Applications of semimartingales and L´evy processes in...

137
Applications of semimartingales and L´ evy processes in finance: duality and valuation Dissertation zur Erlangung des Doktorgrades der Fakult¨ at f¨ ur Mathematik und Physik der Albert-Ludwigs-Universit¨ at Freiburg im Breisgau vorgelegt von Antonis Papapantoleon Dezember 2006

Transcript of Applications of semimartingales and L´evy processes in...

Page 1: Applications of semimartingales and L´evy processes in ...webdoc.sub.gwdg.de/ebook/dissts/Freiburg/Papapantoleon2006.pdf · gales as driving processes. Another one, was to consider

Applications of semimartingalesand Levy processes

in finance:duality and valuation

Dissertation zur Erlangung des Doktorgrades

der Fakultat fur Mathematik und Physik

der Albert-Ludwigs-Universitat Freiburg im Breisgau

vorgelegt von

Antonis Papapantoleon

Dezember 2006

Page 2: Applications of semimartingales and L´evy processes in ...webdoc.sub.gwdg.de/ebook/dissts/Freiburg/Papapantoleon2006.pdf · gales as driving processes. Another one, was to consider

Dekan: Prof. Dr. Jorg Flum

Referenten: Prof. Dr. Ernst Eberlein

Prof. Dr. Dr. h.c. Albert N. Shiryaev (Moscow)

Datum der Promotion: 2. Marz 2007

Abteilung fur Mathematische Stochastik

Albert-Ludwigs-Universitat Freiburg

Eckerstr. 1D-79104 Freiburg im Breisgau

Page 3: Applications of semimartingales and L´evy processes in ...webdoc.sub.gwdg.de/ebook/dissts/Freiburg/Papapantoleon2006.pdf · gales as driving processes. Another one, was to consider

Abstract. The complexity of modern financial derivatives very oftenleads to valuation problems that require the knowledge of the joint dis-tribution of several random variables. This thesis aims to simplify andsolve such valuation problems.

Duality is related to the simplification of the valuation problem.We investigate changes of probability measures in an effort to reducethe multivariate problem to a univariate one. The asset price processesare driven either by general semimartingales or by Levy processes andtheir dynamics are expressed in terms of their predictable characteris-tics. Imposing some very natural conditions on the driving processes, abattery of derivative products – including Asian, lookback and Margrabeoptions – can be simplified considerably.

Valuation is related to the solution of the problem. We provide gen-eral 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. Usingthe Wiener–Hopf factorization, we provide expressions for options on themaximum of a Levy process. Finally, we consider term structure modelsdriven by time-inhomogeneous Levy processes and provide duality andvaluation results.

Page 4: Applications of semimartingales and L´evy processes in ...webdoc.sub.gwdg.de/ebook/dissts/Freiburg/Papapantoleon2006.pdf · gales as driving processes. Another one, was to consider
Page 5: Applications of semimartingales and L´evy processes in ...webdoc.sub.gwdg.de/ebook/dissts/Freiburg/Papapantoleon2006.pdf · gales as driving processes. Another one, was to consider

Contents

Acknowledgements vii

Introduction 1Overview – synopsis 2Remarks 3

Part 1. Exponential semimartingale and Levy models 5

Chapter 1. On the duality principle in option pricing: semimartingalesetting 7

1.1. Introduction 71.2. Semimartingales and semimartingale characteristics 81.3. Exponential semimartingale models 121.4. Martingale measures and dual martingale measures 141.5. The call-put duality in option pricing 28

Chapter 2. On the duality principle in option pricing II:multidimensional PIIAC and α-homogeneous payofffunctions 35

2.1. Introduction 352.2. Time-inhomogeneous Levy processes 372.3. Asset price model 462.4. General description of the method 482.5. Options with α-homogeneous payoff functions 502.6. Options on several assets 56

Chapter 3. Valuation of exotic derivatives in Levy models 673.1. Introduction 673.2. Option valuation: general formulae 683.3. Levy processes and their fluctuations 743.4. Examples of payoff functions 803.5. Applications 82

Part 2. Term structure models 87

Chapter 4. Duality and valuation in Levy term structure models 894.1. Introduction 894.2. Time-inhomogeneous Levy processes 924.3. Levy fixed income models 944.4. Caplet-floorlet duality 994.5. Valuation of compositions 111

v

Page 6: Applications of semimartingales and L´evy processes in ...webdoc.sub.gwdg.de/ebook/dissts/Freiburg/Papapantoleon2006.pdf · gales as driving processes. Another one, was to consider

vi CONTENTS

Appendix A. Transformations 119

Appendix B. An application of Ito’s formula 121

Bibliography 123

Page 7: Applications of semimartingales and L´evy processes in ...webdoc.sub.gwdg.de/ebook/dissts/Freiburg/Papapantoleon2006.pdf · gales as driving processes. Another one, was to consider

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. Andtherein lies the problem. You started looking for a route that did not exist. Youshould have made that route.” Old-Antonio smiles satisfied.

“Then why do you say that we made the route? You made the route, I simplywalked behind you” I said feeling rather uncomfortable.

“Oh, no” continues to smile old-Antonio. “I did not make the route alone. Youalso 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 tothank 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 personalstatement 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 avery natural and subtle way. He shared with me his insights on mathematicsand finance, provided me with the freedom to explore my own – sometimesvague – 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 morethan I could have asked for.

I would like to thank Prof. Dr. Albert N. Shiryaev for several interestingand amazing discussions during his visit at the University of Freiburg. Hisimmense appetite for mathematical research has been a guiding light for meever since.

Many thanks are, of course, due to my doctoral brothers and sister, JanBergenhtum, Wolfgang Kluge and Zorana Grbac. We have discussed withJan several mathematical and non-mathematical ideas and his opinion andadvice was always of great help. I had the pleasure to share my afternoontea with Wolfgang regularly; I would like to thank him for the numerousdiscussions and his patience for my questions. I am particularly glad thesediscussions led to some joint work, which is contained in this thesis. Zoranais a true friend, an attribute of both scientific and human dimensions; I amparticularly grateful to her for thoroughly reading this thesis and some veryhelpful discussions on Chapter 2.

vii

Page 8: Applications of semimartingales and L´evy processes in ...webdoc.sub.gwdg.de/ebook/dissts/Freiburg/Papapantoleon2006.pdf · gales as driving processes. Another one, was to consider

viii ACKNOWLEDGEMENTS

I would also like to thank all my other colleagues from the Departmentof Mathematical Stochastics and especially Monika Hattenbach, for her helpwith accommodation, paperwork and LATEX.

I would like to thank Andreas Kyprianou for his warm hospitality atHeriot–Watt University and for some very helpful discussions.

I am grateful to a number people who have discussed several ideas andshared their insights with me. In no particular order, I thank Jan Kallsen,Thorsten Schmidt, Josef Teichmann, Pierre Patie, Evangelia Petrou, StefanAnkirchner, George Skiadopoulos, Ales Cerny, Ernst August v. Hammer-stein, Mikhail Urusov and Kathrin Glau.

I am grateful to Maria Siopacha for her friendship throughout the yearsand her invaluable support during the last few months.

I would like to thank my parents for all their love and support, and forwhat they have sacrificed for the education of their children. I hope thisthesis makes them happy, although they will not understand its contents.

I thank my sister, Clio, for being my best friend, sincerest advisor andhardest 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 Ihave sincerely enjoyed the numerous discussions with Notis about life, poli-tics 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 Sundaymeetings in ‘Paradies’, his crisp advice and his great analytical skills. Thanksare due to Manolis “Man Of The Year” Havakis, Spyros Koutsoumpos andChristos Lekatsas for their friendship; to Michaella Wenzlaff for her friend-ship 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 goodtimes and the hard lessons.

Many thanks go to my friends from Athens, especially Alexandros Pa-padias, 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 throughthe European Community’s Human Potential Programme under contractHPRN-CT-2000-00100 DYNSTOCH and the financial support provided fromthe Deutsche Forschungsgemeinschaft (DFG).

Now, we will resume the use of the second person in plural and dive intothe cold mathematical universe, where there is little room for emotions.

Page 9: Applications of semimartingales and L´evy processes in ...webdoc.sub.gwdg.de/ebook/dissts/Freiburg/Papapantoleon2006.pdf · gales as driving processes. Another one, was to consider

Introduction

Mathematics is a field of science that has a very subtle but far-reachinginfluence on the life of human beings. Here one could think of number theoryand its applications in encoding, that make internet transactions possible;or, of functional analysis and its applications in quantum mechanics. How-ever, what we have in mind are the applications of mathematics in financialmarkets.

