapter, we will assume that options start at time and mature at T ; therefore we have V = e rt E [ f S 1 T, S 2 T ] We choose, without loss of generality, asset S 1 as the numéraire and express the value of the option in terms of this numéraire, which yields [ ] V = e rt E ST 1 f 1, S2 T ST 1 = S 1 e δ1t E [ e rt S 1 T e δ1t S 1 f ] 1, S2 T ST Now, we can define on Ω, F, F t t T a new probability measure P via the adon Nikodym derivative dp dp = e rt ST 1 e δ1t S 1 =: Z T, 2.34 and EZ T = 1. The valuation problem, under the measure P, becomes [ V = Se 1 δ1t E f 1, S 1,2 T ] 2.35 where we have defined the process S 1,2 = S 1,2 t t T with S 1,2 t = S2 t. St 1 Since the discounted auxiliary process S 1 is a P -martingale, we deduce dp F t dp F t = Z t, t T and because Z > P -a.s., we get that P P. Therefore, we can apply Girsanov s theorem for semimartingales, that allows us to determine the dynamics of S 1,2 under P. If S 1,2, discounted at the new riskless rate δ 1, is a P -martingale, then we have transformed the original valuation problem into a simpler one.

58 5 2. ON THE DUALITY PINCIPLE II: MULTIDIMENSIONAL PIIAC 2.5. Options with α-homogeneous payoff functions 1. The setting we will work in, is that of P1: L = L t t T is the driving real-valued time-inhomogeneous Lévy process, with triplet of predictable characteristics B, C, ν, S 1 = S exp L = S and L 2 = k, such that S 2 = e k = K is the strike price of the option. In accordance with the standard notation in mathematical finance, we will denote by σ 2, instead of c, the local diffusion characteristic, which corresponds to the volatility in the Black Scholes model. Therefore, the characteristic C in 2.2 has the form C = σ2 sds. 2. The main aim of this section is to prove a duality relationship between power options. The payoff of the power call and put option respectively is [ ST K +] α [ and K ST +] α where α N is the power index. We introduce the following notation for the value of a power call option with strike K and power index α CS, K, α; r, δ, C, ν = e rt E [ S T K +] α where the asset price process is modeled as an exponential PIIAC according to 2.21, 2.22 and Similarly, for a power put option we set PS, K, α; r, δ, C, ν = e rt E [ K S T +] α. If the power index equals one, then we have a standard European plain vanilla option and the power index α will be omitted from the notation. Furthermore, if the dividend yield is zero, it will also be omitted. Assumption EM can be replaced by the following weaker assumption, which is the minimal sufficient condition for the duality results to hold. Let D + = D + and D = D. Assumption M. The Lévy measures λ t satisfy T D x λ t dxdt < and T D + xe αx λ t dxdt <. Theorem Assume that the asset price process evolves as an exponential time-inhomogeneous Lévy process according to 2.21, 2.22 and 2.31 and assumption M is in force. Then, we can relate the power call and put option via the following duality: C S, K, α; r, δ, C, ν 1 = K α S α CT P α, K, α; δ, r, C, ν 2.36 S where C α T = exp α 1δ rt + α 2 1 αc T + e αx 1 αe αx +αe α 1x ν T, K = K 1 e C T, C T is given by 2.47 and 1 Ax ν = 1 A xe αx ν. Proof. We observe that, since the payoff function is α-homogeneous for α > 1, the discounted asset price process corrected for dividends cannot serve as a numéraire; that is because [e rt S t ] α t T = [e δ rt S t ] α t T = S α e αδ rt+αlt t T

59 2.5. OPTIONS WITH α-homogeneous PAYOFF FUNCTIONS 51 is not a P -martingale, unless α = 1. Let us denote by L α = L α t t T the martingale part of the exponent, that is t t L α t = ασ s dw s + αxµ L νds, dx. Since L α is an exponentially special semimartingale, using Lemma 2.15 and Theorem 2.18 in Kallsen and Shiryaev 22a, we deduce that its exponential compensator, denoted by CL α = CL α t t T, exists and has the form CL α t = 1 t t α 2 σ 2 2 sds + e αx 1 αxνds, dx. Then, by definition of the exponential compensator and using Eberlein et al. 25, pp. 79-8, we conclude that expl α CL α M. Now, the price of the power call option can be expressed as follows: C = e rt E [ S T K +] α [ e = e δt S α rt ST α E Kα e δt S α [ = e δt S α K α E exp = e δt S α K α C T E ] [ K 1 S 1 T +] α δ rt + αb T + CL α T exp L α T CL α T [ exp L α T CL α T where, using 2.31 and 2.2, we have that log C T := δ rt + αb T + CL α T = α 1r δt + T + [ K 1 S 1 T +] α ] [ K 1 S 1 T +] α αα 1 2 T σ 2 sds ] 2.37 e αx αe x + α 1νds, dx We define on Ω, F, F t t T a new probability measure P, via the adon Nikodym derivative dp dp = exp L α T CL α T = ZT 2.39 and the valuation problem 2.37 becomes C = e δt S α K α C T E [ K S T +] α 2.4 where K := 1 K and S := 1 S. Since the measures P and P are related via the density process Z = Z t t T, which is a positive P -martingale with Z = 1, we immediately

60 52 2. ON THE DUALITY PINCIPLE II: MULTIDIMENSIONAL PIIAC deduce that P P. The density process can be represented in the usual form [ ] dp Z t = E dp F t = exp L α t CL α t t t = exp ασ s dw s + αxµ L νds, dx 1 2 t α 2 σ 2 sds t e αx 1 αxνds, dx, 2.41 or as the stochastic exponential of a suitable time-inhomogeneous Lévy process X α, i.e. Z = EX α, where X α = ασ s dw s + e αx 1 µ L νds, dx; cf. Kallsen and Shiryaev 22a, Lemma 2.6. An application of Girsanov s theorem for semimartingales, cf. Jacod and Shiryaev 23, Theorem III.3.24, yields that the triplet of predictable characteristics of L under P is TL P = B, C, ν, for B = B + β α C + xy α 1 ν C = C 2.42 ν = Y α ν, where we can take the following versions of β α and Y α : β α α and Y α = e αx Indeed, as in part a of the proof of Theorem 1.5 and using Appendix B, we have that Z c, L c = Z s ασ s dw s, σ s dw s = Z s ασsds, 2 therefore, we conclude that β α α. Moreover, we can choose Y α = e αx, because for any non-negative P-measurable function U = Uω; t, x, we get [ T M P µ e αx U = E L [ = E [ T = E t T = M P µ L Z Z U ] e αx Uω; t, xµ L ω; dt, dx e α Ltω Uω; t, L t ω1 { Ltω } ] Z t ω Z t ω Uω; t, xµl ω; dt, dx, ]

61 2.5. OPTIONS WITH α-homogeneous PAYOFF FUNCTIONS 53 since Z Z 1 { Z } = expα L. Now, using Theorem II.4.15 in Jacod and Shiryaev 23, we deduce that a time-inhomogeneous Lévy process remains a time-inhomogeneous Lévy process under the measure P, because the P -characteristics of L are deterministic and satisfy Assumption AC. In addition, the P -canonical decomposition of L is L t = t t b sds + t σ s dw s + xµ L ν ds, dx 2.44 where B t = t b sds = r δt + α 1 t σ 2 sds 2 t + e αx e 1 αx + xν ds, dx Here, W = W ασ sds is a P -Brownian motion and ν = Y α ν is the P compensator of the jumps of L. Define the dual process of L, L := L; its triplet of predictable characteristics TL P = B, C, ν, is given by Lemma 2.14 in terms of TL P. The canonical decomposition of L is t L t = t b sds + t σ s dw s + xµ L ν ds, dx, 2.46 where µ L is the random measure associated with the jumps of L, i.e. 1 A x µ L = 1 A x µ L. We have that S = 1 S = 1 S e L = 1 S e L and from equations 2.46 and 2.45, we can easily deduce that e r δt S t t T is not a P -martingale for α 1. Nevertheless, we can easily construct a martingale, and then take advantage of the fact that the compensating terms of a time-inhomogeneous Lévy process are deterministic. Indeed, adding the appropriate terms, we can re-write L as L = C + L, where C = 1 α σsds 2 e αx e 1 αx + 1 e x ν ds, dx 2.47 and L = L t t T is a time-inhomogeneous Lévy process with triplet of predictable characteristics T L P = B C, C, ν. Define Ŝ = S 1 eb L and then e r δt Ŝ t t T MP.

62 54 2. ON THE DUALITY PINCIPLE II: MULTIDIMENSIONAL PIIAC Therefore, we can conclude the proof, since C = e δt S α K α C T E [ K S T +] α = e δt S α K α C T E [ K e C T ŜT +] α = e δt S α K α C α T E [ K ŜT +] α, where K := K e C T = K 1 e C T and C α T := C T e αc T. Setting α = 1 in the previous Theorem, we immediately get a duality relationship between European plain vanilla call and put options; compare with Theorem Corollary Assume that the asset price evolves as an exponential time-inhomogeneous Lévy process and assumption M is in force. Then, we can relate European call and put options via the following duality: C S, K; r, δ, C, ν 1 = KS P, 1 S K ; δ, r, C, ν 2.48 where 1 A x ν = 1 A xe x ν, A B\{}. emark It is interesting to point out that a different duality relating European and American call and put options, in the Black-Scholes model, was derived by Peskir and Shiryaev 22. They use the mathematical concept of negative volatility and their main result states that CS T, K; σ = P S T, K; σ See also the discussion and the corresponding cartoon! in Haug In this framework, we can easily derive duality relationships between self-quanto and European plain vanilla options. This result is, of course, a special case of Theorem 2.3, that will be proved later. Nevertheless, we give a short proof, since it simplifies considerably due to the driving process being 1-dimensional. The payoff of the self-quanto call and put option is S T S T K + and S T K S T + respectively. Introduce the following notation for the value of the self-quanto call option QCS, K; r, δ, C, ν = e rt E [ S T S T K +] and similarly, for the self-quanto put option we set QPS, K; r, δ, C, ν = e rt E [ S T K S T +]. Assumption EM can be replaced by the following weaker assumption, which is the minimal sufficient condition for the duality results to hold. Assumption M. The Lévy measures λ t satisfy T D x λ t dxdt < and T D + e 2x λ t dxdt <.

63 2.5. OPTIONS WITH α-homogeneous PAYOFF FUNCTIONS 55 Theorem Assume that the asset price evolves as an exponential time-inhomogeneous Lévy process and assumption M is in force. We can relate the self-quanto and European plain vanilla call and put options via the following dualities: QCS, K; r, δ, C, ν = S C T CS, K; δ, r, C, fν 2.5 QPS, K; r, δ, C, ν = S C T PS, K; δ, r, C, fν 2.51 where C T = ec T, C T is given by 2.53, K = Ke C T and fx = e x. Proof. Obviously, we choose asset S as the numéraire, define a new probability measure P via the adon Nikodym derivative in 2.34, and the original valuation problem becomes QC = e δt S E [ S T K +] We want to calculate the P -characteristic triplet of L. Arguing as in the proof of Theorem 2.22, the density process Z = Z t t T has the form 2.41 for α = 1. Hence, the tuple of predictable processes β, Y associated with the process L and the measure P is β 1 and Y = e x. Therefore, similarly to 2.44 and 2.45, L has the P -canonical decomposition t t t L t = b sds + σ s dw s + xµ L ν ds, dx where b s = r δ + σ2 s 2 + e x 1 + xλ sdx. Here, W = W σ sds is a P -Brownian motion and ν = Y ν is the P - compensator of µ L. Consequently, TL P = B, C, ν, which are described above. Notice that e r δt e Lt t T is not a P -martingale; however, we can define L = L t t T where L t := t t σ s dw s + + δ rt t xµ L ν ds, dx σs 2 t 2 ds e x 1 xν ds, dx and then e r δt e L t t T MP. Next, we can express L as L = L +C, where T CT = 2r δt + T σsds 2 + e x + e x 2ν ds, dx. 2.53

64 56 2. ON THE DUALITY PINCIPLE II: MULTIDIMENSIONAL PIIAC Now, by re-arranging the terms in 2.52, using again the fact that the compensating terms are deterministic, the result follows, since QC = e δt S E [ S e L T K + ] = e δt S E [ ] S e L T e CT + K [ = e δt ] S e C T E S e L T Ke CT + = e δt S e C T E [ S e L T K + ], 2.54 where K := Ke C T. The case of the quanto put option is similar Options on several assets 1. The aim in this section is to derive duality relationships between options involving two, or more, assets such as swaps and quanto options and European plain vanilla options. Therefore, similarly to the case for exotic options, the duality principle allows us to reduce a problem involving two, or more, random variables and joint distributions to a problem involving a single random variable and distribution. The latter problem can be treated using the methods developed in Chapter 3. The first results in this direction were obtained by Margrabe 1978 for the option to exchange one asset for another; hence, this option is often referred to as Margrabe option. Schroder 1999 worked in a general semimartingale setting, but provided explicit expressions only in the Black Scholes framework. Fajardo and Mordecki 26b considered homogeneous Lévy processes as the driving processes. The setting we will work in is that of P2: L = L 1,..., L d is the driving d -valued time-inhomogeneous Lévy process with triplet of predictable characteristics TL P = B, C, ν. The vector of asset price processes is S = S 1,..., S d, where we set for convenience S i t = S i exp r δ i t + L i t, i = 1,..., d The canonical decomposition of the time-inhomogeneous Lévy process L = L t t T, which satisfies Assumption EM, is L t = t t b s ds + t c 1/2 s dw s + d xµ L νds, dx 2.56 and the i-th component of the modified drift characteristic is set to be B i t = 1 2 compare with t t c ii x sds e i 1 x i νds, dx; 2.57 d

65 2.6. OPTIONS ON SEVEAL ASSETS The following result is the key tool required to derive the duality relationships in this setting. It provides the characteristic triplet of a 1- dimensional time-inhomogeneous Lévy process defined as the scalar product of a vector with the d-dimensional Lévy process L in terms of the triplet of L under an equivalent change of probability measure. The new measure is defined via an Esscher transformation, cf. Kallsen and Shiryaev 22a. Theorem Let L = L t t T be an d -valued time-inhomogeneous Lévy process that satisfies Assumption EM, with triplet of predictable characteristics TL P = B, C, ν. Let u, v be vectors in d such that v M, M d and u + v [ M, M] d. Define the measure P via the adon Nikodym derivative dp dp = e v,lt E[e v,l T ]. Then, the process L u = L u t t T, where L u t := u, L t, is a 1-dimensional time-inhomogeneous Lévy process with triplet of predictable characteristics TL u P = B u, C u, ν u with b u s = u, b s + u, c s v + u, x e v,x 1 λ s dx d c u s = u, c s u 2.58 λ u s E = λ s{x d : u, x E}, E B\{}, where λ s is a measure defined by λ sa = e v,x λ s dx, A B d \{}. A Proof. There are several equivalent ways to prove the above result, which are represented in the following diagram: where TL P G TL P G U a b e E [e iz u,l ] c d U G TL u P TL u P 2.59 means that we use Girsanov s theorem to calculate the right U side triplet from the left side one and means that we use Proposition 2.1 to calculate the right side triplet from the left side one. The direct proof e relies on the use of the characteristic function of the process L. This was the method used to prove this result in Eberlein and Papapantoleon 25b and will not be pursued further here, since the other methods are more general. We will prove the result using steps a and b, with the proof using steps c and d being analogous.