Ever since the seminal articles of F. Black, M. Scholes, and R. Merton (cf.Black and Scholes 1973 and Merton 1973) and their reformulation in termsof martingale theory by M. Harrison, S. Pliska, and D. Kreps (cf. Harrisonand Kreps 1979, Harrison and Pliska 1981, Kreps 1981), stochastic analysishas become the playground of modern finance. As has already been notedelsewhere, stochastic analysis and martingale theory seem to be tailor-madefor their application in mathematical finance; indeed, the proceeds from theinvestment in an asset can be represented as a stochastic integral, while therational price of an option on an asset equals its discounted expected payoffunder a martingale measure.

Initially, the applications relied on the use of Brownian motion as thedriving process, but empirical evidence showed that this assumption is toorestrictive. One remedy was to consider more general continuous semimartin-gales as driving processes. Another one, was to consider Levy processes asthe 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 gen-eral semimartingales in mathematical finance.

Levy processes are becoming increasingly popular in mathematical fi-nance because they can describe the observed reality of financial marketsin a more accurate way than models based on Brownian motion. In the“real” world, we observe that asset price processes have jumps or spikesand risk-managers have to take them into account. Moreover, the empiricaldistribution of asset returns exhibits fat tails and skewness, behavior thatdeviates from normality; hence, models that accurately fit return distribu-tions are essential for the estimation of profit and loss (P&L) distributions.In the “risk-neutral” world, we observe that implied volatilities are constantneither across strike nor across maturities as stipulated by the Black andScholes (1973) (actually, Samuelson 1965) model. Therefore, traders needmodels that can capture the behavior of the implied volatility smiles moreaccurately, in order to handle the risk of trades. Levy processes and generalsemimartingales provide us with the appropriate framework to adequatelydescribe all these observations, both in the “real” and in the “risk-neutral”world.

1

Page 10: Applications of semimartingales and L´evy processes in ...webdoc.sub.gwdg.de/ebook/dissts/Freiburg/Papapantoleon2006.pdf · gales as driving processes. Another one, was to consider

2 INTRODUCTION

Now, the everyday life of a mathematician working for a bank or anotherfinancial institution consists of four main tasks:

(1) design models that accurately fit return distributions and volatilitysurfaces;

(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 semi-martingales and Levy processes in mathematical finance, especially relatedto the valuation of exotic derivatives.

Our starting point is that the model has been chosen; we assume thatit is driven by a Levy process or a time-inhomogeneous Levy process oreven a general semimartingale. Now, the complexity of some financial prod-ucts, especially of exotic derivatives, means that the joint density of severalrandom variables must be known in order to price the product. The aimis to simplify this problem; for this we exploit the so-called duality prin-ciple. In the simplest case, the duality principle relates a European plainvanilla call option to a European plain vanilla put option. One could thinkof it in the following simple setting: consider an investor trading optionson the Euro/Dollar rate; then, she can intuitively understand that a Eurodenominated call option must equal a Dollar denominated put option on thereciprocal exchange rate.

Nevertheless, if the investor assumes some dynamics for the exchangerate, it is not immediately clear what the dynamics of the reciprocal rateare. 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 ofpredictable characteristics of a semimartingale (cf. Jacod 1979). Then, weexploit several other aspects of this idea, with a view towards simplifyingvaluation problems. A second point of interest is to solve the simplifiedproblem; we provide formulae that allow to price a wide range of productsand which can be evaluated fast. This also has important consequences tothe speed of the calibration algorithms.

Overview – synopsis

The thesis is divided into two parts. In the first part, we consider modelsdriven by general semimartingales and by Levy processes, which correspondto models for the dynamics of stocks or short-dated FX products. In the sec-ond part, we consider term structure models driven by time-inhomogeneousLevy processes; such models are applied for interest rate and long-dated FXderivatives.

Chapter 1 is based on Eberlein, Papapantoleon, and Shiryaev (2006) andEberlein and Papapantoleon (2005a). We develop the appropriate math-ematical tools to study the duality principle in a general semimartingaleframework, and the central result provides the explicit form of the tripletof predictable characteristics of the dual process under the dual martingalemeasure. Several examples are provided, which contain discrete time models,Brownian motion and models driven by Levy processes. Subsequently, weapply these results to option pricing problems. More specifically, we prove

Page 11: Applications of semimartingales and L´evy processes in ...webdoc.sub.gwdg.de/ebook/dissts/Freiburg/Papapantoleon2006.pdf · gales as driving processes. Another one, was to consider

REMARKS 3

a call-put duality for European and American options in the general semi-martingale framework. We also prove duality results between floating andfixed strike Asian and lookback options, and between forward-start and plainvanilla options.

Chapter 2 stems from Eberlein and Papapantoleon (2005b); we provide anew proof of a key result, and detailed proofs of some other results; moreover,the published paper contained a review part, which has been omitted fromthe thesis. Here, we continue our study of the duality principle in two direc-tions: firstly and more importantly, by considering options on several assets;secondly, by considering options with α-homogeneous payoff functions. Thedriving process is a time-inhomogeneous Levy process, although most of theresults can be proved for general semimartingales as driving processes. Weprovide a detailed account of multidimensional time-inhomogeneous Levyprocesses, provide an alternative view of the duality principle and then proveduality relationships for options with α-homogeneous payoff functions. Thefinal section is central in this chapter; the key result provides the triplet ofpredictable characteristics under a change of probability measure and pro-jection of a multidimensional time-inhomogeneous Levy process. This resultis then applied to derive duality relationships between options on severalassets and plain vanilla call and put options.

Chapter 3 deals with valuation problems for vanilla and exotic deriva-tives on assets driven by general semimartingales and by Levy processes.We first provide valuation formulae for single and multi-asset options, wherethe payoff functions can be arbitrary functions and the asset price processis driven by a general semimartingale. Then, we focus on exotic options onassets driven by Levy processes. Using the Wiener–Hopf factorization wederive the characteristic function of the supremum of a Levy process. As anapplication of the developed methods, we consider the pricing of lookbackand one-touch options on Levy-driven assets.

Chapter 4 is based on Eberlein, Kluge, and Papapantoleon (2006) andKluge and Papapantoleon (2006). We present a detailed overview of the threepredominant methods for modeling the term structure of interest rates: aforward rate (HJM) model, a LIBOR model and a forward price model,all driven by time-inhomogeneous Levy processes. Then, we derive dualityrelationships between caplets and floorlets in each of these models; theseresults 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 exoticinterest rate derivative, namely an option on a composition of LIBOR rates,in the forward rate and forward price frameworks.

Remarks

Each chapter of the thesis is self-contained and has its own introduction.This naturally entails several repetitions, especially regarding notation andconventions. However, in the introductory part of Chapter 1, we state severalfacts from stochastic analysis that are used throughout the thesis.

In general, we follow the notation of Jacod and Shiryaev (2003) forstochastic analysis and semimartingale theory; for fluctuation theory of Levyprocesses, we follow Kyprianou (2006).

Page 12: Applications of semimartingales and L´evy processes in ...webdoc.sub.gwdg.de/ebook/dissts/Freiburg/Papapantoleon2006.pdf · gales as driving processes. Another one, was to consider
Page 13: Applications of semimartingales and L´evy processes in ...webdoc.sub.gwdg.de/ebook/dissts/Freiburg/Papapantoleon2006.pdf · gales as driving processes. Another one, was to consider

Part 1

Exponential semimartingale andLevy models

Page 14: Applications of semimartingales and L´evy processes in ...webdoc.sub.gwdg.de/ebook/dissts/Freiburg/Papapantoleon2006.pdf · gales as driving processes. Another one, was to consider

The real research trip does not consist of seeking new land,but of observing with new eyes.

Marcel Proust, In Search of Lost Time.

Page 15: Applications of semimartingales and L´evy processes in ...webdoc.sub.gwdg.de/ebook/dissts/Freiburg/Papapantoleon2006.pdf · gales as driving processes. Another one, was to consider

CHAPTER 1

On the duality principle in option pricing:semimartingale setting

1.1. Introduction

The purpose of this section is to develop the appropriate mathematicaltools for the study of the call-put duality in option pricing. The dynamicsof asset prices are modeled as general exponential semimartingales, hencewe work in the widest possible framework, as far as arbitrage theory isconcerned. The duality principle states that the calculation of the price ofa call option for a model with price process S = eH , with respect to themeasure P , is equivalent to the calculation of the price of a put option fora suitable dual model S′ = eH

′with respect to a dual measure P ′.

From the analysis it becomes clear that appealing to general exponentialsemimartingale models leads to a deeper insight into the essence of theduality principle.

The most standard application of the duality principle relates the valueof 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 moregeneral diffusion models. Chesney and Gibson (1995) considered a two-factordiffusion model and Bates (1997) considered diffusion and jump-diffusionmodels. Schroder (1999) worked in a general semimartingale framework,but calculated the dynamics under the dual measure only in specific exam-ples (diffusion and jump-diffusion models). Fajardo and Mordecki (2006b)considered Levy processes.

These results where used to derive static hedging strategies for someexotic derivatives, using standard European options as hedging instruments;see e.g. Carr, Ellis, and Gupta (1998). They were also used by Bates (1997),and more recently by Fajardo and Mordecki (2006a), to calculate the so-called “skewness premium” from observed market prices.

Naturally, once the duality for European options was derived, researcherslooked into analogous results for American options. The duality betweenAmerican call and put options is even more interesting than its Europeancounterpart, since for American options the put-call parity holds only as aninequality. Carr and Chesney (1996) proved the put-call duality for Americanoptions for general diffusion models, Detemple (2001) studied dualities forAmerican options with general payoffs in diffusion models, while Fajardoand Mordecki (2006b) proved analogous results in Levy models.

The duality principle demonstrates its full strength when consideringexotic derivatives. In certain cases it allows to reduce a problem involvingtwo random variables – for example, the asset price and its supremum –to a problem involving just one random variable – in this example, the

7

Page 16: Applications of semimartingales and L´evy processes in ...webdoc.sub.gwdg.de/ebook/dissts/Freiburg/Papapantoleon2006.pdf · gales as driving processes. Another one, was to consider

8 1. ON THE DUALITY PRINCIPLE: SEMIMARTINGALE SETTING

supremum – under a dual measure. This is the case for Asian, lookback andforward-start options in Levy models. Sometimes, one can solve the probleminvolving a single random variable, see e.g. Borovkov and Novikov (2002)and Benhamou (2002), while the problem involving both random variablesremains very hard to tackle.

Henderson and Wojakowski (2002) showed an equivalence between float-ing and fixed strike Asian options in the Black–Scholes model. Vanmaeleet al. (2006) extended those results to forward-start Asian options in theBlack–Scholes model. Vecer (2002) and Vecer and Xu (2004) used this changeof measure to derive a one-dimensional partial (integro-)differential equationfor floating and fixed strike Asian options in the Black–Scholes and a generalsemimartingale model respectively. Andreasen (1998) also used this changeof measure to derive a one-dimensional partial (integro-)differential equationfor floating and fixed strike lookback options in the Black–Scholes and in ajump-diffusion model.

The connection between the choice of an appropriate numeraire and asubsequent change of measure has been beautifully described in Geman, ElKaroui, and Rochet (1995). Nevertheless, the change of measure methodhas also been used in earlier work, see e.g. Shepp and Shiryaev (1994) andShiryaev et al. (1994).

This chapter is organized as follows: in section 1.2 we collect some factsfrom stochastic analysis, describe the general semimartingale process andintroduce the characteristics of a semimartingale. In section 1.3 we presentthe exponential semimartingale model for the dynamics of a financial assetand 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 dualmartingale measure; several examples are also discussed. Finally, in section1.5 the call-put duality is proved for European, American, lookback, Asianand forward-start options.

1.2. Semimartingales and semimartingale characteristics

In this section we gather some results from stochastic analysis and semi-martingale theory that will be used throughout the thesis. The presentationfollows Jacod and Shiryaev (2003) closely; any unexplained notation is alsoused as in this monograph. Other standard references on these topics areJacod (1979, 1980) and Shiryaev (1999). Peskir and Shiryaev (2006, ChapterII) present a comprehensive overview on stochastic analysis; Kallsen (2006)provides a motivated introduction to the notion of semimartingale charac-teristics.

1. We assume that B = (Ω,F ,F, P ) is a stochastic basis, that is aprobability space (Ω,F , P ) equipped with a filtration F = (Ft)0≤t≤T ; T is afinite time horizon. A filtration is an increasing and right-continuous familyof sub-σ-algebras of F = FT , i.e. Fs ⊂ Ft for all 0 ≤ s ≤ t ≤ T andFt =

⋂s>tFs for all 0 ≤ 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 eachFt contains all P -null sets of F .

Page 17: Applications of semimartingales and L´evy processes in ...webdoc.sub.gwdg.de/ebook/dissts/Freiburg/Papapantoleon2006.pdf · gales as driving processes. Another one, was to consider

1.2. SEMIMARTINGALES AND SEMIMARTINGALE CHARACTERISTICS 9

All stochastic processes H = (Ht)0≤t≤T considered throughout this workhave cadlag trajectories, i.e. they are right continuous for 0 ≤ t < T withleft hand limits for 0 < t ≤ T . As usual, we assume that the process H isadapted to the filtration F = (Ft)0≤t≤T .

Consider the space Ω × [0, T ] = (ω, t) : ω ∈ Ω, t ∈ [0, T ] and a pro-cess Y with left continuous (cag) trajectories. The predictable σ-algebra Pis the σ-algebra on Ω × [0, T ] generated by all cag adapted processes Y ,considered as mappings (ω, t) Yt(ω) on Ω × [0, T ]. An adapted processH = (Ht(ω))0≤t≤T , ω ∈ Ω, that is P-measurable is called a predictableprocess. The optional σ-algebra O is the σ-algebra generated by all cadlagadapted processes Y , considered as mappings (ω, t) Yt(ω). A process Hthat is O-measurable is called an optional process.

Consider the space Ω× [0, T ]× R = (ω; t, x) : ω ∈ Ω, t ∈ [0, T ], x ∈ R.Then, P = P ⊗ B(R) denotes the σ-algebra of predictable sets in Ω =Ω× [0, T ]× R and O = O ⊗ B(R) denotes the σ-algebra of optional sets inΩ. A function W : Ω × [0, T ] × R → R is called predictable, resp. optional,if it is P-measurable, resp. O-measurable.

2. A process H = (Ht)0≤t≤T defined on the stochastic basis (Ω,F ,F, P )is a semimartingale if it admits a representation

H = H0 +M +A (1.1)

where X0 is a finite-valued, F0-measurable random variable, M is a localmartingale with M0 = 0 (M ∈ Mloc) and A is a bounded variation processwith A0 = 0 (A ∈ V). The representation (1.1) is, in general, not unique.

A semimartingale H is a special semimartingale if the process A in therepresentation (1.1) is, in addition, predictable (A ∈ P ∩ V). As a conse-quence of the Doob–Meyer decomposition, we conclude that the canonicalrepresentation (1.1) for a special semimartingale is unique.

The class of semimartingales remains invariant under several transfor-mations such as stopping, localization, change of time, change of filtration,absolutely continuous change of measure, etc. More importantly, it is thewidest class of stochastic processes for which a stochastic integral can bedefined for “reasonable” integrands (i.e. bounded predictable processes).

3. Let L(H) denote the set of predictable processes that are integrablewith respect to the semimartingaleH. LetK ∈ L(H), thenK ·H denotes thestochastic integral

∫ ·0 KsdHs. IfH = A+M , thenK ·H = K ·A+K ·M , where

K · A is the stochastic integral with respect to the bounded variation partof H and K ·M the stochastic integral with respect to the local martingalepart of H.

A random measure µ on [0, T ] × R is a family (µ(ω))ω∈Ω of measureson ([0, T ] × R,B([0, T ]) × B(R)) with µ(ω; 0 × R) = 0 for all ω ∈ Ω. Letµ = µ(ω; dt,dx), be an integer-valued random measure on [0, T ] × R. LetW = W (ω; t, x) be an optional function on Ω× [0, T ]×R; then W ∗µ denotesthe integral process

·∫0

∫R

W (ω; t, x)µ(ω; dt,dx),

Page 18: Applications of semimartingales and L´evy processes in ...webdoc.sub.gwdg.de/ebook/dissts/Freiburg/Papapantoleon2006.pdf · gales as driving processes. Another one, was to consider

10 1. ON THE DUALITY PRINCIPLE: SEMIMARTINGALE SETTING

often written as∫ ·0

∫RWdµ. Let ν denote the predictable compensator of

the random measure µ. For a predictable function W : Ω × [0, T ] × R → Rin Gloc(µ), W ∗ (µ− ν) denotes the stochastic integral

·∫0

∫R

W (ω; t, x)(µ− ν)(ω; dt,dx),

again abbreviated as∫ ·0

∫RWd(µ− ν).