66 58 2. ON THE DUALITY PINCIPLE II: MULTIDIMENSIONAL PIIAC G Step 1: a TL P TL P The process v, L is a 1-dimensional time-inhomogeneous Lévy process on Ω, F, F, P, see Proposition 2.1, a special and exponentially special semimartingale since v M, M d, cf. emark 2.11, and E [ e v,lt ] = EKv t = e Kvt, t T. 2.6 The process Z = Z t t T defined by Z := e v,l Kv satisfies Z > a.s., Z MP and EZ T = 1, therefore the probability measure P defined on Ω, F, F t T via dp = Z T dp is equivalent to P P P and the density is given by Z t = dp F t dp F t = e v,lt Kvt, t T Moreover, using Theorem 2.19 in Kallsen and Shiryaev 22a, we can express Z as follows: Z = E v L c + e v x 1 µ L ν Now, an application of Girsanov s theorem for semimartingales Jacod and Shiryaev 23, Theorem III.3.24, yields that TL P = B, C, ν is B i = B i + Cβ i + x i Y 1 ν, i {1,..., d} C = C 2.63 ν = Y ν where we can take the following versions of β and Y : Indeed, using 2.62, we have that from which we immediately get that β = v and Y = e v,x Z c = Z v L c = Z v L c Z c, L i,c = d j=1 c ij s v j Z s ds; see also Lemma 2.2 in Kallsen and Shiryaev 22a. Similarly, for the verification of Y we can either rely on Lemma 2.2 in Kallsen and Shiryaev 22a, or calculate explicitly, for any P-measurable function U = Uω; t, x

67 on Ω [, T ] d, that [ T M P µ e v,x U = E L 2.6. OPTIONS ON SEVEAL ASSETS 59 d [ = E t T [ = E [ T = E t T d = M P µ L Z Z U ] e v,x Uω; t, xµ L ω; dt, dx e v, Ltω Uω; t, L t ω1 { Ltω } ] e v,ltω Kvt e v,l Uω; t, L t ω1 t ω Kv t { Ltω } ] Z t ω Z t ω Uω; t, xµl ω; dt, dx In addition, since there exists a deterministic version of the P -characteristics of L we conclude, using Theorem II.4.15 in Jacod and Shiryaev 23, that L is a semimartingale with independent increments on Ω, F, F, P. egarding the differential characteristics of L under P, we have that T b s ds T T b s + c s v + d b s + v c s + d v xe x 1λ s dx ds v xe x 1 λ s dx ds < because b s, c s, λ s satisfy Assumption AC, λ s satisfies Assumption EM and v M, M d. Similarly, we have that and T T x 2 x λ sdxds = d c s ds = T K T c s ds < x 2 x e v,x λ s dxds d T D c x 2 λ s dxds+ T D ] x e v,x λ s dxds < where K is a positive constant. Therefore, the differential characteristics satisfy Assumption AC, and we can conclude that L is a P -time-inhomogeneous Lévy process and a special semimartingale.

68 6 2. ON THE DUALITY PINCIPLE II: MULTIDIMENSIONAL PIIAC Step 2: b TL P U TL u P Applying Proposition 2.1 to the time-inhomogeneous Lévy process L under P and the vector U = u, we have that L u = u, L is also a P - time-inhomogeneous Lévy process with triplet of predictable characteristics TL u P = B u, C u, ν u, where b u s = u b s = u b s + u c s v + u v xe x 1λ s dx c u s = u c s u d λ u s E = λ s{x d : u x E}, E B\{}. Note that L u is a special semimartingale, which follows directly from the analogous property of L under P, cf. emark Moreover, since u + v [ M, M] d we can deduce that L u is an exponentially special semimartingale, and the statement is proved. 3. The payoff of a swap option, also coined a Margrabe option or option to exchange one asset for another, is S 1 T S 2 T + and we denote its value by MS 1, S 2 ; r, δ, C, ν = e rt E [ S 1 T S 2 T + ]. The payoff of the quanto call and put option respectively, is S 2 T K + and K S 2 + T S 1 T and we will use the following notation for the value of the quanto call option [ QCS, 1 S, 2 K; r, δ, C, ν = e rt E ST 1 S 2 T K ] + and similarly for the quanto put option QPS 1, S 2, K; r, δ, C, ν = e rt E S 1 T [ ST 1 ] K S 2 + T. The different variants of the quanto option traded in Foreign Exchange markets are explained in detail in Musiela and utkowski 25. The payoff of a digital cash-or-nothing and a correlation digital option respectively, is 1 {ST >K} and S 1 T 1 {S 2 T >K}. Hence, the holder of a correlation digital option receives one unit of the payment asset S 1 at expiration, if the measurement asset S 2 ends up in the money. Of course, this is a generalization of the standard digital asset-or-nothing option, where the holder receives one unit of the asset if it ends up in the money. We denote the value of the digital option by DS, K; r, δ, C, ν = e rt E [ ] 1 {ST >K}

69 2.6. OPTIONS ON SEVEAL ASSETS 61 and the value of the correlation digital option by [ ] CDS, 1 S, 2 K; r, δ, C, ν = e rt E ST 1 1 {S 2 T >K}. emark Notice that in the case of the digital option r, δ, C and ν correspond to a 1-dimensional driving process, while for all other options to a 2-dimensional one. Theorem Assume that the asset price processes evolve as exponential time-inhomogeneous Lévy processes according to 2.21, 2.22 and 2.31, and assumption EM is in force. Then we can relate the value of a swap and a plain vanilla option via the following duality: M S 1, S 2 ; r, δ, C, ν = C P S 2 /S 1, K; δ 1, C, ν 2.66 where K = e δ2t, C = S 1e δ2t and the characteristics C, ν are given by Theorem 2.26 for v = 1, and u = 1, 1. Proof. We will use, without loss of generality, asset S 1 as the numéraire asset; if we used asset S 2 instead, then we would get a duality relationship with a call option. The value of the swap, or Margrabe, option is ] + [ S M = e rt 1 E T ST 2 [ = e δ1t SE 1 e rt ST 1 e δ1t S 1 [ = e δ1t S 1 E = e δ1t S 1 E [ e L1 T e v,l T 1 S2 T 1 S2 T S 1 T S 1 T + ] 1 S2 T S 1 T + ] + ], 2.67 where we have used 2.55 and v = 1,. Moreover, from 2.56 and 2.57, it is immediately obvious that e v,l MP cf. also Define a new measure P via the adon Nikodym derivative dp dp = e v,l T and the valuation problem takes the form M = e δ1t S 1 E [ 1 S2 T S 1 T where, using 2.55 again, we get that St 2 St 1 = S2 e r δ2 t e L2 t S 1 e r δ1 t e L1 t + ] = S2 S 1 e δ1 δ 2 t e u,lt, t T 2.68 where u = 1, 1. Now, applying Proposition III.3.8 in Jacod and Shiryaev 23 and , we obtain that e u,l MP since e u,l e v,l = e L2 MP.

70 62 2. ON THE DUALITY PINCIPLE II: MULTIDIMENSIONAL PIIAC Therefore, we define the process S = S t t T with S t = S 2/S1 eδ1t e u,lt, which once discounted at the new risk-free rate δ 1 becomes a P -martingale. Then, we have that [ ] + M = e δ1t SE 1 1 e δ2t S T [ ] + = e δ1t Se 1 δ2t E e δ2t S T and the proof is completed. Example 2.29 Margrabe Consider two assets, S 1 and S 2, where the dynamics of each asset are S i t = exprt + L i t, i = 1, 2, t T, 2.69 where L i, i = 1, 2, is a Brownian motion with drift. In other words, the characteristics of L = L 1, L 2 are σ 2 C = 1 ρσ 1 σ 2 ρσ 1 σ 2 σ2 2 and ν, where σ 1, σ 2 + and ρ [ 1, 1] is the correlation coefficient of the Brownian motions W 1 and W 2, i.e. W 1, W 2 = ρ. Assume, as in Margrabe 1978, that the assets pay no dividends. According to 2.57, the drift characteristic has the form B = 1 C 11 2 C 22 = 1 σ The price of the option to exchange asset S 1 for asset S 2, according to Theorem 2.28, is equal to the price of a put option with strike 1, on an asset S with characteristics C, ν described by Theorem 2.26 for v = 1, and u = 1, 1. Hence, we get that C = u Cu = 1 1 σ 2 1 ρσ 1 σ 2 ρσ 1 σ 2 σ 2 2 σ = σ σ2 2 2ρσ 1 σ 2 and ν. Therefore we have recovered the original result of Margrabe, cf. Margrabe 1978, p. 179, as a special case in our setting. Moreover, we have that the drift term B of S has the form B = u B + u Cv = 1 σ σ σ1 2 ρσ 1 σ 2 1 ρσ 1 σ 2 σ2 2 = 1 2 σ2 1 + σ 2 2 2ρσ 1 σ 2 = 1 2 C, as was expected, since S discounted is a P -martingale. Theorem 2.3. Assume that the asset price processes evolve as exponential time-inhomogeneous Lévy processes according to 2.21, 2.22 and 2.31, and assumption EM is in force. Then we can relate the value of a quanto and a European call option via the following duality: QC S 1, S 2, K; r, δ, C, ν = X C S 2, K; δ 1, C, ν 2.7

71 2.6. OPTIONS ON SEVEAL ASSETS 63 where K = Ke δ1 +δ 2 rt e K1 T, X = S 1er δ1 δ 2 T e K1 T and the characteristics C, ν are given by Theorem 2.26 for v = 1, and u =, 1. Proof. Obviously, we will use asset S 1 as the numéraire asset, and similarly to the proof of Theorem 2.28, we have [ QC = e rt E ST 1 S 2 T K ] + [ = e δ1t S 1 E e L1 T S 2 T K ] + [ = e δ1t S 1 E e v,l T ST 2 K ] +, 2.71 where v = 1, and e v,l MP. Define a measure P via the adon Nikodym derivative dp dp = e v,l T and the valuation problem takes the form QC = e δ1t S 1 E [ S 2 T K + ]. Notice that, using 2.55 again, we have that S 2 t = S 2 e r δ2 t e L2 t = S 2 e r δ2 t e u,lt, t T 2.72 where u =, 1. An application of Proposition III.3.8 in Jacod and Shiryaev 23 and , yield that e u,l / MP since e u,l e v,l = e 1,L / MP. Nevertheless, since e 1,L is an exponentially special semimartingale, we get from 2.24 and 2.26 that e 1,L MP. ek1 Hence, using Proposition III.3.8 in Jacod and Shiryaev 23 again, we get that e u,l e K1 MP since e u,l e K1 e v,l = e 1,L MP. ek1 Therefore, we define the process S = S t t T with S t = S 2 e δ1t e u,lt e K1t, which once discounted at the new risk-free rate δ 1 becomes a P -martingale. Now, using again that the exponential compensator is a deterministic process, we can conclude that [ ] + QC = e δ1t SE 1 e r δ1 δ 2 T e K1 T S T K [ = e δ1t XE S T K ] + where X := S 1 er δ1 δ 2 T e K1 T and K := Ke δ1 +δ 2 rt e K1 T.

72 64 2. ON THE DUALITY PINCIPLE II: MULTIDIMENSIONAL PIIAC Theorem Assume that the asset price processes evolve as exponential time-inhomogeneous Lévy processes according to 2.21, 2.22 and 2.31, and assumption EM is in force. Then we can relate the value of a correlation digital and a digital option via the following duality: CD S 1, S 2, K; r, δ, C, ν = S 1 D S 2, K; δ 1, C, ν 2.73 where K = Ke δ1 +δ 2 rt e K1 T and the characteristics C, ν are given by Theorem 2.26 for v = 1, and u =, 1. Proof. The proof follows along similar lines to the proof of Theorem 2.3 and is therefore omitted. 4. As a final application, we will treat an option that depends on three assets which will be called, for obvious reasons, a quanto-swap option. The payoff of the quanto-swap option is S 1 T S 2 T S 3 T + and can be interpreted as a swap option struck in a foreign currency. Let us denote its value by [ QMS, 1 S, 2 S; 3 r, δ, C, ν = e rt E ST 1 S 2 T ST 3 ] +. Theorem Assume that the asset price processes evolve as exponential time-inhomogeneous Lévy processes according to 2.21, 2.22 and 2.31, and assumption EM is in force. Then we can relate the value of a quanto-swap and a plain vanilla put option via the following duality: QM S 1, S 2, S 3 ; r, δ, C, ν = Z P S 3 /S 2, K; δ 1 + δ 2, C, ν 2.74 where K = e δ1 +δ 3 T e Kv T Kw T, Z = S 1 S2 er δ1 δ 3 T e Kw T, w = 1,, 1 and the characteristics C, ν are given by Theorem 2.26 for v = 1, 1, and u =, 1, 1. Proof. Instead of changing measure once using S 1 as the numéraire and then once more using either S 2 or S 3, we will combine S 1 and S 2 or S 3 directly. We have that QM = e rt E [ S 1 T S 2 T S 3 T + ] = e δ1t S 1 e δ2t S 2 e rt E = e r δ1 δ 2 T S 1 S 2 E [ [ e v,l T e L1 T e L 2 T 1 S3 T 1 S3 T S 2 T S 2 T + ] + ], 2.75 where v = 1, 1,. Note immediately that e v,l / MP, but since v M, M 3 we conclude that e v,l is an exponentially special semimartingale. Hence, we define a measure P on Ω, F, F t t T via the adon Nikodym derivative dp dp = e v,lt e Kv T

73 and the valuation problem takes the form [ 2.6. OPTIONS ON SEVEAL ASSETS 65 QM = e r δ1 δ 2 T S 1 S 2 E Now, using 2.55 again, we get that St 3 St 2 = S3 e r δ3 t e L3 t S 2 e r δ2 t e L2 t e Kv T + ] 1 S3 T ST 2. = S3 S 2 e δ2 δ 3 t e u,lt, t T 2.76 where u =, 1, 1. Moreover, denote by w := v +u = 1,, 1 and then, using Proposition III.3.8 in Jacod and Shiryaev 23, we get that e u,l e Kw Kv MP e u,l e v,l e w,l since e Kw Kv = MP. ekv ekw Therefore, we define the process S = S t t T with S t = S3 S 2 e δ1 +δ 2 t e u,lt e Kvt Kwt, which once discounted at the new risk-free rate δ 1 + δ 2 becomes a P - martingale. Now, using again that the exponential compensator is a deterministic process, we can conclude that [ ] + QM = e δ1 +δ 2 T SS 1 e 2 rt E e Kv T S T e δ1 +δ 3 T e Kw T = e δ1 +δ 2 T Z E [ K S T + ] where Z := S 1 S2 er δ1 δ 3 T e Kw T and K := e δ1 +δ 3 T e Kv T e Kw T.

74

75 CHAPTE 3 Valuation of exotic derivatives in Lévy models 3.1. Introduction The purpose of this chapter is twofold: on the one hand we derive general formulae for the valuation of vanilla and exotic options on single or multiple assets; on the other hand, as the title already dictates, we are particularly interested in the valuation of exotic options on assets driven by Lévy processes. The first part of this chapter is devoted to the derivation of general valuation formulae. We consider the framework of Chapter 1, where the asset is driven by a general semimartingale and aim at pricing vanilla and exotic options with arbitrary payoff functions. The options we consider are European style, in the sense that they cannot be exercised or terminated before maturity. Other than that, the variable that determines the payoff can be the asset price at maturity or some functional of the asset price, e.g. the supremum over the lifetime of the option; the sole requirement, is that the characteristic function of this variable is known. The payoff function is also an arbitrary function, subject to some mild integrability conditions; the examples include all the commonly traded contracts in financial markets, such as standard call and put options, digital options and double digital options. These formulae allow the fast pricing of European plain vanilla options, and hence the calibration to market data, for a large variety of driving processes, such as Lévy processes and stochastic volatility models driven by Lévy processes. When considering the valuation of exotic options, the situation becomes much more delicate since, in most cases, the characteristic function is not known in advance. Therefore, in the second part of this chapter we concentrate on deriving the characteristic function of the supremum of a Lévy process. This will allow us to price exotic path-dependent options on assets driven by Lévy processes, such as one-touch and lookback options. Indeed, making use of classical results from fluctuation theory, more specifically of the celebrated Wiener Hopf factorization of a Lévy process, we present an expression for the characteristic function of the supremum and subsequently apply it to valuation problems. Of course, we can similarly derive the characteristic function of the infimum of a Lévy process; this result would also have interesting applications in credit risk. However, for the sake of brevity, we have shied away from doing so. It should be pointed out that Lévy processes are becoming standard tools in financial modeling, since they can describe the observed reality in financial markets in a quite accurate way. The recent volume of Kyprianou 67

76 68 3. EXOTIC DEIVATIVES IN LÉVY MODELS et al. 25 contains an up-to-date account of applications of Lévy processes in finance. For more general considerations about Lévy processes one could refer to the books of Bertoin 1996, Sato 1999, and Applebaum 24. The books of Schoutens 23 and Cont and Tankov 23 discuss Lévy processes with particular focus on their applications in mathematical finance. This chapter is organized as follows: in section 3.2 we present the valuation formulae, while in section 3.3 some basic facts on Lévy processes and their fluctuations are presented and the characteristic function of the supremum of a Lévy process is obtained. In section 3.4 we present some examples of payoff functions commonly used in financial markets. Finally, in section 3.5 we combine all the previously obtained results and provide, as examples, valuation formulae for plain vanilla, lookback, one-touch and multi-asset options 3.2. Option valuation: general formulae 1. Let B = Ω, F, F, P be a stochastic basis where F = F T and F = F t t T. We model the price process of a financial asset, e.g. a stock or an FX rate, as an exponential semimartingale S = S t t T, i.e. a stochastic process with representation S t = e Ht, t T 3.1 shortly: S = e H, where H = H t t T is a semimartingale with H =. Every semimartingale H = H t t T admits a canonical representation H = H + B + H c + hx µ ν + x hx µ, 3.2 where B = B t t T is a predictable process of bounded variation, H c = H c t t T is the continuous martingale part of H, h = hx is a truncation function, µ = µω; ds, dx is the random measure of jumps of H and ν = νω; ds, dx is the predictable compensator of µ with respect to P. The continuous martingale part H c of H has predictable quadratic characteristic H c which will be denoted by C = C t t T. For the processes B, C, and the measure ν we use the notation TH P = B, C, ν which is called the triplet of predictable characteristics of the semimartingale H with respect to the probability measure P. Let MP, resp. M loc P, denote the class of all martingales, resp. local martingales, on the given stochastic basis B = Ω, F, F, P. 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. Subject to this assumption, we can deduce that see Chapter 1 or Eberlein, Papapantoleon, and Shiryaev 26 for details S = e H M loc P B + C 2 + ex 1 hx ν =. 3.3 Throughout this work, we assume that P is an equivalent martingale measure for the asset S. We do not elaborate on how this measure has been chosen; it can be the outcome of the minimization of a distance e.g. L 2 -distance,

77 3.2. OPTION VALUATION: GENEAL FOMULAE 69 Hellinger distance, entropic distance, etc. to the physical measure or the result of a calibration to market data of vanilla options. emark 3.1. 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 price process S = e H is a special semimartingale. 2. Let X = X t t T be any stochastic process on the given stochastic basis. We denote by X = X t t T and X = X t t T, where X t = sup X u and X t = inf X u u t u t the supremum and infimum processes of X respectively. Notice that since the exponential function is monotonically increasing, the supremum processes of S and H are related via S T = sup S t = sup t T t T e H t = e sup t T H t = e H T. 3.4 Similarly, the infimum processes of S and H are related via S T = e H T We derive and prove a general option pricing formula, analogous to aible s option pricing formula cf. aible 2, Chapter 3, which does not require the existence of a Lebesgue density for the driving process. The proof relies on the Fourier transform of the payoff function, generalizing the idea of Borovkov and Novikov 22. A similar representation of the payoff function is used for hedging purposes by Hubalek, Kallsen, and Krawczyk 26 and Černý 27. Of course, our method also has similarities and generalizes the method of Carr and Madan Note that Carr and Madan 1999 and aible 2 consider the Fourier transform of the option price instead of the payoff function. Moreover, we would like to tackle plain vanilla options, such as European call and put options, and exotic path-dependent options, such as lookback and one-touch options, in a unified framework. Therefore, assume we want to price an option on an asset S, where S = e H, with payoff fx T, where fx T = fh t, t T is an F T - measurable functional and F T = FT S = σs t, t T ; in other words, X T is a random variable that depends on the distribution of H, or some functional of H, at maturity T. The functionals we consider are European style, in the sense that the option writer or holder do not have the right to exercise or terminate the option before maturity. Nevertheless, the payoff functional can depend on the whole history of the price process and not just on the value at maturity. More specifically, the payoff functional consists of two parts: a the underlying process: it can be the asset price process or the supremum/infimum of the asset price process or an average of the asset price process. This will always be denoted by X i.e. X = H or X = H or X = H, etc.; b the payoff function: it is an arbitrary function f : +, for example fx = e x K + or fx = 1 {e x >B}, for K, B +.

78 7 3. EXOTIC DEIVATIVES IN LÉVY MODELS We want to derive a valuation formula for an option with an arbitrary payoff function, and impose the following conditions. 1: Assume that e x fxdx < for all I 1. 2: Assume that M XT z, the moment generating function of X T, exists for all z I 2. 3: Assume that I 1 I 2. Theorem 3.2. Assume that the asset price process is modeled as an exponential semimartingale process according to and conditions 1 3 are in force. Then, the price V f X of an option on S = S t t T with payoff function f = fx T is given by V f X = 1 2π ϕ XT u if f u + idu, 3.6 where ϕ XT denotes the extended characteristic function of X T and F f denotes the Fourier transform of f. Proof. According to the first fundamental theorem of asset pricing see Delbaen and Schachermayer 1994, 1998 and references therein the price of an integrable contingent claim is equal to its discounted expected payoff with respect to a martingale measure. Without loss of generality, we assume that interest rates are equal to zero. Introduce the dampened payoff function gx = e x fx, for an I 1 I 2. Since S = e H M loc P, we have V f X = E [fx T ] = fx T dp = e X T gx T dp = Ω Ω e x gxp XT dx. 3.7 Under assumption 1, we have that g L 1 = gx dx <, 3.8 hence g L 1 and the Fourier transform of g F g u = e iux gxdx is well defined for every u. Using Theorem 9.6 in udin 1987, the Fourier transform is continuous and F g L g L Applying Theorem 3.8 in udin 1987, we get that F g L 1 F g L and since g L 1 we conclude, using 3.9 and 3.8, that F g L 1. Therefore, the prerequisites of the Inversion Theorem cf. udin 1987, Theorem 9.11 are satisfied and the Fourier transform of g can be inverted; we

79 get 3.2. OPTION VALUATION: GENEAL FOMULAE 71 gx = 1 2π e ixu F g udu. 3.1 Now, returning to the valuation problem 3.7 we get that V f X = e x gxp XT dx = = 1 2π = 1 2π e x 1 e ixu F g udu P XT dx 2π e i u ix P XT dx F g udu ϕ XT u if f u + idu, 3.11 where for the third equality we have applied the Fubini theorem and for the last equality we have F g u = e iux gxdx = e iux e x fxdx = e iu+ix fxdx = F f u + i Finally, the application of Fubini s theorem is justified as follows; define the function for any x, u. Then F x, u dup XT dx = F x, u = e x e iux F g u, = K e x e iux F g u dup XT dx e x 1 F g u dup XT dx F g u du e x P XT dx e x P XT dx = KM XT <, where for the second inequality we used that F g L 1 and the finiteness of M XT for I 1 I 2 follows from Assumptions 2 and 3. Hence, we can conclude that F L 1 λ\ P XT. Theorem 3.2 is proved.

80 72 3. EXOTIC DEIVATIVES IN LÉVY MODELS emark 3.3. If the moment generating function of a random variable exists in some interval I 2, then using standard arguments from complex analysis, one can extend the characteristic function to complex arguments with imaginary part in J 2, where J 2 = I 2 where for a set [a, b] we denote by [a, b] := [ b, a]. emark 3.4. The valuation formulae of Theorem 3.2 bears, as was mentioned already before, great similarities to aible s option pricing formula; compare with Theorem 3.2 in aible 2. In addition, the prerequisites of Theorem 3.2 in aible 2 are the same as 1 3, requiring additionally the existence of a Lebesgue density for the random variable X T cf. p. 63 in aible 2. The latter is needed to establish the convolution representation of the option price, which we do not use. Let us also point out the following; the proof of the result, that the Laplace transform of a convolution equals the product of the Laplace transforms of the convolution factors, which is crucial in aible s work, relies essentially on the Fubini theorem. 4. Next, we would like to establish some valuation formulae for options that depend on two functionals of the driving process. Examples of such options are barrier options, with payoff S T K + 1 {ST >B}, K, B + and slide-in or corridor options, with payoff N S T K + 1 {L<STi <H}, H, K, L +. i=1 It should be pointed out that these options cannot be treated with the duality methods developed in Chapters 1 and 2. The setting is the familiar one: S = e H is the asset price process, where H is a semimartingale with canonical decomposition 3.2, that satisfies Assumption ES and 3.3. We also consider two payoff functions f and g, which are arbitrary mappings from to +, and two functionals X and Y of the driving process H e.g. X = H and Y = H. Moreover, we impose the following condition on the payoff functions and the underlying processes. T1: Assume that e 1x fxdx < for all 1 I 1. T2: Assume that e 2x gxdx < for all 2 I 2. T3: Assume that M XT,Y T u, v, the moment generating function of the random vector X T, Y T, exists for all u I 3 and all v I 4. T4: Assume that I 1 I 3 and I 2 I 4. Theorem 3.5. Assume that the asset price process is modeled as an exponential semimartingale process according to and conditions T1 T4 are in force. Then, the price V f,g X, Y of an option on S = S t t T with payoff function fx T gy T is given by V f,g X, Y = 1 4π 2 ϕ XT,Y T u i 1, v i 2 F g v + i 2 F f u + i 1 dvdu, 3.13

81 3.2. OPTION VALUATION: GENEAL FOMULAE 73 where ϕ XT,Y T denotes the extended characteristic function of the random vector X T, Y T. Proof. Introduce the dampened payoff function g x = e 2x gx for an 2 I 2 I 4. Define the random variable Y := gy T E[gY T ] and let us note that E[Y] = 1; in addition, we trivially have that gy T = e 2Y T g Y T. Moreover, we denote by Π := E[gY T ]; it can evaluated using Theorem 3.2 and therefore is finite. Now, we have that V f,g X, Y = E [fx T gy T ] [ ] gy T = ΠE fx T E[gY T ] = ΠE [fx T ] 3.14 where we have defined the measure P on Ω, F, F t t T via dp = YdP. We can apply the valuation formulae of Theorem 3.2 to the problem 3.14 and all we need to know is the characteristic function of X T with respect to P, denoted by ϕ X T. Using that dp = YdP, conditions T1 and T2, 3.1, 3.12 and Fubini s theorem, we get that, for all u ϕ X T u = E [ e iux ] T = e iux T dp Ω = e iux gy T T E[gY T ] dp Ω = 1 e iux e 2y g yp XT,Y Π T dx, dy = 1 Π = 1 1 Π 2π = 1 1 Π 2π e iux e 2y 1 2π e ivy F g vdv e iux vy i 2y P XT,Y T dx, dy P XT,Y T dx, dy F g v + i 2 dv ϕ XT,Y T u, v i 2 F g v + i 2 dv Therefore, returning to the valuation problem 3.14 and using 3.6, 3.15 and Assumptions T2 T4, we can conclude that V f,g X, Y = ΠE [fx T ] = Π 1 ϕ X 2π T u i 1 F f u + i 1 du = 1 4π 2 ϕ XT,Y T u i 1, v i 2 where 1 I 1 I 3. F g v + i 2 dvf f u + i 1 du, 3.16

82 74 3. EXOTIC DEIVATIVES IN LÉVY MODELS Finally, the application of the Fubini theorem is justified as follows: define the function Gx, y, v = e iux e 2 ivy F g v for x, y, v and 2 I 2 I 4. Then, we have that Gx, y, vdvp XT,Y T dx,dy= e iux e 2 ivy F g v dvp XT,Y T dx,dy e 2y F g v dv P XT,Y T dx,dy K e 2y P XT,Y T dx,dy < where K is a positive constant; hence, G L 1 λ\ P XT,Y T. The statement is proved. emark 3.6. Theorem 3.5 states that the knowledge of the joint characteristic function is sufficient for the valuation of options that depend on two functionals of the driving process. Unfortunately, such joint characteristic functions are not readily available for most of the interesting cases Lévy processes and their fluctuations The focal point of this section is to exploit results from fluctuation theory for Lévy processes to derive the characteristic function of the supremum of a Lévy process. The characteristic function of the infimum can be derived analogously. 1. Let L = L t t T be a Lévy process on the stochastic basis B = Ω, F, F, P, i.e. L is a semimartingale with independent and stationary increments PIIS. We denote the triplet of predictable characteristics of L with respect to the probability measure P by TL P = B, C, ν; moreover, we denote the triplet of local characteristics of L by b, c, λ and using Jacod and Shiryaev 23, II.4.2, the two triplets are related via B t ω = bt, C t ω = ct, νω; dt, dx = dt λdx. The triplet of predictable characteristics of a semimartingale with independent increments determines the law of the random variables generating the process. More specifically, for a Lévy process we know from the Lévy Khintchine formula that E [ e iult] = exp t iub iuhxλdx + u2 2 c + e iux 1 for all t [, T ] and all u. In the sequel we will assume that the following condition is in force. Assumption EM. There exists a constant M > 1 such that e ux λdx <, u [ M, M]. { x >1}

83 3.3. LÉVY POCESSES AND THEI FLUCTUATIONS 75 It is immediately clear that assumption EM is a mild strengthening of assumption ES. Moreover, subject to EM L is a special and exponentially special semimartingale and the use of the truncation function will be omitted cf. also emark 2.3. Applying Theorem 25.3 in Sato 1999, we have that E [ e ult] <, u [ M, M], t [, T ]. Moreover, applying Lemma 12 in Keller 1997, we have that the characteristic function of L t is holomorphic in the horizontal strip {z C : M < Iz < M} and can be represented as a Fourier integral in the complex plane; hence, for z C with M < Iz < M we have that ϕ Lt z = E [ e izlt] = e izx P Lt dx. Therefore, assumption 2 of Theorem 3.2 is satisfied with I 2 = [ M, M]. We model the asset price process S = S t t T as an exponential Lévy process, that is as a stochastic process with representation S t = S exp L t, t T 3.17 where, subject to Assumption EM, L is a special semimartingale with canonical decomposition Jacod and Shiryaev 23, II.2.38 L t = bt + t cw t + xµ L ν L ds, dx Using Assumption EM and 3.3 we have that S M loc P b + c 2 + e x 1 xνdx = 3.19 and by Lemma 4.4 in Kallsen 2, we can even conclude that S MP. Let us denote the supremum and infimum processes of L = L t t T by L = L t t T and L = L t t T respectively and, as in the previous section, we have that S = e L and S = e L. 2. We establish some results regarding the existence and analyticity of the characteristic function of the supremum of a Lévy process. The next theorem will be useful for our considerations. Theorem 3.7. Let L = L t t T be a Lévy process and define L t = sup L u, t T. u t Let gr be a submultiplicative, non-negative, continuous function on [,, increasing to as r. Then, the following are equivalent: a E[gL t ] <, t [, T ], b E[g L t ] <, t [, T ]. Proof. See Theorem 25.18, pp in Sato 1999.