4. Every semimartingale H = (Ht)0≤t≤T admits a canonical represen-tation

H = H0 +B +Hc + h(x) ∗ (µ− ν) + (x− h(x)) ∗ µ (1.2)or, in detail

Ht = H0 +Bt +Hct +

t∫0

∫R

h(x)d(µ− ν) +

t∫0

∫R

(x− h(x))dµ, (1.2′)

where(a) h = h(x) is a truncation function, i.e. a bounded function with compact

support that behaves as h(x) = x in a neighborhood of zero; a canon-ical choice of h is h(x) = x1|x|≤a where 1A(x), or 1(A), denotes theindicator of the set A;

(b) B = (Bt)0≤t≤T is a predictable process of bounded variation;(c) Hc = (Hc

t )0≤t≤T is the continuous martingale part of H;(d) ν = ν(ω; dt,dx) is the predictable compensator of the random measure

of jumps µ = µ(ω; dt,dx) of H; for clarity we write also νH and µH

instead of ν and µ.The continuous martingale part Hc of any semimartingale H is uniquely

defined (up to indistinguishability). The predictable quadratic variation〈Hc〉 of the continuous martingale Hc will be denoted by C = (Ct)0≤t≤T . Anapplication of the Doob–Meyer decomposition yields that (Hc)2 − 〈Hc〉 ∈Mloc (actually Mc

loc, the space of continuous local martingales).The random measure of jumps µ = µ(ω; dt,dx) of the semimartingale

H is an integer-valued random measure of the form

µ(ω; dt,dx) =∑s

1∆Hs(ω) 6=0ε(s,∆Hs(ω))(dt,dx),

where ∆Hs = Hs −Hs− and εa 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 thatfor every non-negative P-measurable function W = W (ω; t, x) on Ω

E[ T∫

0

∫R

W (ω; t, x)µ(ω; dt,dx)]

= E[ T∫

0

∫R

W (ω; t, x)ν(ω; dt,dx)]. (1.3)

Equivalently, we have that the process·∫

0

∫R

W (ω; t, x)µ(ω; dt,dx)−·∫

0

∫R

W (ω; t, x)ν(ω; dt,dx) ∈Mloc. (1.4)

Page 19: Applications of semimartingales and L´evy processes in ...webdoc.sub.gwdg.de/ebook/dissts/Freiburg/Papapantoleon2006.pdf · gales as driving processes. Another one, was to consider

1.2. SEMIMARTINGALES AND SEMIMARTINGALE CHARACTERISTICS 11

In addition, we have that the process·∫

0

∫R

(x2 ∧ 1)ν(ω; dt,dx) ∈ A+loc. (1.5)

The processesB, C, and the measure ν are called the triplet of predictablecharacteristics of the semimartingale H with respect to the probability mea-sure P , and will be denoted by

T(H|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) ofa semimartingale H depends on the selected truncation function h = h(x).However, the characteristics C and ν do not depend on the choice of h whileB = B(h) does. If h and h′ are two truncation functions, then

B(h)−B(h′) = (h− h′) ∗ ν.In the sequel, we assume that the truncation function h = h(x) satisfies

the following antisymmetry property:

h(−x) = −h(x).We will see that this property simplifies many formulae. Note that, for ex-ample, the canonical choice h(x) = x1|x|≤a satisfies this property.

6. An equivalent way to define the characteristics of a semimartingaleH, which reveals some additional properties, is the following (cf. TheoremII.2.42 and Corollary II.2.48 in Jacod and Shiryaev 2003). Let B be a real-valued, predictable process in V, C a non-negative-valued predictable processin V and ν the predictable compensator of the random measure of jumps ofH. Then, (B,C, ν) is called the triplet of predictable characteristics of H ifand only if

eiuH − eiuH− ·K(iu) (1.6)

is a (complex-valued) local martingale for all u ∈ R, whereK is the cumulantprocess of H

K(u) = uB +u2

2C + (eux − 1− uh(x)) ∗ ν. (1.7)

Equivalently, we have that T(H|P ) = (B,C, ν) if and only if for all u ∈ ReiuH

G(iu)∈Mloc(P ), (1.8)

where G(u) = E(K(u)), assuming it never vanishes. Here, E(·) denotes thestochastic exponential, cf. (1.14) and (1.15).

In addition, there exist an increasing predictable process A, predictableprocesses b and c and a transition kernel F from (Ω×[0, T ],P) into (R,B(R))such that

Bt =

t∫0

bsdAs, Ct =

t∫0

csdAs, ν([0, t]× E) =

t∫0

∫E

Fs(dx)dAs, (1.9)

Page 20: Applications of semimartingales and L´evy processes in ...webdoc.sub.gwdg.de/ebook/dissts/Freiburg/Papapantoleon2006.pdf · gales as driving processes. Another one, was to consider

12 1. ON THE DUALITY PRINCIPLE: SEMIMARTINGALE SETTING

where E ∈ B(R). Moreover, we have that K(u) = κ(u) ·A, where

κ(u) = ub+u2

2c+

∫R

(eux − 1− uh(x))F (dx). (1.10)

In several applications, the characteristics (B,C, ν) are absolutely contin-uous, in which case we can choose the process At = t. Then, we call thetriplet (b, c, F ) the differential characteristics of H.

1.3. Exponential semimartingale models

1. Let B = (Ω,F ,F, P ) be a stochastic basis and S = (St)0≤t≤T be anexponential semimartingale i.e. a stochastic process with representation

St = eHt , 0 ≤ t ≤ T (1.11)

(shortly: S = eH), where H = (Ht)0≤t≤T is a semimartingale, H0 = 0.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 analysisof S, the following simple interest representation

St = E(H)t, 0 ≤ t ≤ T (1.12)

(shortly: S = E(H)) with some suitable semimartingale H = (Ht)0≤t≤T , isconvenient for the study of the process S by martingale methods; see detailsin Shiryaev (1999).

In (1.12) we used the standard notation E(X) = (E(X)t)0≤t≤T for thestochastic exponential of a semimartingale, defined as the unique strongsolution of the stochastic differential equation

dE(X)t = E(X)t−dXt, X0 = 0, (1.13)

that has the following explicit solution

E(X)t = eXt− 12〈Xc〉t

∏0<s≤t

(1 + ∆Xs)e−∆Xs , (1.14)

where 〈Xc〉 is the predictable quadratic characteristic of the continuous mar-tingale part Xc of X and ∆Xs = Xs −Xs−.

From (1.11) and (1.12) it follows that the process H should satisfy theequation

eHt = E(H)t, 0 ≤ t ≤ T (1.15)

(shortly: eH = E(H)) which implies ∆H > −1. In other words,

Ht = log E(H)t, 0 ≤ t ≤ T (1.16)

and vice versaHt = Log(eHt), 0 ≤ t ≤ T (1.17)

(shortly: H = Log(eH)) where LogX denotes the stochastic logarithm of apositive process X = (Xt)0≤t≤T :

LogXt =

t∫0

dXs

Xs−. (1.18)

Page 21: Applications of semimartingales and L´evy processes in ...webdoc.sub.gwdg.de/ebook/dissts/Freiburg/Papapantoleon2006.pdf · gales as driving processes. Another one, was to consider

1.3. EXPONENTIAL SEMIMARTINGALE MODELS 13

Note that for a positive process X with X0 = 1 we have for LogX

LogX = logX +1

2X2−· 〈Xc〉 −

∑0<s≤·

(log(1 +

∆Xs

Xs−

)− ∆Xs

Xs−

); (1.19)

for details see Kallsen and Shiryaev (2002a) or Jacod and Shiryaev (2003,Chapter II).

From (1.15)–(1.18) one may get the following useful formulae:

H = H +12〈Hc〉+

∑0<s≤·

(e∆Hs − 1−∆Hs) (1.20)

andH = H − 1

2〈Hc〉+

∑0<s≤·

(log(1 + ∆Hs)−∆Hs). (1.21)

If µH = µH(ω; ds,dx) and µ eH = µeH(ω; ds,dx) are the random measures

of jumps of H and H, then the formulae (1.20) and (1.21) may be writtenin the form

H = H +12〈Hc〉+ (ex − 1− x) ∗ µH (1.20′)

andH = H − 1

2〈Hc〉+ (log(1 + x)− x) ∗ µ eH . (1.21′)

2. It is useful to note that discrete time sequences H = (Hn)n≥0 withH0 = 0 and Fn-measurable random variables Hn can be considered as asemimartingale H = (Ht)t≥0 in continuous time, where Ht = Hn for t ∈[n, n+1), given on the stochastic basis B = (Ω,F , (F t)t≥0, P ) with F t = Fnfor t ∈ [n, n+ 1).

In the discrete time setting S = (Sn)n≥0 has a compound interest repre-sentation

Sn = eHn , Hn = h1 + · · ·+ hn, n ≥ 1, (1.22)S0 = 1, where hn are random variables with h0 = 0; the analogue of thesimple interest representation has the form

Sn = E(H)n =∏

0≤k≤n(1 + hk) (1.23)

with hk = ehk − 1, Hk = h1 + · · ·+ hk, k ≥ 1, H0 = 0. We see that

∆Sn = Sn−1∆Hn

where ∆Sn = Sn − Sn−1, ∆Hn = Hn − Hn−1 = hn (compare with (1.13)).