84 76 3. EXOTIC DEIVATIVES IN LÉVY MODELS Making use of the assumption on exponential moments EM, we immediately have that In particular, we get that E [ e ult] <, u [ M, M], t [, T ]. Therefore, we can easily conclude that E [ e M Lt ] = e M x P Lt dx = E [ e MLt] <, and 3.2 E [ e MLt] <, t [, T ] e Mx P Lt dx + e Mx P Lt dx < 3.22 for all t [, T ].,, Lemma 3.8. Let L = L t t T be a Lévy process that satisfies assumption EM. Then, the moment generating function of L t is well defined for all u, M], for all t [, T ]. Proof. Consider the function gx = e Mx, x [,. It is a submultiplicative, non-negative, continuous function on [,, which increases to as x. Hence, it satisfies the prerequisites of Theorem 3.7. Using equation 3.22, we have that condition b of Theorem 3.7 is satisfied for this choice of g, i.e. Applying Theorem 3.7, we get that Now, it suffices to notice that E [ e M Lt ] <, t [, T ]. E [ e ML t ] <, t [, T ] L t = sup L u sup L u = L t u t u t and since M >, we have that ML t ML t. Since the exponential function is monotone increasing, we have that e MLt e ML t, therefore E [ e MLt] E [ e ML t ] < Moreover, we trivially have that for any M > and any t [, T ] E [ e M L t ] E[1] <, 3.25 because L is an increasing process starting at zero, i.e. the measure of L t is supported on [,. Lemma 3.9. Let L = L t t T be a Lévy process that satisfies assumption EM. Then, the characteristic function ϕ Lt of L t is holomorphic in the

85 3.3. LÉVY POCESSES AND THEI FLUCTUATIONS 77 half plane {z C : M < Iz < } and can be represented as a Fourier integral in the complex domain ϕ Lt z = E [ e izlt] = e izx P Lt dx. Proof. We will follow the proof of Lemma 12 in Keller Using assumption EM and Lemma 3.8, we have that the moment generating function M Lt of L t exists for u, M]. Hence, M Lt is a real analytic function and can be expanded in a power series around zero, cf. Gut 1995, emark III.3.4, that is M Lt u = n= u n n! E[L t n ], where the radius of convergence is M < u < M. Using the Extension Theorem for power series, we obtain that M Lt is holomorphic within {z C : M < Iz < M}, hence the above function can be applied to complex arguments. Taking ϕ Lt u = M Lt iu for M < u < M, we have that ϕ Lt is analytic and thus can be extended to a holomorphic function in {z C : M < Iz < M }. Now, applying Theorem in Lukacs 197, the required result is proved. 3. The next step is to establish a relationship between the characteristic function of the supremum at a fixed and at an independent and exponentially distributed time. Independent exponential times play a fundamental role in fluctuation theory for Lévy processes. Let θ denote an exponentially distributed random variable with parameter q >, independent of L = L t t T. We denote by L θ the supremum process of L sampled at an independent and exponentially distributed time, that is L θ = sup L u. u θ Lemma 3.1. Let L = L t t T be a Lévy process that satisfies assumption EM and consider β C with β [ M,. The Laplace transforms of L t, t [, T ] and L θ, θ Expq are related via E [ e βl θ ] = q Moreover, the Laplace transform of L θ [ M,. e qt E [ e βlt] dt is finite for β C with β Proof. It suffices to notice that an application of Fubini s theorem yields E [ e βl ] [ ] θ = E qe qt e βlt dt = q e qt E [ e βlt] dt and the finiteness of E[e βl θ] is a direct consequence of Fubini s theorem see e.g. Corollary 13.9 in Schilling 25.

86 78 3. EXOTIC DEIVATIVES IN LÉVY MODELS The application of Fubini s theorem is justified as follows. Define the function Ht, x = qe qt e βx for t >, q > and β = γ + ic C with β = γ [ M,. Then Ht, x dtp Lt dx = = = q qe qt e βx dtp Lt dx qe qt e γx dtp Lt dx e qt dt e γx P Lt dx e γx P Lt dx <, using Lemma 3.8. Hence, H L 1 λ\ P Lt and the result is proved. emark For an alternative proof of the existence of moments of the supremum process, we refer to Kyprianou and Surya 25, Lemma Next, we present some facts from fluctuation theory for Lévy processes that will be useful in deriving the characteristic function of the supremum of a Lévy process. Fluctuation identities for Lévy processes originate from analogous results for random walks, based on combinatorial methods, see e.g. Spitzer 1964 or Feller Greenwood and Pitman 198a,198b proved those results for random walks and Lévy processes using excursion theory. Our presentation relies on the beautiful recent book Kyprianou 26. The most celebrated result concerning fluctuation identities for Lévy processes is the so-called Wiener Hopf factorization, which serves as a common reference to a multitude of statements regarding the distributional decomposition of the excursions of a Lévy process sampled at an independent and exponentially distributed time. A statement often referred to as the Wiener Hopf factorization, relates the characteristic function of the supremum, the infimum and the Lévy process itself, sampled at an independent and exponentially distributed time, and reads as follows: E [ e izl ] θ = E [ e izl ] θ E [ e izl ] θ or equivalently, q q ψz = ϕ+ q zϕ q z, z where ψ denotes the characteristic exponent of L and ϕ + q, ϕ q are the socalled Wiener Hopf factors. In the sequel, we will make use of the following form of the Wiener Hopf factorization.

87 3.3. LÉVY POCESSES AND THEI FLUCTUATIONS 79 Theorem 3.12 Wiener Hopf factorization. Let L be a Lévy process but not a compound Poisson process. The Laplace transform of L θ at an independent and exponentially distributed time θ can be identified from the Wiener Hopf factorization of L via E [ e βl θ ] = κq, κq, β The Laplace exponent of the ascending ladder process κα, β, for α, β and k >, is given by κα, β = k exp e t e αt βx 1 t P L t dxdt Moreover, κα, β can be analytically extended to α, β C with a and β M. Proof. The first part follows directly from Theorem 6.16 ii in Kyprianou 26. The second part follows from Theorem 6.16 iii in Kyprianou 26 and Lemma The final step is to invert the Laplace transform in Theorem 3.12 to recover the characteristic function of L t. This is done in Theorem 3.14, after an auxiliary lemma. Lemma The map t E [ e βlt] is continuous for all β C with β [ M,. Proof. Since the Lévy process L is stochastically continuous and L is an increasing process, we get that L s L t a.s. as s t. We first consider real arguments and have to distinguish two cases. If β > then e βls e βlt a.s. as s t and we have that e βls 1 a.s. because L s a.s. Applying the dominated convergence theorem, we get that E [ e βls] E [ e βlt] as s t. If M β then e βls e βlt a.s. as s t. Because the random variables e βls are positive, we can apply the monotone convergence theorem to conclude E [ e βls] E [ e βlt] as s t. The proof for complex arguments is analogous, considering the real and imaginary parts separately. Theorem Let L = L t t T be a Lévy process but not a compound Poisson process. The Laplace transform of L t at a fixed time t, t [, T ], is given by E [ e βlt] = 1 ety +iv κy + iv, dv, π Y + iv κy + iv, β for Y >. Moreover, the Laplace transform can be extended to the complex plane for β C with β [ M,.

88 8 3. EXOTIC DEIVATIVES IN LÉVY MODELS Proof. Combining equations 3.26 and 3.27, we get that e qt E [ e βlt] dt = 1 q κq, κq, β. 3.3 In addition, the Laplace transform on the right hand side is convergent on the half plane q >. Therefore, applying Satz in Doetsch 195, we have that this Laplace transform can be inverted and E [ e βlt] = 1 2π = 1 2π Y +i Y i e tz z κz, κz, β dz ety +iv κy + iv, dv, 3.31 Y + iv κy + iv, β a.e. for Y >. Now, using the continuity of the map t E [ e βlt], cf. Lemma 3.13, the result follows Examples of payoff functions Almost every payoff function commonly used in mathematical finance satisfies assumption 1. Here we list the most representative examples together with their Fourier transform and the range of definition I 1. Example 3.15 Call and put option. The payoff of a call option is fx = e x K +, K +. Let z C with Iz 1,, then the Fourier transform of the payoff function of the call option is F f z = e izx e x K + dx = e izx e x K1 {x>ln K} dx Therefore, = ln K F f u + i = e 1+izx dx K ln K e izx dx = iz e1+iz ln K + K ln K eiz iz = K 1+iz iz + K Kiz iz = K1+iz iz1 + iz. K 1+iu iu 1 + iu, I 1 = 1, Similarly, for a put option, where fx = K e x +, K +, we have that F f u + i = K 1+iu iu 1 + iu, I 1 =,. 3.33

89 3.4. EXAMPLES OF PAYOFF FUNCTIONS 81 Example 3.16 Digital option. The payoff of a digital call option is 1 {e x >B}, B +. Let z C with Iz,, then the Fourier transform of the payoff function of the digital call option is Therefore, F f z = e izx 1 {x>ln B} dx = = B iz 1 iz. ln B e izx dx = 1 ln B eiz iz F f u + i = Biu iu, I 1 =, Similarly, for a digital put option, where fx = 1 {e x <B}, B +, we have that F f u + i = Biu iu, I 1 =, A variant of the digital option is the so-called asset-or-nothing digital, where the option holder receives one unit of the asset, instead of currency, if the underlying reaches, or not, some barrier. Hence, the payoff of the asset-or-nothing digital call option is fx = e x 1 {e x >B}, B +, and the Fourier transform, for z C with Iz 1,, similarly to the case of the standard digital call, is F f z = e izx e x 1 {x>ln B} dx = B 1+iz iz. Therefore, F f u + i = B1+iu 1 + iu, I 1 = 1, Similarly, for an asset-or-nothing digital put option, where fx = e x 1 {e x <B}, B +, we have that F f u + i = B1+iu 1 + iu, I 1 =, Example 3.17 Double digital option. The payoff of a double digital call option is 1 {B<e x <B}, B, B +. Let z C with Iz,,, then the Fourier transform of the double digital call option is Therefore, F f u + i = F f z = ln B e izx 1 {B<e x <B} dx = e izx dx = 1 iz eiz ln B 1 iz = 1 B iz B iz. iz eiz ln B ln B 1 B iu B iu, I 1 = \{} iu

90 82 3. EXOTIC DEIVATIVES IN LÉVY MODELS Example 3.18 Self-quanto option. The payoff of a self-quanto call option is fx = e x e x K +, K +. Let z C with Iz 2,, then the Fourier transform of the payoff function of the self-quanto call option, similarly to the case of the standard call option, is F f z = e izx e x e x K1 {x>ln K} dx Therefore, F f u + i = = ln K e 2+izx dx K ln K e 1+izx dx = K 2+iz iz + K K1+iz 1 + iz K 2+iz = 1 + iz2 + iz. K 2+iu 1 + iu 2 + iu, I 1 = 2, Similarly, for a self-quanto put option, where fx = K e x +, K +, we have that F f u + i = K 2+iu 1 + iu 2 + iu, I 1 =, Applications A. European options. Assume we are interested in pricing a European plain vanilla option on the asset S = e H, e.g. a call or put option or a digital option. Then, it is sufficient to know the characteristic function of the random variable X T H T, and H T must possess a moment generating function in [, 1] I 2. Note that the restriction [, 1] I 2 stems from no-arbitrage considerations and is directly related to Assumption ES. Examples of options that can be treated include plain vanilla call and put options, with payoffs S T K + and K S T +, digital cash-or-nothing and asset-or-nothing options, with payoffs 1 {ST >B} and S T 1 {ST >B}, double digital options, with payoff 1 {B<ST <B}, and self-quanto options. The price of the option follows by simply using Theorem 3.2 with the given ϕ HT and the Fourier transform of the corresponding payoff function from section 3.4; therefore, we do not elaborate further. Below we describe some characteristic examples of models used in mathematical finance. Example 3.19 Black Scholes model. In the standard Black and Scholes 1973 model, the driving process H = H t t T follows a Brownian motion with LawH 1 P = Normalµ, σ 2, σ >. The characteristic function of H t, t [, T ], is ϕ Ht u = exp iuµt u2 σ 2 2 t, 3.41 and the moment generating function exists for I 2 =.

91 3.5. APPLICATIONS 83 Example 3.2 Lévy models. In exponential time-inhomogeneous Lévy models, the driving process is a time-inhomogeneous Lévy process with triplet of semimartingale characteristics B, C, ν. The triplet of characteristics determines the law of the random variables generating the process. Indeed, using the Lévy Khintchine formula we have that, for all t [, T ] E [ e iult] t = exp iub s + u 2 c s 2 + e iux 1 iuhxλ s dx ds Moreover, subject to a moment condition such as Assumption EM, we can extend the characteristic function to the complex plane, with I 2 = [ M, M]. Therefore, the methods developed here are particularly suitable for the class of Lévy models. 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. The characteristic function of H 1 is ϕ H1 u = e iuµ α 2 β 2 α 2 β + iu 2 λ 2 Kλ δ α 2 β + iu 2, 3.43 K λ δ α 2 β2 where K λ denotes the Bessel function of the third kind with index λ cf. Abramowitz and Stegun 1968 and the moment generating function exists for I 2 = α β, α β. The class of generalized hyperbolic distributions is not closed under convolution, hence the law or the characteristic function of the random variable H t is not known in closed form; of course, using the properties of Lévy processes, we have that ϕ Ht u = ϕ H1 u t. The only exception is the class of normal inverse Gaussian distributions, where λ = 1 2 cf. Barndorff-Nielsen In that case, LawH t P = NIGα, β, δt, µt and the characteristic function resumes the form ϕ Ht u = e iuµt expδt α 2 β 2 expδt α 2 β + iu 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 see e.g. Cont and Tankov 23. The characteristic function of H t, t [, T ], is ϕ Ht u = exp tcγ Y [ M iu Y + G + iu Y M Y G Y ] 3.45 where C, G, M > and Y < 2, and the moment generating function exists for I 2, depending on the parameters C, G, M, Y. For example, if Y = then we recover the bilateral gamma distribution Küchler and Tappe 26 and the moment generating function exists for I 2 = G, M.

92 84 3. EXOTIC DEIVATIVES IN LÉVY MODELS 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 characteristic function H t, t [, T ], is cos β 2δt 2 ϕ Ht u = cosh αu iβ and the moment generating function exists for I 2, depending on the parameters α, β, δ. Example 3.21 Stochastic volatility Lévy models. This class of models was proposed by Carr, Geman, Madan, and Yor 23 and further investigated in Schoutens 23. Let X = X t t T be a pure jump Lévy process and Y = Y t t T be an increasing process, independent of X. The process Y acts as a stochastic clock measuring activity in business time and has the form Y t = t y s ds where y = y s s T is a positive process. Carr et al. 23 consider the CI process as a candidate for y, i.e. the solution of the stochastic differential equation dy t = Kη y t dt + λy 1 2 t dw t, where W = W t t T is a standard Brownian motion. For other choices of Y see Schoutens 23, Chapter 7. The stochastic volatility Lévy process is defined by time-changing the Lévy process X with the increasing process Y, that is H t = X Yt, t T. If we denote the characteristic functions of X t and Y t by ϕ Xt and ϕ Yt respectively, then the characteristic function of H t, t [, T ], is given by ϕ Ht u = ϕ Y t iϕ Xt u ϕyt iϕ Xt i iu 3.47 cf. Schoutens 23, section 7.3. The moment generating function exists for I 2, depending on the choice of X and Y. Example 3.22 BNS model. The BNS model incorporates stochastic volatility and jump effects in the asset price process; it was introduced by Barndorff-Nielsen and Shephard 21. The driving process satisfies the SDE where dh t = µ + βσ 2 t dt + σ t dw t + ρdz λt, H = dσ 2 t = λσ 2 t dt + dz λt, σ 2 >, t T. Here, µ, β, λ > and ρ, W = W t t T is a standard Brownian motion and Z = Z t t T is a subordinator with triplet,, ν.

93 3.5. APPLICATIONS 85 The moment generating function of H T was derived in Nicolato and Venardos 23 and is given by M HT u=exp zµt + z 2 + 2βz ɛt T 2 σ2 + λ ds, + e fs,zx 1νdx where ɛt = 1 e λt λ and fs, z = ρz z2 + 2βzɛT. An efficient numerical algorithm for calculating this function has been developed in Keller-essel 26. Moreover, the moment generating function can be analytically extended in the complex domain, to an open strip of the form {z C : z θ, θ + }; see Nicolato and Venardos 23, p. 45. B. Lookback options. The results concerning the characteristic function of the supremum of a Lévy process, cf. section 3.3, allow us to price lookback options in models driven by Lévy processes. Indeed, assuming that the asset price evolves according to 3.17, 3.18 and 3.19, a fixed strike lookback call option with payoff S T K + can easily be viewed as a call option with driving process the supremum of the underlying Lévy processes L. Therefore, combining Theorem 3.2, Theorem 3.14 and Example 3.15, we get that where C T S; K = 1 2π ϕ LT u i = 1 2π K 1+iu ϕ LT u i du, 3.48 iu 1 + iu et Y +iv κy + iv, dv, 3.49 Y + iv κy + iv, iu for 1, M] and Y >. Of course, floating strike lookback options can be treated by the same formulae, making use of the duality relationships in Chapter 1, cf. Theorem 1.2. C. One-touch options. Similarly, we can treat one-touch options on assets driven by Lévy processes. Assuming that the asset price evolves according to 3.17, 3.18 and 3.19, a one-touch call option with payoff 1 {ST >B} can easily be priced using the developed methods, i.e. as a digital call option where the driving process is the supremum of the underlying Lévy process. Indeed, combining Theorem 3.2, Theorem 3.14 and Example 3.16, we get that DC T S; B = 1 4π 2 where, M] and Y >. et Y +iv Y + iv κy + iv, dvdu, 3.5 κy + iv, iu iu B iu

94 86 3. EXOTIC DEIVATIVES IN LÉVY MODELS emark There are a number of articles that address the pricing of exotic options, such as touch, barrier and lookback options in jump-diffusion and general Lévy models. Kou and Wang 23, 24 derived formulas for the value of lookback and barrier options in a jump diffusion model where the jumps are double-exponentially distributed; similar results where also proved by Lipton 22. Boyarchenko and Levendorskiǐ 22 applied methods from potential theory and pseudodifferential operators, while Nguyen-Ngoc and Yor 25 use a probabilistic approach based on excursion theory. Finally, Nguyen-Ngoc 23 takes a similar probabilistic approach, combined with the method of Carr and Madan 1999; nevertheless, the formulae he derives for lookback options are quite more involved than D. Options on two assets. As a final application we consider a twoasset correlation option. These options have the payoff of a vanilla option on the payment asset, denoted by S 1, if the measurement asset, denoted by S 2, ends up in the money; thus, a special case of these options is the correlation digital option, considered in Chapter 2. In the case of the correlation call option, the payoff is S 1 T K + 1 {S 2 T >B}. Assume, for simplicity, that the process driving the two assets is an 2 -valued time-inhomogeneous Lévy process L = L 1, L 2, that satisfies Assumption EM, cf. p. 38. The characteristic function of L t, t [, T ], is described by 2.1 and the dynamics of the asset price processes S 1 and S 2 are described by 2.21, 2.22 and Then, applying Theorem 3.5, Example 3.15 and Example 3.16, we get that TAC T S 1, S 2 ; K, B = 1 4π 2 ϕ LT u i 1, v i 2 where 1 1, M] and 2, M]. K 1+iu 1 B iu 2 iu iu 1 2 iu dvdu, 3.51

95 Part 2 Term structure models

96 The musical notes are only five in number but their melodies are so numerous that one cannot hear them all. The primary colors are only five in number but their combinations are so infinite that one cannot visualize them all. The flavors are only five in number but their blends are so various that one cannot taste them all. In battle there are only the normal and extraordinary forces, but their combinations are limitless; none can comprehend them all. Sun Tzu, The Art of War.

97 CHAPTE 4 Duality and valuation in Lévy term structure models 4.1. Introduction The modeling of the term structure of interest rates is a contemporary topic of research that has attracted the attention of academics and practitioners alike. On the one hand, the market of over-the-counter OTC interest rate derivative products represents the largest part of the OTC traded derivatives; indeed, out of an estimated total amount of 284,819 billion US$ in December 25, an amount of 215,237 billion US$ corresponds to interest rate contracts see BIS Quarterly eview, September 26, p. A13. Therefore, appropriate models for the term structure of interest rates are of vital interest for fund managers and derivatives traders operating in these markets. On the other hand, interest rate theory represents a unique challenge for researchers in mathematical finance; contrary to stock markets that consist of a finite number of traded assets, bond markets consist of the entire term structure of interest rates: an infinite dimensional object. Another difference between stock markets and bond markets is the modeling object; in stock markets, the individual traded stocks are subject to modeling. In bond markets, there are different quantities that can be modeled. We start by describing these quantities and outlining some of the models proposed in the literature. Of course, we do not aim at presenting a selfcontained introduction to interest rate theory here, there are many excellent textbooks available for that purpose; let us just refer to Björk 24, Brigo and Mercurio 26, Hunt and Kennedy 24 and Musiela and utkowski 25. A zero coupon bond is a financial instrument that pays off one unit of currency at maturity. The time-t price of a zero coupon bond with maturity T t T is denoted by Bt, T. Instantaneous continuously compounded forward rates are mathematical objects, rather than traded interest rates; however, they have proved to be very convenient objects for modeling purposes. Let ft, T denote the instantaneous forward rate at time t for investing money over an infinitesimal time period starting at time T. Instantaneous forward rates are formally defined by ft, T := T log Bt, T. Then, zero coupon bond prices can be deduced from forward rates using the relationship Bt, T = exp T t ft, udu. The instantaneous forward rate prevailing at time t for immediate borrowing or lending is called short or spot rate. It is usually denoted by r t and the following relationship holds: r t = ft, t. An amount of one unit of currency that is continuously reinvested at the short rate, from time zero 89

98 9 4. LÉVY TEM STUCTUE MODELS until time T, yields the amount BT M = exp T r udu. The quantity B M is called the money market account or discount factor. The LIBO rate is a rate where interest accrues according to a discrete grid, the tenor structure. The LIBO rate for investing at time T for a period of length δ, is denoted LT, T and is related to zero coupon bond prices via LT, T := 1 BT,T δ BT,T +δ. 1 The forward LIBO rate Lt, T is the time-t LIBO for investing one currency unit from time T until time T +δ. Finally, F t, T, U denotes the forward price at time t of the T -maturity zero coupon bond with settlement date U. Summarizing, the prices of zero coupon bonds, the forward LIBO rate and the forward price are related in the following way: 1 + δlt, T = Bt, T = F t, T, T + δ. 4.1 Bt, T + δ The different approaches to modeling the term structure of interest rates correspond to different choices of modeling objects from the above mentioned quantities. The most classical models used the short rate as modeling object; however, this approach is not followed here. A standard approach to modeling the term structure of interest rates is that of Heath, Jarrow, and Morton In the Heath Jarrow Morton henceforth HJM framework subject to modeling are instantaneous continuously compounded forward rates, or equivalently bond prices, which are driven by a d-dimensional Wiener process. However, data from bond markets do not support the use of the normal distribution. Empirical evidence for the non-gaussianity of daily returns from bond market data can be found in aible 2, chapter 5; the fit of the normal inverse Gaussian distribution to the same data is particularly good, supporting the use of Lévy processes for modeling interest rates. Similar evidence appears in the risk-neutral world, i.e. from caplet implied volatility smiles and surfaces; see Eberlein and Kluge 26a. The Lévy forward rate model was developed in Eberlein and aible 1999 and extended to time-inhomogeneous Lévy processes in Eberlein, Jacod, and aible 25. In these models, forward rates are driven by a time inhomogeneous Lévy process; therefore, the model allows to accurately capture the empirical dynamics of interest rates, while it is still analytically tractable, so that closed form valuation formulas for liquid derivatives can be derived. Valuation formulas for caps, floors, swaptions and range notes have been derived in Eberlein and Kluge 26a, 26b. The method relies on a convolution representation of the option value and the use of Laplace transforms, beautifully described in aible 2. Estimation and calibration methods are discussed in Eberlein and Kluge 26a, 27. Eberlein, Jacod, and aible 25 provide a complete classification of all equivalent martingale measures in the Lévy forward rate model. They also prove that in certain situations essentially, if the driving process is 1- dimensional the set of equivalent martingale measures becomes a singleton. This result might not be too surprising; roughly speaking, there are infinitely many risk factors, the jumps, and infinitely many hedging instruments, the entire term structure of interest rates, which render the market complete.

99 4.1. INTODUCTION 91 There have been several other extensions of the HJM framework proposed in the literature. We just mention here Björk et al. 1997, where forward rates are driven by a d-dimensional Wiener process and a general random measure, and Özkan and Schmidt 25, where forward rates are driven by an infinite dimensional Lévy process. The main pitfall of the HJM framework is the assumption of continuously compounded rates, while in real markets interest accrues according to a discrete grid, the tenor structure. LIBO market models, that is, arbitragefree term structure models on a discrete tenor, were constructed in a series of articles by Sandmann et al. 1995, Miltersen et al. 1997, Brace et al. 1997, and Jamshidian The backward induction construction of the log-normal LIBO market model was carried out by Musiela and utkowski In addition, LIBO market models are consistent with the market practice of pricing caps and floors using Black s formula Black Nevertheless, a familiar phenomenon appears: since the model is driven by a Brownian motion, it cannot be calibrated accurately to the whole term structure of volatility smiles. As a remedy, Eberlein and Özkan 25 developed a LIBO model driven by time inhomogeneous Lévy processes. Valuation methods for caps and floors using an approximation already employed by Schlögl 22 and other authors and aible s method, were presented in Eberlein and Özkan 25. Kluge 25 uses other approximations together with aible s method to value caps, floors and swaptions in the Lévy LIBO model. Calibration issues for this model are discussed in Eberlein and Kluge 27; see also Belomestny and Schoenmakers 26. The Lévy forward price model is a market model based on the forward price rather than the LIBO rate and driven by time inhomogeneous Lévy processes; it was put forward by Eberlein and Özkan 25, pp A detailed construction of the model is presented in Kluge 25, Chapter 3; there, it is also shown how this model can be embedded in the Lévy forward rate model. Although the LIBO rate and the forward price differ only by an additive and a multiplicative constant, see 4.1, the two specifications lead to models with very different qualitative and quantitative behavior. In the LIBO model, LIBO rates change by an amount relative to their current level, while in the forward price model changes do not depend on the actual level cf. Kluge 25, p. 6. There are authors who claim that models based on the forward process also coined arithmetic or Bachelier LI- BO models are able to better describe the dynamics of the market than classical LIBO models; see Henrard 25. Another advantage of the forward price model is that the driving process remains a time-inhomogeneous Lévy process under each forward measure, hence this model is particularly suitable for practical implementation. The downside is that negative LIBO rates can occur, like in an HJM model; this comes as no surprise, since this model can be embedded in the Lévy forward rate framework. The aim of this chapter is two-fold; on the one hand, we derive duality results relating caplets and floorlets in term structure models driven by Lévy processes. On the other hand, we derive valuation formulas for an exotic path

100 92 4. LÉVY TEM STUCTUE MODELS dependent interest rate derivative, the option on the composition of LIBO rates. This chapter is organized as follows: in section 4.2 we briefly describe time-inhomogeneous Lévy processes; for a detailed account, we refer to section 2.2. In section 4.3 we review three different approaches to modeling interest rates based on Lévy processes; namely, an HJM forward rate model, a LIBO model and a model for forward prices. In section 4.4 we provide caplet-floorlet dualities in each of these models and in section 4.5 we derive valuation formulas for the option on the composition in the forward rate and forward price models Time-inhomogeneous Lévy processes Let Ω, F, F, IP be a complete stochastic basis, where F = F T and the filtration F = F t t [,T ] satisfies the usual conditions; we assume that T + is a finite time horizon. The driving process L = L t t [,T ] is a time-inhomogeneous Lévy process, or a process with independent increments and absolutely continuous characteristics, in the sequel abbreviated PIIAC. Therefore, according to Definition 2.1, L is an adapted, càdlàg, real-valued stochastic process with independent increments, starting from zero, where the law of L t, t [, T ], is described by the characteristic function IE [ e iult] t = exp ib s u c s 2 u2 + e iux 1 iuxλ s dx ds, 4.2 where b t, c t + and λ t is a Lévy measure, i.e. satisfies λ t {} = and 1 x 2 λ t dx <, for all t [, T ]. In addition, the process L satisfies Assumptions AC and EM given below. Assumption AC. The triplets b t, c t, λ t satisfy T b t + c t + 1 x 2 λ t dx dt <. 4.3 Assumption EM. There exist constants M, ε > such that for every u [ 1 + εm, 1 + εm] T { x >1} expuxλ t dxdt <. 4.4 Moreover, without loss of generality, we assume that { x >1} eux λ t dx < for all t [, T ] and u [ 1 + εm, 1 + εm]. These assumptions render the process L = L t t T a special semimartingale, therefore it has the canonical decomposition cf. Jacod and Shiryaev 23, II.2.38, and Eberlein et al. 25 L t = t t b s ds + t cs dw s + xµ L νds, dx, 4.5

101 4.2. TIME-INHOMOGENEOUS LÉVY POCESSES 93 where µ L is the random measure of jumps of the process L and W = W t t T is a IP-standard Brownian motion. The triplet of predictable or semimartingale characteristics of L with respect to the measure P, TL P = B, C, ν, is B t = t b s ds, C t = t c s ds, ν[, t] A = t A λ s dxds, 4.6 where A B. The triplet b, c, λ represents the local or differential characteristics of L. In addition, the triplet of semimartingale characteristics B, C, ν determines the distribution of L, cf. Lemma 2.5. We denote by θ s the cumulant associated with the infinitely divisible distribution with Lévy triplet b s, c s, λ s, i.e. for z [ 1 + εm, 1 + εm] θ s z := b s z + c s 2 z2 + e zx 1 zxλ s dx. 4.7 In addition, we can extend θ s to the complex domain C, for z C with z [ 1 + εm, 1 + εm] and the characteristic function of L t can be written as IE [ e iult] t = exp θ s iuds. 4.8 If L is a time-homogeneous Lévy process, then b s, c s, λ s and thus also θ s do not depend on s. In that case, θ equals the cumulant log-moment generating function of L 1. Lemma 4.1. Let L = L t t T be a time-inhomogeneous Lévy process satisfying assumption EM and f : + C a continuous function such that f M. Then [ t ] t IE exp fsdl s = exp θ s fs ds. 4.9 The integrals are to be understood componentwise for real and imaginary part. Proof. The proof is similar to the proof of Lemma 3.1 in Eberlein and aible 1999; see also Kluge 25, Proposition 1.9. Lemma 4.2. Let L = L t t T be a time-inhomogeneous Lévy process with characteristic triplet B, C, ν, satisfying assumption EM. Then L := L is again a time-inhomogeneous Lévy process satisfying assumption EM, with triplet B, C, ν, where B = B C = C A x ν = 1 A x ν, A B\{}. Proof. See Lemma 2.14.