3. From formulae (1.20) and (1.21) it is not difficult to find the rela-tionships between the triplets T(H|P ) = (B,C, ν) and T(H|P ) = (B, C, ν)(with respect to the same truncation function h):

B = B +C

2+ (h(ex − 1)− h(x)) ∗ ν

C = C (1.24)

1A(x) ∗ ν = 1A(ex − 1) ∗ ν, A ∈ B(R\0)

Page 22: Applications of semimartingales and L´evy processes in ...webdoc.sub.gwdg.de/ebook/dissts/Freiburg/Papapantoleon2006.pdf · gales as driving processes. Another one, was to consider

14 1. ON THE DUALITY PRINCIPLE: SEMIMARTINGALE SETTING

and

B = B − C

2+ (h(log(1 + x))− h(x)) ∗ ν

C = C (1.25)

1A(x) ∗ ν = 1A(log(1 + x)) ∗ ν, A ∈ B(R\0);

for more details we refer to Kallsen and Shiryaev (2002a) and Jacod andShiryaev (2003).

In particular, if H is a Levy process with triplet of local characteristics(b, c, F ), then H will also be a Levy process with the triplet (b, c, F ) forwhich

b = b+c

2+∫R

(h(ex − 1)− h(x))F (dx)

c = c (1.24′)

F (A) =∫R

1A(ex − 1)F (dx), A ∈ B(R\0).

Correspondingly,

b = b− c

2+∫R

(h(log(1 + x))− h(x)

)F (dx)

c = c (1.25′)

F (A) =∫R

1A(log(1 + x))F (dx), A ∈ B(R\0).

1.4. Martingale measures and dual martingale measures

1. Let Mloc(P ) be the class of all local martingales on a given stochasticbasis B = (Ω,F ,F, P ). It is known (and easily follows from the canonicalrepresentation (1.2)) that if T(H|P ) = (B,C, ν) then

H ∈Mloc(P ) ⇔ B + (x− h(x)) ∗ ν = 0. (1.26)

Similarly, for the process H = Log(eH) we have

H ∈Mloc(P ) ⇔ B + (x− h(x)) ∗ ν = 0. (1.26)

In the sequel, we will assume that the following condition is in force.

Assumption (ES). The process 1x>1ex ∗ ν has bounded variation.

Under Assumption (ES) the property (1.26) can be rewritten, takinginto account (1.24), in the following form:

H ∈Mloc(P ) ⇔ B +C

2+ (ex − 1− h(x)) ∗ ν = 0. (1.27)

Remark 1.1. The assumption that the process 1x>1ex∗ν has boundedvariation is equivalent, by Kallsen and Shiryaev (2002a, Lemma 2.13), tothe assumption that the semimartingale H is exponentially special, i.e. the

Page 23: Applications of semimartingales and L´evy processes in ...webdoc.sub.gwdg.de/ebook/dissts/Freiburg/Papapantoleon2006.pdf · gales as driving processes. Another one, was to consider

1.4. MARTINGALE MEASURES AND DUAL MARTINGALE MEASURES 15

price process S = eH is a special semimartingale. This justifies to call itAssumption (ES).

Since H ∈Mloc(P ) if and only if E(H) ∈Mloc(P ), we get from (1.27)

E(H) ∈Mloc(P ) ⇔ B +C

2+ (ex − 1− h(x)) ∗ ν = 0, (1.28)

and, therefore, using (1.15)

S = eH ∈Mloc(P ) ⇔ B +C

2+ (ex − 1− h(x)) ∗ ν = 0. (1.29)

2. In the sequel we shall assume that S is not only a local martingalebut also a martingale (S ∈ M(P )) on [0, T ]. Thus EST = 1, which allowsus to define on (Ω,F , (Ft)0≤t≤T ) a new probability measure P ′ with

dP ′

dP= ST . (1.30)

Since S is a martingale

d(P ′|Ft)d(P |Ft)

= St, 0 ≤ t ≤ T (1.31)

and since S > 0 (P -a.s.), we have P P ′ and

dPdP ′ =

1ST

. (1.32)

Let us introduce the process

S′ =1S. (1.33)

Then, denoting by H ′ the dual of the semimartingale H, i.e. H ′ = −H, wehave

S′ = eH′. (1.34)

The following simple but, as we shall see, useful lemma plays a crucialrole in the problem of duality between call and put options. It also explainsthe name of dual martingale measure for the measure P ′.

Lemma 1.2. Suppose S = eH ∈ M(P ) i.e. S is a P -martingale. Thenthe process S′ ∈M(P ′) i.e. S′ is a P ′-martingale.

Proof. The proof follows directly from Proposition III.3.8 in Jacod andShiryaev (2003), which states that if Z = dP ′

dP then S′ ∈ M(P ′) iff S′Z is aP -martingale. In our case Z = S and S′S ≡ 1. Thus S′ ∈M(P ′).

3. The next theorem is crucial for all calculations of option prices onthe basis of the duality principle (see Section 1.5). We first prove an aux-iliary proposition of independent interest, about the characteristics of thestochastic integral process

∫ ·0 fdH.

Page 24: Applications of semimartingales and L´evy processes in ...webdoc.sub.gwdg.de/ebook/dissts/Freiburg/Papapantoleon2006.pdf · gales as driving processes. Another one, was to consider

16 1. ON THE DUALITY PRINCIPLE: SEMIMARTINGALE SETTING

Proposition 1.3. Let f be a predictable, bounded process. The tripletof predictable characteristics of the stochastic integral process J =

∫ ·0 fdH,

denoted by T(J |P ) = (BJ , CJ , νJ), is

BJ = f ·B + [h(fx)− fh(x)] ∗ ν (1.35)CJ = f2 · C (1.36)

1A(x) ∗ νJ = 1A(fx) ∗ ν, A ∈ B(R\0). (1.37)

Proof. The last two statements follow directly from the properties ofthe stochastic integral J = f ·H:

Jc = f ·Hc (1.38)

and∆J = f∆H. (1.39)

Indeed, (1.36) follows directly from (1.38) and Jacod and Shiryaev (2003,I.4.41):

CJ = 〈Jc〉 = f2 · 〈Hc〉 = f2 · C.From (1.39) we deduce

1A(x) ∗ µJ = 1A(fx) ∗ µH , A ∈ B(R\0) (1.40)

which gives for νJ , the compensator of the random measure of jumps µJ ofJ , the relation (1.37).

For the proof of relation (1.35) we recall the canonical representation ofthe semimartingale H:

H = H0 +B +M + (x− h(x)) ∗ µH (1.41)

where M is a local martingale (in fact M = Hc + h(x) ∗ (µH − ν)) and thecanonical representation of the semimartingale J :

J = J0 +BJ + Jc + h(y) ∗ (µJ − νJ) + (y − h(y)) ∗ µJ . (1.42)

From the definition J = f ·H and the representation (1.41) we get

J = f ·B + f ·M + (fx− fh(x)) ∗ µH (1.43)

which gives, together with (1.40), the following formula:

J − (y − h(y)) ∗ µJ =

= f ·B + f ·M + (fx− fh(x)) ∗ µH − (fx− h(fx)) ∗ µH

= f ·B + f ·M + (h(fx)− fh(x)) ∗ µH . (1.44)

The process J − (y − h(y)) ∗ µJ has bounded jumps. Hence this processis a special semimartingale (Jacod and Shiryaev 2003, Lemma 4.24, p. 44)and by Proposition 4.23(iii), again from Jacod and Shiryaev (2003, p. 44),we conclude that the process f · B + (h(fx) − fh(x)) ∗ µH) ∈ Aloc, i.e.it is a process with locally integrable variation. Note now that the processf ·B belongs also to the class Aloc since it is a predictable process of locallybounded variation (Jacod and Shiryaev 2003, Lemma 3.10, p. 29). Hence theprocess (h(fx) − fh(x)) ∗ µH ∈ Aloc and using Jacod and Shiryaev (2003,Theorem 3.18, p. 33) there exists a compensator of this process given by

Page 25: Applications of semimartingales and L´evy processes in ...webdoc.sub.gwdg.de/ebook/dissts/Freiburg/Papapantoleon2006.pdf · gales as driving processes. Another one, was to consider

1.4. MARTINGALE MEASURES AND DUAL MARTINGALE MEASURES 17

the formula (h(fx) − fh(x)) ∗ ν (Jacod and Shiryaev 2003, Theorem 1.8,pp. 66–67). As a result we get from (1.44) that

J − (y − h(y)) ∗ µJ = [f ·B + (h(fx)− fh(x)) ∗ ν]+ [f ·M + (h(fx)− fh(x)) ∗ (µH − ν)]

= f ·B + (h(fx)− fh(x)) ∗ ν+ f ·Hc + h(fx) ∗ (µH − ν). (1.45)

Comparing the decomposition (1.45) of the special semimartingale J −(y−h(y))∗µJ with the representation of J−(y−h(y))∗µJ from the canonicalrepresentation (1.42) we conclude, by the uniqueness of the representationof a special semimartingale (Jacod and Shiryaev 2003, I.4.22), that the pro-cesses BJ and f · B + (h(fx) − fh(x)) ∗ ν are indistinguishable; cf. Jacodand Shiryaev (2003, p. 3). Therefore, formula (1.35) is proved.