102 94 4. LÉVY TEM STUCTUE MODELS 4.3. Lévy fixed income models In this section we review three different approaches to modeling the term structure of interest rates, where the driving process is a time-inhomogeneous Lévy process The Lévy forward rate model. In the Lévy forward rate framework for modeling the term structure of interest rates, the dynamics of forward rates are specified and the prices of zero coupon bonds are then deduced. Let T be a fixed time horizon and assume that for every T [, T ], there exists a zero coupon bond maturing at T traded in the market; in addition, let U [, T ]. The forward rates are driven by a time-inhomogeneous Lévy process L = L t t [,T ] on the stochastic basis Ω, F, F, IP with semimartingale characteristics B, C, ν or local characteristics b, c, λ. The dynamics of the instantaneous continuously compounded forward rates for T [, T ] is given by t ft, T = f, T + t αs, T ds σs, T dl s, t T The initial values f, T are deterministic, and bounded and measurable in T. In general, α and σ are real-valued stochastic processes defined on Ω [, T ] [, T ] that satisfy the following conditions: A1: for s > T we have αω; s, T = and σω; s, T =. A2: ω, s, T αω; s, T, σω; s, T are P B[, T ]-measurable. A3: Sω := sup s,t T αω; s, T + σω; s, T <. Then, 4.11 is well defined and we can find a joint version of all ft, T such that ω; t, T ft, T ω1 {t T } is O B[, T ]-measurable. Taking the dynamics of the forward rates as the starting point, explicit expressions for the dynamics of zero coupon bond prices and the money market account can be deduced cf. Proposition 5.2 in Björk et al From Eberlein and Kluge 26b, 2.6, we get that the time-t price of a zero coupon bond maturing at time U is B, U BT, U = B, T exp T T Σs, T, UdL s where the following abbreviations are used: and Σs, T, U := Σs, U Σs, T, As, T, U := As, U As, T, As, T, Uds, 4.12 As, T := T s T αs, udu and Σs, T := σs, udu. T s T

103 4.3. LÉVY FIXED INCOME MODELS 95 Similarly, using Eberlein and Kluge 26b, 2.5, we have for the money market account BT M 1 T T = B, T exp As, T ds Σs, T dl s In the sequel we will consider only deterministic volatility structures. Therefore, Σ and A are assumed to be deterministic real-valued functions defined on := {s, T [, T ] [, T ]; s T }, whose paths are continuously differentiable in the second variable. Moreover, they satisfy the following conditions. B1: The volatility structure Σ is continuous in the first argument and bounded in the following way: for s, T we have Σs, T M, where M is the constant from Assumption EM; for the duality results, we will assume that Σs, T M 2. Furthermore, Σs, T for s < T and ΣT, T = for T [, T ]. B2: The drift coefficients A, T are given by As, T = θ s Σs, T, 4.14 where θ s is the cumulant associated with the triplet b s, c s, λ s, s [, T ]. emark 4.3. The drift condition 4.14 guarantees that bond prices discounted by the money market account are martingales; hence, IP is a martingale measure. In addition, from Theorem 6.4 in Eberlein et al. 25, we know that the martingale measure is unique The Lévy LIBO model. In the Lévy LIBO model, the forward LIBO rate is modeled directly. Let = T < T 1 < T 2 < < T N < T N+1 = T denote a discrete tenor structure where δ i = T i+1 T i, i {, 1,..., N}; since the model is constructed via backward induction, we denote by T j := T N+1 j for j {, 1,..., N + 1} and δ j := δ N+1 j for j {1,..., N + 1}. Consider a complete stochastic basis Ω, F, F, IP T and let L = L t t [,T ] be a time-inhomogeneous Lévy process satisfying Assumption EM. L has semimartingale characteristics, C, ν T or local characteristics, c, λ T and its canonical decomposition is L t = t t cs dw T s + xµ L ν T ds, dx, 4.15 where W T is a IP T -standard Brownian motion, µ L is the random measure associated with the jumps of L and ν T is the IP T -compensator of µ L. We further assume that the following conditions are in force. L1: For any maturity T i there exists a bounded, continuous, deterministic function λ, T i : [, T i ], which represents the

104 96 4. LÉVY TEM STUCTUE MODELS volatility of the forward LIBO rate process L, T i. Moreover, N λs, Ti M, i=1 for all s [, T ], where M is the constant from Assumption EM and λs, T i = for all s > T i. L2: The initial term structure B, T i, 1 i N + 1, is strictly positive and strictly decreasing. Consequently, the initial term structure of forward LIBO rates is given, for 1 i N, by L, T i = 1 δ i B, Ti B, T i + δ i 1 The construction starts by postulating that the dynamics of the forward LIBO rate with the longest maturity L, T1 is driven by the timeinhomogeneous Lévy process L, and evolve as a martingale under the terminal forward measure IP T. Then, the dynamics of the LIBO rates for the preceding maturities are constructed by backward induction; therefore, they are driven by the same process L and evolve as martingales under their associated forward measures. Let us denote by IP T j 1 the forward measure associated to the settlement date Tj 1, j {1,..., N + 1}. The dynamics of the forward LIBO rate L, Tj, for an arbitrary T j, is given by t t Lt, Tj = L, Tj exp b L s, Tj, Tj 1ds + λs, Tj dl T j 1 s, where L T j 1. is a special semimartingale with canonical decomposition t L T j 1 t = t cs dw T j 1 s xµ L ν T j 1ds, dx Here W T j 1 is a IP T j 1 -standard Brownian motion, and ν T j 1 is the IP T j 1 - compensator of µ L. The dynamics of an arbitrary LIBO rate again evolves as a martingale under its corresponding forward measure; therefore, we specify the drift term of the forward LIBO process L, T j as b L s, Tj, Tj 1 = 1 2 λs, T j 2 c s e λs,tj x 1 λs, Tj x λ T j 1 s dx The forward measure IP T j 1, which is defined on Ω, F, F t t T j 1, is related to the terminal forward measure IP T via dip T j 1 dip T = j 1 k=1 j 1 B, Tj δ k LT j 1, T k 1 + δ k L, T k = B, T k=1 1 + δ k LT j 1, T k.

105 4.3. LÉVY FIXED INCOME MODELS 97 Additionally, W T j 1 is a IP T j 1 -Brownian motion which is related to the IP T -Brownian motion via where W T j 1 t = W T j 2 t = W T t t t j 1 αs, T j 1, T j 2 c s ds =... αs, Tk, T k 1 cs ds, k=1 αt, T k, T k 1 = δ klt, T k 1 + δ k Lt, T k λt, T k. Similarly, ν T j 1 is the IP T j 1 -compensator of µ L and is related to the IP T - compensator of µ L via where ν T j 1ds, dx = βs, x, Tj 1, Tj 2ν T j 2ds, dx =... j 1 = βs, x, Tk, T k 1 ν T ds, dx, 4.19 k=1 βt, x, Tk, T k 1 = δ klt, Tk 1 + δ k Lt, Tk e λt,t k x emark 4.4. Notice that the process L T j 1, driving the forward LIBO rate L, Tj, and L = LT have the same martingale parts and differ only in the finite variation part drift. An application of Girsanov s theorem for semimartingales yields that the IP T j 1 -finite variation part of L is j 1 c s k=1 αs, Tk, T k 1 ds + x j 1 k=1 βs, x, T k, T k 1 1 ν T ds, dx. emark 4.5. The process L = L T driving the most distant LIBO rate L, T1 is by assumption a time-inhomogeneous Lévy process. However, this is not the case for any of the processes L T j 1 driving the remaining δlt,tk LIBO rates, because the random terms 1+δLt,Tk enter into the compensators ν T j 1 during the construction; see equations 4.19 and The Lévy forward price model. In the Lévy forward price model the dynamics of forward prices, i.e. ratios of successive bond prices, are specified. Let = T < T 1 < T 2 < < T N < T N+1 = T denote a discrete tenor structure where δ i = T i+1 T i, i {, 1,..., N}; the model is again constructed via backward induction, hence we denote by T j := T N+1 j for j {, 1,..., N + 1} and δ j := δ N+1 j for j {1,..., N + 1}. Consider a complete stochastic basis Ω, F, F, IP T and let L = L t t [,T ] be a time-inhomogeneous Lévy process satisfying Assumption EM. L has

106 98 4. LÉVY TEM STUCTUE MODELS semimartingale characteristics, C, ν T or local characteristics, c, λ T and its canonical decomposition is L t = t t cs dw T s + xµ L ν T ds, dx, 4.21 where W T is a IP T -standard Brownian motion, µ L is the random measure associated with the jumps of L and ν T is the IP T -compensator of µ L. Moreover, we assume that the following conditions are in force. FP1: For any maturity T i there exists a bounded, continuous, deterministic function λ, T i : [, T i ], which represents the volatility of the forward price process F, T i, T i + δ i. Moreover, we require that the volatility structure satisfies i λs, T k M, k=1 i {1,..., N}, for all s [, T ], where M is the constant from Assumption EM and λs, T i = for all s > T i. FP2: The initial term structure B, T i, 1 i N + 1 is strictly positive. Consequently, the initial term structure of forward price processes is given, for 1 i N, by F, T i, T i + δ i = B, T i B, T i + δ i. The construction starts by postulating that the dynamics of the forward process with the longest maturity F, T1, T are driven by the timeinhomogeneous Lévy process L, and evolve as a martingale under the terminal forward measure IP T. Then, the dynamics of the forward processes for the preceding maturities are constructed by backward induction; therefore, they are driven by the same process L and evolve as martingales under their associated forward measures. Let us denote by IP T j 1 the forward measure associated with the settlement date Tj 1, j {1,..., N + 1}. The dynamics of the forward price process F, Tj, T j 1 is given by F t, Tj, Tj 1=F, Tj, Tj 1 exp where t L T j 1 t = t t cs dw T j 1 s + bs, T j, T j 1ds+ t xµ L ν T j 1ds, dx is a time-inhomogeneous Lévy process. Here W T j 1 λs, Tj dl T j 1 is a IP T j 1 -standard Brownian motion and ν T j 1 is the IP T j 1 -compensator of µ L. The forward s

107 4.4. CAPLET-FLOOLET DUALITY 99 price process evolves as a martingale under its corresponding forward measure, hence, we specify the drift of the forward price process to be bs, Tj, Tj 1 = 1 2 λs, T j 2 c s e λs,t j x 1 λs, Tj x λ T j 1 s dx The forward measure IP T j 1, which is defined on Ω, F, F t t T j 1, is related to the terminal forward measure IP T via dip T j 1 dip T = j 1 k=1 j 1 B, Tj 1 F T j 1, T k, T k 1 F, T k, T k 1 = B, T k=1 F T j 1, T k, T k 1. In addition, the IP T j 1 -Brownian motion is related to the IP T -Brownian motion via W T j 1 t = W T j 2 t = W T t t t j 1 λs, T j 1 c s ds =... λs, Tk cs ds Similarly, the IP T j 1 -compensator of µ L is related to the IP T -compensator of µ L via k=1 ν T j 1ds, dx = exp λs, Tj 1x ν T j 2ds, dx =... j 1 = exp x λs, Tk ν T ds, dx k=1 emark 4.6. The process L = L T, driving the most distant forward price, and L T j 1, driving the forward price F, Tj, T j 1, are both timeinhomogeneous Lévy processes, sharing the same martingale parts and differing only in the finite variation parts. Applying Girsanov s theorem for semimartingales yields that the IP T j 1 -finite variation part of L is j 1 c s λs, Tk ds + k=1 j 1 x exp x λs, Tk 1 ν T ds, dx. k= Caplet-floorlet duality The well-known caplet-floorlet parity relates caps and floors of the same strike and time to maturity; let LT, T denote the LIBO rate for the period [T, T +δ], then the values of a caplet and a floorlet with strike K and payoff δlt, T K + and δk LT, T + respectively, are related via C K, T = F K, T + B, T 1 + δkb, T + δ,

108 1 4. LÉVY TEM STUCTUE MODELS where B, T denotes the price of a zero coupon bond maturing at T and C K, T and F K, T denote the present value of a caplet and a floorlet respectively, with strike rate K maturing at T. This section aims at providing duality relationships between caplets and floorlets with different strikes but the same time of maturity and moneyness, in term structure models driven by time-inhomogeneous Lévy processes. By moneyness of a caplet resp. floorlet, we mean the ratio of the initial forward LIBO rate over the strike resp. the reciprocal of this ratio. In equity markets there is a long list of articles discussing similar results, with driving processes of increasing generality; see the literature review in the introduction of Chapter 1. Apart from providing better understanding of valuation formulas and simplifying computational work, such results are applied for statically hedging other usually exotic derivatives; see, e.g. Carr et al The proofs are based on the choice of a suitable numéraire and the subsequent change of the probability measure; this method was pioneered by Geman et al Three different approaches to modeling interest rates are considered: a Heath Jarrow Morton forward rate model, a model for the LIBO, and a model for the forward price, driven by time inhomogeneous Lévy processes, as described in the previous section Duality in the Lévy forward rate model. In this section we derive a duality relationship between call and put options on zero coupon bonds. As a direct corollary of this result, we obtain a duality relating caplets and floorlets in the Lévy forward rate model. For the duality result, we define the constant D via [ ] BT, U D := IE B M 2 T = IE [ B, UB, T exp exp T = B, UB, T exp exp T T Σs, U + Σs, T dls As, U + As, T ds T ] θ s Σs, T, U ds θ s Σs, U + θ s Σs, T ds where we used the abbreviation Σs, T, U := Σs, U+Σs, T. In addition, we define a measure ĨP via the adon Nikodym derivative dĩp dip BT, U := D BT M 2 = BT, U [ B M 2 T IE BT,U BT M 2, ] = Z T, 4.25

109 [ noting that IE BT,U 4.4. CAPLET-FLOOLET DUALITY 11 ] = 1 and that the two measures, IP and ĨP, are DBT M 2 equivalent since Z T is strictly positive. The density process Z = Z t t [,T ] related to this change of measure is given by the restriction of the adon Nikodym derivative to the σ-field F t, i.e. for t T, we get [ ] BT, U Z t = IE 2 F t = IE = exp D BT M [ exp t T T Σs, T, UdL s Σs, T, UdL s t θ s Σs, T, U ds F t ] θ s Σs, T, U ds. emark 4.7. Notice that using B1, we have that Σs, T dl s is well defined. Moreover, from B1 and EM we get that Σs, T dl s is exponentially special cf. Kallsen and Shiryaev 22a, Definition Applying Theorem 2.18 in Kallsen and Shiryaev 22a we have that [ t t ] exp Σs, T, UdL s θ s Σs, T, U ds t [,T ] is a martingale and the last equality follows. Alternatively, this follows from Lemma 4.1 and Assumption B1. Now, using the canonical decomposition of L and 4.7, we can rewrite the density process in the usual form Z t = exp t + Σs, T, U t c s dw s t t 2 cs Σs, T, U 2 ds xσs, T, Uµ L νds, dx e xσs,t,u 1 xσs, T, U νds, dx. Equivalently, we can express Z as the stochastic exponential of a suitable time-inhomogeneous Lévy process denoted by X, that is Z = EX where X := Σs, T, U c s dw s + e xσs,t,u 1 µ L νds, dx; here, we have applied Lemma 2.6. in Kallsen and Shiryaev 22a.

110 12 4. LÉVY TEM STUCTUE MODELS Proposition 4.8. The triplet of local characteristics of L = L t t [,T ] under the measure ĨP is bs = b s + β s c s + xy s, x 1λ s dx c s = c s 4.26 λ s dx = Y s, xλ s dx, where the tuple β, Y of predictable processes associated to the process L under this change of measure is β s = Σs, T, U and Y s, x = e xσs,t,u Proof. Applying Girsanov s theorem for semimartingales, cf. Theorem III.3.24 in Jacod and Shiryaev 23, we get that the ĨP-semimartingale characteristics of L are B = B + β C + xy 1 ν C = C 4.28 ν = Y ν using the notation of Chapter 1. Taking into account the relationship between the local and semimartingale characteristics in 4.6, the result is proved. Now, it remains to verify that we can use the versions of β and Y as described in 4.27, where β = β s ω and Y = Y ω; s, x are defined by the following formulae cf. Jacod and Shiryaev 23, III.3.28: and Z c, L c = Z β C 4.29 Z Y = M IP µ P. 4.3 L Z In equation 4.3 P = P B denotes the σ-field of predictable sets on Ω [, T ] and M IP = µ L ω; dt, dxipdω is the positive measure µ L on Ω [, T ], F B[, T ] B defined by M IP µ L W = IEW µ L T 4.31 for measurable nonnegative functions W = W ω; t, x on Ω [, T ]. The conditional expectation M IP ZZ µ P is, by definition, the M IP -a.s. unique L µ L P-measurable function Y satisfying Z M IP µ U = M IP L Z µ Y U 4.32 L for all nonnegative P-measurable functions U = Uω; t, x. An application of Itô s formula, cf. Appendix B, yields that the continuous martingale part of the density process is Z c = Z s dx c s; 4.33

111 4.4. CAPLET-FLOOLET DUALITY 13 using in addition I.4.41 in Jacod and Shiryaev 23, we get Z c, L c t = = = = t t t Z s dx c s, L c Z s d X c, L c s Z s d Z s Σs, T, Uc s ds. t Σu, T, U c u dw u, cu dw u Therefore, we conclude that β s = Σs, T, U. Finally, we prove that we can choose Y s, x = e xσs,t,u. We have M IP µ e xσs,t,u U = IE L [ T ] e xσt,t,u Uω; t, xµ L ω; dt, dx [ ] = IE e LtωΣt,T,U Uω; t, L t ω1 { Ltω } t T [ T ] Z t ω = IE Z t ω Uω; t, xµl ω; dt, dx = M IP Zs µ U, L Z s 4.34 s because Zω Z ω 1 { Zω } = expσ, T, U Lω. We denote the value of a call option, with maturity T and strike K, on a bond with maturity U, by [ ] B, U 1 C B, T ;, K; C, ν = IE B, T BT M BT, U K +, where B, T is the discount factor associated with the option s maturity date T and B, U/B, T is the initial value of the forward price process B, U/B, T. The dynamics of BT M and BT, U are given by equations 4.13 and 4.12 respectively and the drift terms A are determined by the two characteristics of the driving process C, ν and the volatility structures Σ, according to equation Similar notation is used for the put option on a zero coupon bond. Theorem 4.9. Assume that bond prices are modeled according to the Lévy forward rate model. Then, we can relate the value of a call and a put

112 14 4. LÉVY TEM STUCTUE MODELS option on a bond via the following duality: B, U B, U C B, T ;, K; C, ν = P B, T ; K, B, T B, T ; C, ν where fs, x = exp Σs, U + Σs, T x and 1 A x ν = 1 A xf ν, A B\{}. Proof. The price of a call option with maturity T and strike K, on a bond with maturity U, is given by [ ] 1 C = IE BT, U K + B M T [ ] KBT, U = IE K 1 BT, U 1 + B M T [ ] BT, U = IE D BT M 2 KDBT M K 1 BT, U 1 + and changing measure from IP to ĨP, we get that C = ĨE [ KDB M T K 1 BT, U 1 +]. This can be re-written as [ C = ĨE B, T B, U DBM T [ = ĨE 1 B M T B, U B, T ] + K BT, U ] B, U + K BT, U 1 B, T, 4.35 for B M T 1 := B,T B,U DBM T, K := B,U B,T and BT, U := K B,U B,T BT, U 1. We will calculate the ĨP-dynamics of BT, U. Firstly, by its definition and 4.12, we get that BT, U = K exp T T Σs, T Σs, U dls + As, U As, T ds Keeping in mind that the local characteristics of L under ĨP are given by Proposition 4.8, we define the time-inhomogeneous Lévy process L := L. The local characteristics of L, using Lemma 4.2, are b, c, λ, where 1A xλdx = 1 A x λdx. Then we get BT, U = K exp T T Σs, U Σs, T d Ls + As, U As, T ds..

113 exp 4.4. CAPLET-FLOOLET DUALITY 15 Secondly, for the deterministic terms we have T As, Uds = IE [ T exp Σs, UdL s ] = ĨE [ D B M T 2 BT, U exp [ T = ĨE exp T Σs, UdL s ] Σs, T, UdL s T T ] exp Σs, UdL s exp θ s Σs, T, U ds = ĨE [ exp T Σs, T dl s ] exp T θ s Σs, T, U ds [ T ] T = ĨE exp Σs, T d L s exp θ s Σs, T, U ds = exp T Ãs, T ds exp T θ s Σs, T, U ds, 4.36 where Ãs, T := θ s Σs, T and θ s is the cumulant associated with the Lévy triplet b s, c s, λ s. Similarly, for the other term we have exp T As, T ds = IE [ = exp T exp T Σs, T dl s ] Ãs, Uds exp T θ s Σs, T, U ds. Therefore, the ĨP-dynamics of BT, U is BT, U = K exp T T Σs, U Σs, T d Ls + Ãs, T Ãs, U ds. 4.37

114 16 4. LÉVY TEM STUCTUE MODELS Finally, for the term corresponding to the money-market account, we have T 1 = B, T B, UB, T exp θ s Σs, T, U ds B, U B M T B M T exp T T 1 B, T exp = B, T exp exp T θ s Σs, U + θ s Σs, T ds T T θ s Σs, T ds Σs, T d L s exp θ s Σs, T, U ds and using equation 4.36, we get T T 1 = B, T exp Σs, T d L s exp exp T = B, T exp θ s Σs, T, U ds exp T T Σs, T d L s T T Ãs, T ds Σs, T dl s As, Uds θ s Σs, T, U ds Ãs, T ds In view of equations 4.35, 4.37 and 4.38, the result is proved. becomes a ĨP- emark 4.1. Note that BT, U discounted by B T M martingale. emark The change of measure from IP to ĨP is not structurepreserving for time-homogeneous processes, e.g. Lévy processes. Therefore, even if we had modeled bond prices as exponentials of Lévy processes under IP, the process driving the bond prices under ĨP would have been a time-inhomogeneous Lévy process; the driving process would remain timehomogeneous only if the jump part vanished or in some pathetic cases e.g. Σ, T, T [, T ]. This is obvious from the structure of the function f in Theorem 4.9. A similar phenomenon does not occur when modeling equities with Lévy processes compare with Theorem 1.18 or Corollary 4.2 in Eberlein and Papapantoleon 25b. Expressing the payoff of a caplet resp. floorlet as a put resp. call option on a zero coupon bond, cf. Appendix A, we get a duality directly relating the values of caplets and floorlets in the Lévy forward rate model.

115 4.4. CAPLET-FLOOLET DUALITY 17 We denote the value of a floorlet with strike K maturing at time T i that settles in arrears at T i+1, by [ ] 1 + FL B, T i ; L, T i, K; C, ν = IE BT M δ i K LT i, T i i+1 [ ] 1 + = 1 + δ i KIE BT M BT i, T i+1 K, i where L, T i = 1 δ i B,Ti B,T i+1 1 is the initial value of the forward LIBO rate and the strike K := 1/1 + δ i K. Similar notation is used for a caplet. Corollary Assume that bond prices are modeled according to the Lévy forward rate model. Then, we can relate the value of a caplet and a floorlet via the following duality: FL B, T i ; L, T i, K; C, ν = C CL B, T i ; K, L, T i ; C, ν where C := 1+δ ik 1+δ i L,T i, fs, x = exp Σs, T i + Σs, T i+1 x and 1 A x ν = 1 A xf ν, A B\{}. Proof. We simply use the result of Appendix A to express a floorlet as a call option on a zero coupon bond, then apply Theorem 4.9 and then the formula of Appendix A in the other direction, to express a put option on a zero coupon bond as a caplet. We get FL B, T i ; L, T i, K; C, ν = K 1 C B, T i ; B, T i+1 B, T i, K; C, ν = K 1 P = CCL B, T i ; K, B, T i+1 B, T i ; C, ν B, T i ; K, L, T i ; C, ν Duality in the Lévy LIBO model. The aim of this section is to provide a duality relationship between caplets and floorlets in the Lévy LIBO model. This result generalizes Theorem 5.1 in Eberlein, Kluge, and Papapantoleon 26 since we do not approximate the random compensator by a deterministic one. Instead, we build on the results of Chapter 1 and deal with the most general case directly. The payoff of a caplet with strike K, that is settled in arrears at time Tj 1, is δ j LT j, T j K+ ; similarly, the payoff of a floorlet with the same settlement date and strike is δj K LT j, T j +. Assuming that LIBO rates are modeled according to the Lévy LIBO model, we denote the value of a caplet with strike K, by C L, Tj, K; C, ν T j 1 = B, Tj 1 [ IE IPT δ j LTj, Tj K +], j 1 where L, Tj is the initial value of the forward LIBO process. Notice that the drift term is determined by the other two characteristics of the driving process C, ν T j 1 and the volatility structure λ, Tj, according to.

116 18 4. LÉVY TEM STUCTUE MODELS equation Moreover, the discount factor B, Tj 1 corresponds to the settlement date Tj 1. Similar notation is used for a floorlet. Theorem Let the LIBO rate be modeled according to the Lévy LIBO model. We can relate the values of caplets and floorlets via the following duality C L, T j, K; C, ν T j 1 = F K, L, Tj ; C, ν T j 1 where fs, x = expλs, Tj x and 1 Ax ν T j 1 B\{}. = 1 A xf ν T j 1, A Proof. From the time-t value of a caplet settled at time Tj 1, we get C = B, T j 1 IE IPT j 1 [ δ j LT j, T j K +] = B, T j 1 IE IPT j 1 [ δ j KLT j, T j K 1 LT j, T j 1 +] = B, Tj 1KL, Tj [ LT j, Tj IE ] IPT j 1 L, Tj δ j K 1 LTj, Tj Define the measure ĨP Tj 1 derivative on Ω, F, F t t T j 1 via the adon Nikodym and the valuation problem 4.39, reduces to dĩp T j 1 dip T j 1 = LT j, T j L, T j = Z 4.4 C = B, T j 1KL, T j IE eipt j 1 [ δ j K 1 LT j, T j 1 +] The density process is given by the restriction of the adon Nikodym derivative to the σ-field F t, and because the forward LIBO process is a IP T j 1 - martingale, we get [ ] d ĨP T Z t = IE j 1 IPT j 1 dip T F t j 1 t = exp + = Lt, T j L, T j λs, Tj c 1/2 s dw T j 1 s 1 t λs, T 2 j 2 cs ds t t xλs, T j µ L ν T j 1ds, dx e λs,t j x 1 λs, T j x ν T j 1ds, dx. 4.42

117 4.4. CAPLET-FLOOLET DUALITY 19 Equivalently, the density process Z can be expressed as the stochastic exponential of a semimartingale X, that is Z = EX where X := λs, Tj cs 1/2 dw T j 1 s + e xλs,t j 1 µ L ν T j 1ds, dx. Adapting the results of Proposition 4.8 to the present setting, it follows that the tuple of predictable processes associated to the process L and this change of measure is β s = λs, T j and Y s, x = e xλs,t j Additionally, using Girsanov s theorem for semimartingales, we immediately recognize W T j 1 = W T j 1 λs, T j c1/2 s ds as a ĨP Tj 1 -Brownian motion and ν T j 1dt, dx = e xλt,t j ν T j 1dt, dx as the ĨP Tj 1 -compensator of µl. Hence, the ĨP T j 1 -local characteristics of LT j 1 λ T j 1 s bt j 1 s = β s c s + c T j 1 s = c s dx = Y s, xλ T j 1 s dx are xy s, x 1λ T j 1 s dx and the ĨP T j 1 -canonical decomposition of the semimartingale LT j 1 L T j 1 = j 1 bt s ds + cs 1/2 d W T j 1 s is xµ L ν T j 1ds, dx Let L M,T j 1 denote the martingale part of L T j 1, i.e. L M,T j 1 is a semimartingale with predictable characteristics TL M,T j 1 ĨP T j 1 =, C, νt j 1. Now, the dynamics of L, T j 1 under ĨP T j 1 Lt, T j 1 = L, T j 1 exp = L, T j 1 exp t t is b L s, T j, T j 1ds bl s, T j, T j 1ds + t t λs, T j dl T j 1 s λs, T j d L T j 1 s =: Lt, T j, 4.46 where L T j 1 := L M,T j 1 is the dual process of L M,T j 1 and its triplet of semimartingale characteristics T L Tj 1 ĨP Tj 1, using Proposition 1.3 with f 1, is, C, ν T j 1, where 1 A x ν T j 1 = 1 A x ν T j

118 11 4. LÉVY TEM STUCTUE MODELS Furthermore, we define the drift term bl s, T j, T j 1 := b L s, T j, T j 1 λs, T j b T j 1 s The following simple calculation shows that the drift term b L s, T j, T j 1 corresponding to Lt, Tj, is of the same form as in Keeping in mind the form of ν T j 1 or, equivalently λ T j 1, we have bl s, Tj, Tj = 1 λs, T j 2 cs + e λs,t j x 1 xλs, Tj e λs,t j x λ T j 1 s dx 4.44 = 1 λs, T 2 j 2 cs e λs,t j x 1 + xλs, Tj j 1 λt dx 4.47 = 1 λs, T 2 j 2 cs e λs,t j x 1 xλs, Tj λ T j 1 s dx This concludes the proof, since C = B, Tj 1KL, Tj IE eipt [δ j K 1 LTj, Tj 1 ] + j 1 = B, T j 1 IE eipt j 1 [δ j L, T j LT j, T j + ], where LT j, T j := KL, T j LT j, T j and noting that the dynamics of L, Tj is given by 4.46 and Duality in the Lévy forward price model. In this section, we state a duality relationship between call and put options on the forward price. Since a call option on the forward is equivalent to a caplet, see equation A.1, this result can also be viewed as a duality between caplets and floorlets in the forward price model. We denote the time-t value of a call option on the forward price with strike K, which is settled in arrears at time Tj 1, by C FT j, K; C, νt j 1 = B, Tj 1 [ IE IPT F T j, Tj, Tj 1 K +] j 1 where FT j := F, T j, T j 1. Note that the drift characteristic of the driving process is determined by the other two characteristics C, ν T j 1 and the volatility structure λ, Tj, using equation Similar notation will be used for a put option on the forward price. Theorem Assume that the forward process is modeled according to the Lévy forward price model. Then, we can relate the values of call and s