Remark 1.4. Variants of Proposition 1.3 are stated in Jacod and Shiryaev(2003, IX.5.3) and Kallsen and Shiryaev (2002b, Lemma 3).

Theorem 1.5. The triplet T(H ′|P ′) = (B′, C ′, ν ′) can be expressed viathe triplet T(H|P ) = (B,C, ν) by the following formulae:

B′ = −B − C − h(x)(ex − 1) ∗ νC ′ = C (1.46)

1A(x) ∗ ν ′ = 1A(−x)ex ∗ ν, A ∈ B(R\0).

Proof. We give two proofs which are of interest here, since these proofscontain some additional useful relationships between different triplets. Thestructure of these proofs can be represented by the following diagram:

T(H|P ′)

(c)

(−)

))SSSSSSSSSSSSSS

T(H|P )

(G)

(a)

55llllllllllllll

(b)

(−) ))RRRRRRRRRRRRRRT(H ′|P ′)

T(H ′|P )

(d)

(G)

55kkkkkkkkkkkkkk

(1.47)

where(G) // means that we use Girsanov’s theorem for calculating the

right side triplet from the left side one and(−) // means that we consider

the dual of the semimartingale on the left side.

(a) T(H|P )(G) // T(H|P ′).

For the calculation of the triplet T(H|P ′) = (B+, C+, ν+) from thetriplet T(H|P ) = (B,C, ν), we use Girsanov’s theorem for semimartingales(Jacod and Shiryaev 2003, pp. 172–173) which states that

B+ = B + β+ · C + h(x)(Y + − 1) ∗ ν (1.48)C+ = C (1.49)ν+ = Y + · ν. (1.50)

Page 26: Applications of semimartingales and L´evy processes in ...webdoc.sub.gwdg.de/ebook/dissts/Freiburg/Papapantoleon2006.pdf · gales as driving processes. Another one, was to consider

18 1. ON THE DUALITY PRINCIPLE: SEMIMARTINGALE SETTING

Here β+ = β+t (ω) and Y + = Y +(ω; t, x) are defined by the following formu-

lae (see Jacod and Shiryaev 2003, p. 173):

〈Sc,Hc〉 = (S−β+) · C (1.51)

and

Y + = MPµH

(S

S−

∣∣∣P) . (1.52)

In equation (1.52) MPµH = µH(ω; dt,dx)P (dω) is the positive measure

on (Ω× [0, T ]× R,F ⊗ B([0, T ])⊗ B(R)) defined by

MPµH (W ) = E(W ∗ µH)T (1.53)

for measurable non-negative functions W = W (ω; t, x) on Ω× [0, T ]× R.The conditional expectation MP

µH

(SS−

∣∣P) is, by definition, the MPµH -a.s.

unique P-measurable function Y + with the property

MPµH

(S

S−U

)= MP

µH

(Y +U

)(1.54)

for all non-negative P-measurable functions U = U(ω; t, x).We show that in our special case S = eH , where evidently S

S−= e∆H ,

one may take the following versions of β+ and Y +:

β+ ≡ 1 and Y + = ex. (1.55)

Indeed, for S = eH , we get applying Ito’s formula to eH , see AppendixB, that

(eH)c =

·∫0

eHs−dHcs

and, therefore,

〈Sc,Hc〉 = 〈(eH)c,Hc〉 =⟨ ·∫

0

eH−dHc,Hc⟩

=

·∫0

eH−d〈Hc〉 =

·∫0

eH−dC = S− · C (1.56)

and

(S−β+) · C =

·∫0

eH−β+dC. (1.57)

From this formula and the equality (1.51) we see that one may take β+ ≡ 1.For the proof that one may choose Y + = ex we need to verify (1.54)

with this version of Y +.

Page 27: Applications of semimartingales and L´evy processes in ...webdoc.sub.gwdg.de/ebook/dissts/Freiburg/Papapantoleon2006.pdf · gales as driving processes. Another one, was to consider

1.4. MARTINGALE MEASURES AND DUAL MARTINGALE MEASURES 19

We have, using that µH is the random measure of jumps of H:

MPµH (exU) = E

[ T∫0

∫R

exU(ω; t, x)µH(ω; dt,dx)]

= E

[ ∑0≤t≤T

e∆Ht(ω)U(ω; t,∆Ht(ω))1∆Ht(ω) 6=0

]

= E

[ T∫0

∫R

St(ω)St−(ω)

U(ω; t, x)µH(ω; dt,dx)]

= MPµH

( SS−

U). (1.58)

Consequently in (1.48)–(1.50) one may put β+ ≡ 1 and Y + = ex which givesthe following result:

B+ = B + C + h(x)(ex − 1) ∗ νC+ = C (1.59)ν+ = ex · ν.

Remark 1.6. It is useful to note that for the discrete time case the rela-tion dν+ = exdν can be proved (with the obvious notation) in the followingsimple way.

Let hn = ∆Hn and µn = µn(ω; ·) be the random measure of jumps of Hat time n, i.e.

µn(ω;A) = 1(hn(ω)∈A) for A ∈ B(R\0).

The compensator νn = νn(ω; ·) of µn(ω; ·) has here the simple formνn(ω;A) = P (hn∈A|Fn−1)(ω) (see Jacod and Shiryaev (2003, p. 92) forthe definition of the compensator in the discrete time case). If ν+

n (ω;A) =P ′(hn∈A|Fn−1)(ω) then, applying the already used Proposition III.3.8 inJacod and Shiryaev (2003) or, equivalently, applying Bayes’ formula (alsocalled the conversion formula; see Shiryaev 1999, p. 438) we find that

ν+n (ω;A) = E′[1A(hn)|Fn−1](ω)

= E[1A(hn)ehn |Fn−1](ω) =∫A

exνn(ω; dx).

Therefore,

ν+n νn and

dν+n

dνn(ω;x) = ex (νn-a.e.).

(b) T(H|P )(−) // T(H ′|P ).

Because H ′ = −H the triplet T(H ′|P ) = T(−H|P ). Now, applyingProposition 1.3 to the function f ≡ −1 (i.e. J = −H) and assuming that

Page 28: Applications of semimartingales and L´evy processes in ...webdoc.sub.gwdg.de/ebook/dissts/Freiburg/Papapantoleon2006.pdf · gales as driving processes. Another one, was to consider

20 1. ON THE DUALITY PRINCIPLE: SEMIMARTINGALE SETTING

h(x) = −h(−x), we get

B− = −BC− = C (1.60)

1A(x) ∗ ν− = 1A(−x) ∗ ν, A ∈ B(R\0).

(c) T(H|P ′)(−) // T(H ′|P ′).

The triplet T(H|P ′) = (B+, C+, ν+) is given by the formulae (1.59).Then from (1.60) (with the necessary adaptation of the notation) we get

B′ = −B+ = −B − C − h(x)(ex − 1) ∗ νC ′ = C+ = C (1.61)

1A(x) ∗ ν ′ = 1A(−x) ∗ ν+ = 1A(−x)ex ∗ ν,

so, the proof using steps (a) and (c) leads to the formulae (1.46).

(d) T(H ′|P )(G) // T(H ′|P ′).

Here T(H ′|P ) = T(−H|P ) = (B−, C−, ν−) and T(H ′|P ′) = T(−H|P ′) =(B′, C ′, ν ′). Similarly to the case (a) we have the following formulae (com-pare with (1.48)):

B′ = B− + β− · C− + h(x)(Y − − 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 (comparewith (1.51) and (1.52))

〈Sc, (−H)c〉 = (S−β−) · C− (1.65)

and

Y − = MPµ−H

(S

S−

∣∣∣P) . (1.66)

Since

〈Sc, (−H)c〉 =⟨(eH)c,−Hc

⟩=⟨ ·∫

0

eH−dHc,−Hc⟩

= −·∫

0

eH−d〈Hc〉

= −·∫

0

eH−d〈(−H)c〉 = −·∫

0

eH−dC−

= (−S−) · C−, (1.67)

comparing (1.65) and (1.67) we see that one may take β− ≡ −1.