119 4.5. VALUATION OF COMPOSITIONS 111 put options on the forward price via the following duality: C FT j, K; C, νt j 1 = P K, FT j ; C, νt j 1 where fs, x = expλs, Tj x and 1 Ax ν T j 1 B\{}. = 1 A xf ν T j 1, A Proof. The proof is analogous to the proof of Theorem 4.13 and therefore omitted for the sake of brevity Valuation of compositions We begin by describing the structure of the composition and the payoff of an option on the composition. Consider a discrete tenor structure = T < T 1 < < T N < T N+1 = T, where the accrual factor for the time period [T i, T i+1 ] is δ i = T i+1 T i, i {, 1,..., N}. The composition pays a floating rate, typically the LIBO, compounded on several consecutive dates. The rates are fixed at the dates s i T i and the composition is N 1 + δi Ls i, T i ; i=1 therefore, the composition equals an investment of one currency unit at the LIBO rate for N consecutive periods. The value of the composition is subjected to a cap or floor denoted by K and is settled in arrears, at time T. Hence, a cap on the composition pays off at maturity N 1 + δi Ls i, T i + K, i=1 while the payoff of a floor on the composition is N K 1 + δi Ls i, T i +. i=1 Notice that without the cap resp. floor, the payoff of the composition would simply be that of a floating rate note. Similarly, if we only consider a single compounding date, then we are dealing with a caplet resp. floorlet, with strike K := K 1 δ. In the following sections, we present methods for the valuation of a cap on the composition in the Lévy-driven forward rate and forward price frameworks. The value of a floor on the composition can either be deduced via analogous valuation methods cf. also Chapter 3 or via the cap-floor parity for compositions, which reads CT ; K = FT ; K + B, T 1 KB, T. Here CT ; K and FT ; K denote the time-t value of a cap, resp. floor, on the composition with cap, resp. floor, equal to K.

120 LÉVY TEM STUCTUE MODELS Forward rate framework. In this section we derive an explicit formula for the valuation of a cap on the composition in the Lévy forward rate model, making use of the methods developed in Chapter 3. As a special case, we get valuation formulae for caplets in the Lévy forward rate framework that generalize the results of Eberlein and Kluge 26a, since we do not require the existence of a Lebesgue density cf. emark 3.4. Firstly, we calculate the quantity that appears in the composition. By an elementary calculation, we have that Bs i, T i Bs i, T i+1 = B, T i s i s i B, T i+1 exp As, T i, T i+1 ds Σs, T i, T i+1 dl s. Using the fact that 1 + δ i Ls i, T i = Bs i,t i Bs i,t i+1 N 1+δi Ls i, T i N Bs i, T i = Bs i, T i+1 i=1 i=1 = B, T 1 B, T N exp s i i=1 As, T i, T i+1 ds we immediately get N s i i=1 Σs, T i, T i+1 dl s. Next, we define the forward measure associated with the date T via the adon Nikodym derivative dip T dip := 1 BT M B, T T = exp As, T ds + T Σs, T dl s. The measures IP and IP T are equivalent, since the density is strictly positive; moreover, we immediately note that IE [ ] 1 BT M B,T = 1. The density process related to the change of measure is given by the restriction of the adon Nikodym derivative to the σ-field F t, t T, therefore [ dipt ] IE F t = Bt, T dip Bt M B, T t t = exp As, T ds + Σs, T dl s. This allows us to determine the tuple of functions that characterize the process L under this change of measure and we can conclude, using Theorems III.3.24 and II.4.15 in Jacod and Shiryaev 23, that the driving process L = L t t [,T ] remains a time-inhomogeneous Lévy process under the measure IP T. According to the fundamental theorem of asset pricing the price of an option on the composition is equal to its discounted expected payoff under

121 4.5. VALUATION OF COMPOSITIONS 113 the martingale measure. Combined with the forward measure defined above, this gives CT ; K = IE IP 1 N + Bs i, T i BT M Bs i=1 i, T i+1 K N + = B, T IE IPT Bs i, T i Bs i, T i+1 K = B, T IE IPT i=1 where the random variable H is defined as H := log B, T 1 B, T + N s i i=1 [ exp H K + ], As, T i, T i+1 ds N s i i=1 Σs, T i, T i+1 dl s. Let us denote by M T H the moment generating function of H under the measure IP T. The next theorem provides an analytical expression for the value of a cap on the composition. Before that, we provide an expression for z for suitable complex arguments z. M T H Lemma Let M and ε be suitably chosen such that Σs, T M for all s, T [, T ] and Σs, T i+1 1 [si,s i+1 ]s M for all s, s i, T i+1 [, T ], where < M < M < M and M M > N + 1. Then, for each I 2 = [1 M M N+1 M +M N+1, 1 + M M M +M N+1], we have that M T H < and for every z C with z = H z = Z z exp M T where Z := B,T 1 B,T. T z N i=1 +θ s Σs, T z As, T i, T i+1 1 [,si ]s θ s Σs, T N Σs, T i, T i+1 1 [,si ]s ds Proof. Fix an I 2. Then, for z C with z =, and denoting by Σs, T i+1 = N i= Σs, T i+11 [si,s i+1 ]s, we get that z N i=1 Σs, T i, T i+1 1 [,si ]s i= i=1 + Σs, T N = z Σs, T i+1 1 [si,s i+1 ]s zσs, T + Σs, T = 1 z Σs, T Σs, T i+1 + Σs, T i+1 1 Σs, T Σs, T i+1 + Σs, T i+1 M M N + 1 M + M N + 1 M + M N M N + 1 = M. 4.5

122 LÉVY TEM STUCTUE MODELS Now, define the constants Z := exp z log B, T 1 B, T + := Z exp T and Z 1 function of H is [ M T H z = IE IPT ] expzh = IE IPT exp z = exp T T N ]sds As, T i, T i+1 1 [,si i=1 As, T ds. Hence, the moment generating log B, T 1 B, T + N N s i i=1 As, T ds Z N IE IP exp z = Z 1 IE IP exp = Z 1 exp T T z θ s z s i i=1 s i i=1 As, T i, T i+1 ds Σs, T i, T i+1 dl s Σs, T i, T i+1 dl s + T Σs, T dl s N Σs, T i, T i+1 1 [,si ]s + Σs, T dl s i=1 N Σs, T i, T i+1 1 [,si ]s + Σs, T ds, i=1 where for the last equality we have applied Lemma 4.1, which is justified by 4.5. In addition, we get that M T H < for I 2 and therefore the moment generating function of H can be extended to the complex plane for z C with z I 2. Theorem Assume that bond prices are modeled according to the Lévy forward rate model. The price of a cap on the composition is CT ; K = B, T M T K 1+iu H iu 2π iu 1 + iu du, where M T H is given by Lemma 4.15 and I 1 I 2 = 1, 1 + M M N+1 M +M N+1 ]. Proof. Since the prerequisites of Theorem 3.2 are satisfied for I 1 I 2, we immediately have that [ e CT ; K = B, T IE H IPT K ] + = B, T M T K 1+iu H iu 2π iu 1 + iu du,

123 4.5. VALUATION OF COMPOSITIONS 115 where we have also used Example 3.15, hence I 1 = 1, Forward price framework. The aim of this section is to derive an explicit formula for the valuation of a cap on the composition in the Lévy forward price model. Once again, the valuation formulae will be based on the methods developed in Chapter 3. We begin by noticing that the quantity that appears in the composition can be expressed in terms of forward prices, since 1 + δ i L, T i = F, T i, T i+1, and the forward prices are the modeling object in this framework. We know that each forward price process evolves as a martingale under its corresponding forward measure; moreover, we know that all forward price processes are driven by the same time-inhomogeneous Lévy process see also emark 4.6. Therefore, we will carry out the following program to arrive at the valuation formulae: 1 lift all forward price processes from their forward measure to the terminal forward measure; 2 calculate the product of the composition factors; 3 price the composition as a call option on this product. Appealing to the structure of the forward price process and the connection between the Brownian motions and the compensators under the different measures, cf. equations 4.23 and 4.24, we get that F t, Tj, Tj 1 = F, Tj, Tj 1 exp = F, Tj, Tj 1 exp t t bs, T j, T j 1ds + bs, T j, T ds + t t λs, Tj dl T j 1 λs, Tj dl T. s s 4.51 Here L T is the driving time-inhomogeneous Lévy process with IP T -canonical decomposition t L T t = t cs dw T s + xµ L ν T ds, dx, 4.52 and the drift term of the forward process F, Tj, T j 1 under the terminal measure IP T, is bs, Tj, T 1 = c s 2 λs, T j 1 j 2 + λs, Tj λs, Tk k=1 e xλs,t j 1 e x P j 1 k=1 λs,t k xλs, Tj λ T s dx. 4.53

124 LÉVY TEM STUCTUE MODELS It is immediately obvious from 4.51, 4.52 and 4.53 that F, T j, T j 1 is not a IP T -martingale, unless j = 1 where we use the convention that j=1 =. Now, the composition takes the following form N 1 + δi Ls i, T i = i=1 N F s j, Tj, Tj 1 j=1 = B, T N B, T N exp s j j=1 bs, T j, T ds + N s j j=1 where s j = s N+1 j, j {1,, N}. Define the random variable H := log B, T N B, T + N s j j=1 bs, T j, T ds + N s j j=1 λs, T j dl T s λs, T j dl T s 4.55 and now we can express the option on the composition as an option depending on this random variable. The next theorem provides a formula for the valuation of the composition. Theorem Let the forward prices be modeled according to the Lévy forward process framework. Then, the price of a cap on the composition is CT ; K = B, T K 1+iu M H iu du π iu 1 + iu where the moment generating function of H is given by Lemma 4.18 and 1, M M ]. Proof. The option on the composition is priced under the terminal forward martingale measure IP T. Using 4.54 and 4.55, we can express the cap on the composition as a call option depending on the random variable H. Then we get + N CT ; K = B, T IE IPT F s j, Tj, Tj 1 K j=1 [ e = B, T IE H IPT K ] + = B, T K 1+iu M H iu 2π iu 1 + iu du where we have applied Theorem 3.2 and used Example Lemma Let M and ε be suitably chosen such that N k=1 λs, T k M for some M < M and for all s [, T ]. Then, for each [, M M ] we

125 4.5. VALUATION OF COMPOSITIONS 117 have that M H <, and for every z C with z [, M M ] the moment generating function of H is T N N M H z = Z z exp z bs, Tj, T + θ T s z λs, Tj ds, where Z = B,T N B,T, c s, λ T s. Then and θ T s j=1 j=1 is the cumulant associated with the triplet Proof. Fix an [, M M ] and then for z C with z = we get N z λs, T k = N λs, T k M M M = M k=1 Now, define the constant Z 2 := B, T N B, T 1 k=1 z exp z [ ] M H z = IE IPT expzh N = Z 2 IE IPT exp z = Z 2 IE IPT exp = Z 2 exp T θ T s j=1 T z z N j=1 N s j j=1 s j N j=1 bs, Tj, T ds. λs, T j dl T s λs, T j 1 [,s j ]sdl T s λs, Tj ds where for the last equality we have applied Lemma 4.1, which is justified by Note also that λs, Tj = for s > s j, which is the fixing date for the rate; accordingly, bs, Tj, T = for s > s j, cf In addition, we get that M H < for [, M M ] and therefore the moment generating function of H can be extended to the complex plane for z C with z [, M M ].

126

127 APPENDIX A Transformations We use the well-known relationships between the LIBO, the forward price and the bond price, to transform a caplet into a call option on the forward price or a put option on a bond. Similarly, a floorlet is transformed into a put option on the forward price or a call option on a bond. Let T < T 1 < T 2 < < T N < T N+1 = T denote a discrete tenor structure where δ i = T i+1 T i, i {, 1,..., N}. The time-t i+1 payoff of a caplet settled in arrears at time T i+1, is Nδ i LT i, T i K + where K is the strike rate and N is the notional amount. Now, using the relationship between the LIBO and the forward price, i.e. F T i, T i, T i+1 = 1 + δ i LT i, T i, we can rewrite the payoff of a caplet as a call option on the forward price. We have Nδ i LT i, T i K + = Nδ i F Ti, T i, T i+1 1 δ i + K = NF T i, T i, T i+1 K +, A.1 where K = 1 + δ i K. Moreover, the payoff NF T i, T i, T i+1 K + settled at time T i+1 is equal to the payoff NBT i, T i+1 F T i, T i, T i+1 K +, settled at time T i. Using the relationship between forward and bond prices, i.e. F T i, T i, T i+1 = BT i, T i /BT i, T i+1, we have NBT i, T i+1 F T i, T i, T i+1 K + BTi, T i + = NBT i, T i+1 BT i, T i+1 K where K = K 1 and N = NK. = N1 KBT i, T i+1 + = NK BT i, T i+1 +, A.2 119

128

129 APPENDIX B An application of Itô s formula We apply Itô s formula for semimartingales, cf. Theorem I.4.57 in Jacod and Shiryaev 23 or Lemma A.5 in Goll and Kallsen 2, to calculate the canonical decomposition of an exponential semimartingale. 1. Consider a stochastic basis Ω, F, F t t T, P and a semimartingale H = H t t T with triplet of predictable characteristics TH P = B, C, ν. Assume that H is an exponentially special semimartingale and e H M loc P. Therefore, applying Theorem 2.19 in Kallsen and Shiryaev 22a, we have that e H = E H c + ex Ŵ µh ν, B.1 where Ŵt := e x 1ν{t} dx. Applying Itô s formula to the function fx = e x, since f C 2, and using the canonical decomposition of H, cf. 1.2, we get e H = 1 + = + e H s dh s e H s db s e H s d H c s e H s +x e H s e H s x µ H ds, dx e H s d H c s e H s +x e H s e H s hx µ H ds, dx e H s dhs c + e H s hxµ H νds, dx. B.2 Therefore, we can conclude that e H c = e H s dh c s. B.3 121

130 122 B. AN APPLICATION OF ITÔ S FOMULA 2. Consider the setting of Chapter 2, section 2.5. We have that Z = EX α, where X α = ασ s dw s + e αx 1µ L νds, dx, B.4 where L = L t t T is a time-inhomogeneous Lévy process. Therefore, we immediately get that Z c = Z s dxs α,c = Z s ασ s dw s. B.5 3. Finally, consider the setting of Chapter 4, section 4.4, where we have that Z = EX, where X = Σs, T, U c s dw s + e xσs,t,u 1µ L νds, dx, B.6 where L = L t t T is again a time-inhomogeneous Lévy process. Similarly, we can immediately conclude that Z c = Z s dx c s = Z s Σs, T, U c s dw s. B.7

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