Page 29: Applications of semimartingales and L´evy processes in ...webdoc.sub.gwdg.de/ebook/dissts/Freiburg/Papapantoleon2006.pdf · gales as driving processes. Another one, was to consider

1.4. MARTINGALE MEASURES AND DUAL MARTINGALE MEASURES 21

Similarly to the calculations in (a) we deduce

MPµ−H (e−xU) = E

[ T∫0

∫R

e−xU(ω; t, x)µ−H(ω; dt,dx)]

= E

[ ∑0≤t≤T

e−∆(−Ht(ω))U(ω; t,∆(−Ht(ω)))1∆(−Ht(ω)) 6=0

]

= E

[ ∑0≤t≤T

e∆Ht(ω)U(ω; t,∆(−Ht(ω)))1∆(−Ht(ω)) 6=0

]

= E

[ T∫0

∫R

St(ω)St−(ω)

U(ω; t, x)µ−H(ω; dt,dx)]

= MPµ−H

( SS−

U). (1.68)

Therefore one may take Y − = e−x in (1.66) and from (1.62)–(1.64) and(1.60) we find that

B′ = −B − C + h(x)(e−x − 1

)∗ ν− (1.69)

C ′ = C (1.70)ν ′ = e−x · ν− (1.71)

where ν− is such that 1A(x) ∗ ν− = 1A(−x) ∗ ν, A ∈ B(R\0). Hence, asone easily sees

1A(x) ∗ ν ′ = 1A(x) ∗(e−x · ν−

)= 1A(−x)ex ∗ ν. (1.72)

In addition, if h(−x) = −h(x)

h(x)(e−x − 1

)∗ ν− = h(−x)(ex − 1) ∗ ν

= −h(x)(ex − 1) ∗ ν. (1.73)

From (1.69)–(1.73) we conclude that the triplet T(H ′|P ′) = (B′, C ′, ν ′),obtained using steps (b) and (d) is given by formulae (1.46). Hence, Theorem1.5 is proved.

Remark 1.7. Note that under Assumption (ES) we can conclude fromformulae (1.46) that (|x|2 ∧ 1) ∗ ν ′ ∈ Aloc, because

(|x|2 ∧ 1) ∗ ν ′ ≤ K|x|21|x|≤1 ∗ ν + 1x<−1 ∗ ν + ex1x>1 ∗ ν.

HereK is a constant and the processes on the right-hand side are predictableprocesses of bounded variation, hence belong to Aloc (cf. Jacod and Shiryaev2003, Lemma I.3.10). Similarly, we get that ν ′ satisfies Assumption (ES),because

1x>1ex ∗ ν ′ = 1x<−1e

−xex ∗ ν = 1x<−1 ∗ ν

and 1x<−1 ∗ ν ∈ Aloc.

Page 30: Applications of semimartingales and L´evy processes in ...webdoc.sub.gwdg.de/ebook/dissts/Freiburg/Papapantoleon2006.pdf · gales as driving processes. Another one, was to consider

22 1. ON THE DUALITY PRINCIPLE: SEMIMARTINGALE SETTING

Corollary 1.8. Suppose that H is a P -Levy process with local charac-teristics (b, c, F ). Then the process H ′ is a P ′-Levy process with local char-acteristics (b′, c′, F ′) given by the formulae (we take h(−x) = −h(x)):

b′ = −b− c−∫R

h(x)(ex − 1)F (dx)

c′ = c

F ′(A) =∫R

1A(−x)exF (dx), A ∈ B(R\0).

Proof. The proof follows from Theorem 1.5 and Jacod and Shiryaev(2003, Corollary 4.19, p. 107).

4. Lemma 1.2 states that S′ is a P ′-martingale. Now, there exists analternative path to verify this result.

Remark 1.9. The formulae (1.46) provide a simple way to confirm thatthe process S′ = eH

′ ∈ Mloc(P ′). Indeed, by (1.29) it is sufficient to checkthat

B′ +C ′

2+ (ex − 1− h(x)) ∗ ν ′ = 0. (1.74)

From (1.46) (with −h(−x) = h(x)) we get

B′ +C ′

2+ (ex − 1− h(x)) ∗ ν ′

=(−B − C − h(x)(ex − 1) ∗ ν

)+C

2+(e−x − 1− h(−x)

)ex ∗ ν

= −(B +

C

2+ (ex − 1− h(x)) ∗ ν

)= 0

where the last equality follows from the assumption S = eH ∈Mloc(P ) andcriterion (1.29).

5. Now we consider some examples that show how to calculate the tripletT(H ′|P ′) from the triplet T(H|P ) and for which particular models in financeAssumption (ES) is satisfied.

Example 1.10 (Brownian case). Suppose ν ≡ 0. From (1.29) S = eH ∈Mloc(P ) iff B + C

2 = 0. If S ∈ M(P ) then by Theorem 1.5 the tripletT(H ′|P ′) = (B′, C ′, 0) with B′ = −(B + C) and C ′ = C. So, B′ + C′

2 =−(B + C

2 ) = 0 which implies S′ ∈Mloc(P ′). In particular, if

St = eσWt−σ2

2t,

i.e. Ht = σWt − σ2

2 t where W = (Wt)0≤t≤T is a standard Brownian motion(Wiener process) then Bt = −σ2

2 t, Ct = σ2t. Evidently B + C2 = 0 which

implies by (1.29) that S ∈ Mloc(P ). In fact, S ∈ M(P ). Note also thatdSt = σStdWt.

The process S′ = eH′= e−H has stochastic differential

dS′t = −σS′t(dWt − σdt). (1.75)

Page 31: Applications of semimartingales and L´evy processes in ...webdoc.sub.gwdg.de/ebook/dissts/Freiburg/Papapantoleon2006.pdf · gales as driving processes. Another one, was to consider

1.4. MARTINGALE MEASURES AND DUAL MARTINGALE MEASURES 23

Since S′ ∈Mloc(P ′) from formula (1.75) one can deduce that the processW ′t = Wt − σt, 0 ≤ t ≤ T is a P ′-local martingale. This is a particular case

of the classical Girsanov theorem which can be easily checked directly us-ing the fact (already mentioned before, in Lemma 1.2) that W ′ ∈Mloc(P ′)iff W ′S ∈ Mloc(P ). The last property follows from calculating d(W ′S) byIto’s formula. The fact that W ′ is a P ′-Brownian motion follows also fromLevy’s characterization of a Brownian motion (Revuz and Yor 1999, Theo-rem IV.(3.6)). So, dS′t = −σS′tdW ′

t .

Example 1.11 (Poissonian case). Consider S = eH with

Ht = απt − λ(eα − 1)t, α 6= 0 (1.76)

where π = (πt)0≤t≤T is a Poisson process with parameter λ > 0 (Eπt = λt).Take h(x) ≡ 0. Then the corresponding triplet (B,C, ν) has the followingform:

Bt = −λ(eα − 1)tCt = 0 (1.77)

ν(dt,dx) = λ1α(dx)dt.

By (1.29) S ∈ Mloc(P ) ⇔ B + (ex − 1) ∗ ν = 0. With the process given in(1.76)

Bt + (ex − 1) ∗ νt = −λ(eα − 1)t+ λ(eα − 1)t = 0.Therefore, S ∈ Mloc(P ) and even S ∈ M(P ); moreover, P is the uniquemartingale measure for the Poisson model (cf. e.g. Corcuera et al. 2005, pp.120–121). In addition, with respect to the measure P ′ the process S′ is alocal martingale; this follows directly from criterion (1.29)

B′ + (ex − 1) ∗ ν ′ = 0. (1.78)

By Theorem 1.5B′t = λ(eα − 1)t

and(ex − 1) ∗ ν ′t = (e−x − 1)ex ∗ νt = λ(1− eα)t.

Hence, the property (1.78) does hold and S′ ∈Mloc(P ′).

Example 1.12 (Discrete time, CRR-model). In the binomial model ofCox, Ross and Rubinstein (CRR-model), asset prices are modeled by Sn =eHn , with Hn = h1 + · · · + hn, n ≥ 1, H0 = 0, where (hn)n≥1 is a P -i.i.d.sequence of random variables which have only two values.

If hn = ehn − 1 then Sn =∏k≤n(1 + hk) and Sn = (1 + hn)Sn−1, n ≥ 1,

with S0 = 0. For simplicity let us assume that the random variables hn takethe values lnλ 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(hn = ln

)= P (hn = a) =

b

b− a=

λ

1 + λ(1.79)

andP (hn = lnλ) = P (hn = b) =

−ab− a

=1

1 + λ(1.80)

Page 32: Applications of semimartingales and L´evy processes in ...webdoc.sub.gwdg.de/ebook/dissts/Freiburg/Papapantoleon2006.pdf · gales as driving processes. Another one, was to consider

24 1. ON THE DUALITY PRINCIPLE: SEMIMARTINGALE SETTING

then we find thatEehn = E(1 + hn) = 1.

This means that the measure P is a martingale measure for the sequenceS = (Sn)n≥0. Indeed it is the unique martingale measure for the CRR-model;see Shiryaev (1999, Example V.3.2, pp. 477–480).

With the truncation function h(x) = x and the martingale measureP we easily find that the triplet T(H|P ) = (B, 0, ν) where (with ∆Bn =Bn −Bn−1)

∆Bn = Ehn =1− λ

1 + λlnλ (1.81)

and (with νn(A) = ν(n ×A))

νn(lnλ) = P (hn = lnλ) =1

1 + λ

νn

(ln

)= P

(hn = ln

)=

λ

1 + λ.

(1.82)

Note that from (1.81) and (1.82) we get ∆Bn+(ex−1−x)∗νn = 0, whichis another derivation of the martingale property for S under the measure Pgiven by (1.79) and (1.80).

Based on formulae (1.46) we find directly that

∆B′n = ∆Bn, ν ′n = νn (1.83)

and from the previous note and (1.29) it follows that S′ ∈Mloc(P ′) (in factS′ ∈M(P ′)).

Example 1.13 (Purely discontinuous Levy models). In this class of mod-els, asset prices are modeled as S = eH , where H = (Ht)0≤t≤T is a purelydiscontinuous Levy process with triplet T(H|P ) = (B, 0, ν). We can alsowork with the triplet of differential characteristics, denoted by (b, 0, F ),which using Jacod and Shiryaev (2003, II.4.20), is related for our case tothe triplet of semimartingale characteristics via

Bt(ω) = bt, ν(ω; dt,dx) = dtF (dx).

Since S = eH ∈Mloc(P ), the characteristic b resumes the form

b = −∫R

(ex − 1− h(x))F (dx).

and criterion (1.29) is satisfied. Now, we have in addition that S ∈ M(P ),cf. Lemma 4.4 in Kallsen (2000). Then, we can apply Theorem 1.5 and thetriplet T(H ′|P ′) = (B′, 0, ν ′) is given by

1A(x) ∗ ν ′ = 1A(−x)ex ∗ ν (1.84)

and B′ = −B−h(x)(ex−1)∗ν = −(ex−1−h(x))∗ν ′. Therefore, using (1.29)again, or alternatively Remark 1.9, we have that S′ = eH

′ ∈Mloc(P ′).When considering parametric models it is very convenient to represent

the Levy measure F = F (dx) in the form

F (dx) = eϑxf(x)dx (1.85)

Page 33: Applications of semimartingales and L´evy processes in ...webdoc.sub.gwdg.de/ebook/dissts/Freiburg/Papapantoleon2006.pdf · gales as driving processes. Another one, was to consider

1.4. MARTINGALE MEASURES AND DUAL MARTINGALE MEASURES 25

where ϑ ∈ R and f is an even function, i.e. f(x) = f(−x). In that case, thetriplet of local characteristics of the dual process H ′ is (b′, 0, F ′) where∫

1A(x)F ′(dx) =∫

1A(−x)e(1+ϑ)xf(x)dx

and, of course, b′ = −∫

R(ex − 1− h(x))F ′(dx).

Examples of parametric models are:

Example 1.13.1 (Generalized hyperbolic model). Let H = (Ht)0≤t≤Tbe a generalized hyperbolic process with Law(H1|P ) = GH(λ, α, β, δ, µ),cf. Eberlein (2001, p. 321) or Eberlein and Prause (2002). Then the Levymeasure of H admits the representation (1.85) with parameters ϑ = β,0 ≤ |β| < α and

f(x) =1|x|

∞∫0

exp(−√

2y + α2 |x|)π2y(J2

|λ|(δ√

2y ) + Y 2|λ|(δ

√2y ))

dy + λe−α|x|1λ>0,

where α > 0, δ > 0, λ ∈ R, µ ∈ R, cf. Eberlein (2001, p. 323). Here Jλ andYλ are the modified Bessel functions of first and second kind respectively.The moment generating function exists for u ∈ (−α− β, α− β), hence, As-sumption (ES) is satisfied. The class of generalized hyperbolic distributionscontains several other distributions as subclasses, for example hyperbolicdistributions (Eberlein and Keller 1995), normal inverse Gaussian distribu-tions (Barndorff-Nielsen 1998) or limiting classes (e.g. variance gamma). Werefer to Eberlein and v. Hammerstein (2004) for an extensive survey.

Example 1.13.2 (CGMY model). Let H = (Ht)0≤t≤T be a CGMYLevy process, cf. Carr, Geman, Madan, and Yor (2002); another name forthis process is (generalized) tempered stable process. The Levy measure ofthis process admits the representation (1.85) with the following parameters

ϑ =

G, x < 0−M, x > 0 and f(x) =

C

|x|1+Y,

where C > 0, G > 0, M > 0, and Y < 2.The CGMY processes are closely related to stable processes; in fact, the

function f coincides with the Levy measure of the stable process with indexα ∈ (0, 2), cf. Samorodnitsky and Taqqu (1994, Def. 1.1.6). Due to the ex-ponential tempering of the Levy measure, the CGMY distribution has finitemoments of all orders. Moreover, the moment generating function exists,hence Assumption (ES) is satisfied. Again, the class of CGMY distributionscontains several other distributions as subclasses, for example the variancegamma distribution (Madan and Seneta 1990) and the bilateral gamma dis-tribution (Kuchler and Tappe 2006).

Example 1.13.3 (Meixner model). Let H = (Ht)0≤t≤T be a Meixnerprocess with Law(H1|P ) = Meixner(α, β, δ), α > 0, −π < β < π, δ > 0, cf.Schoutens and Teugels (1998) and Schoutens (2002). The Levy measure ofthe Meixner process admits the representation (1.85) with ϑ = β

α and

f(x) =δ

x sinh(πxα ).

Page 34: Applications of semimartingales and L´evy processes in ...webdoc.sub.gwdg.de/ebook/dissts/Freiburg/Papapantoleon2006.pdf · gales as driving processes. Another one, was to consider

26 1. ON THE DUALITY PRINCIPLE: SEMIMARTINGALE SETTING

The Meixner distribution possesses finite moments of all orders. Moreover,the moment generating function exists, hence, Assumption (ES) is againsatisfied.

Remark 1.14. Note that H ′, in all the parametric examples considered,remains a P ′-Levy process from the same class of processes, just with a newparameter ϑ in representation (1.85).

Remark 1.15. Theorem 1.5 cannot be applied, for example, to stableprocesses, because they do not satisfy Assumption (ES). In fact, stable pro-cesses may not even have finite first moment, cf. Samorodnitsky and Taqqu(1994, Property 1.2.16). This fact makes them particularly unsuitable foroption pricing, although these models are applied for risk management pur-poses, cf. Rachev (2003).

Example 1.16 (Stochastic volatility Levy models). This class of modelswas proposed by Carr, Geman, Madan, and Yor (2003) and further investi-gated in Schoutens (2003).

Let X = (Xt)0≤t≤T be a pure jump Levy process and Y = (Yt)0≤t≤T bean increasing process, independent of X. The process Y acts as a stochasticclock measuring activity in business time and has the form

Yt =∫ t

0ysds

where y = (ys)0≤s≤T is a positive process. Carr, Geman, Madan, and Yor(2003) consider the CIR process as a candidate for y, i.e. the solution of thestochastic differential equation

dyt = K(η − yt)dt+ λy12t dWt,

where W = (Wt)0≤t≤T is a standard Brownian motion. For other choices ofY see Schoutens (2003).

The stochastic volatility Levy process is defined by time-changing theLevy process X with the increasing process Y , that is

Zt = XYt , 0 ≤ t ≤ T .

The process Z is a pure jump semimartingale with canonical decomposition

Z = Z0 +BZ + h(x) ∗ (µZ − νZ) + (x− h(x)) ∗ µZ ,where the compensator of the random measure of jumps of Z has the formνZ(ds,dx) = y(s)νX(dx)ds, where νX denotes the Levy measure of X.

Asset prices are modeled as S = eH , where H is a semimartingale suchthat νH(ds,dx) = νZ(ds,dx) = y(s)νX(dx)ds and S ∈ Mloc(P ), therefore,T(H|P ) = (B, 0, νH), where

B = −(ex − 1− h(x)) ∗ νH .If S ∈ M(P ), cf. Kallsen (2006, Proposition 4.1) for a sufficient condition,then applying Theorem 1.5, we get that T(H ′|P ′) = (B′, 0, ν ′) with

1A(x) ∗ ν ′ = 1A(−x)ex ∗ νH

=∫

1A(−x)exνX(dx)y(s)ds

and B′ = −(ex − 1− h(x)) ∗ ν ′.

Page 35: Applications of semimartingales and L´evy processes in ...webdoc.sub.gwdg.de/ebook/dissts/Freiburg/Papapantoleon2006.pdf · gales as driving processes. Another one, was to consider

1.4. MARTINGALE MEASURES AND DUAL MARTINGALE MEASURES 27

Example 1.17 (Local volatility models). Local volatility models wereintroduced by Dupire (1994); we refer to Skiadopoulos (2001) for a surveyof this literature. The dynamics of the asset price process is given by thestochastic differential equation

dSt = Stσ(t, St)dWt, S0 = 1, (1.86)

where W = (Wt)0≤t≤T is a standard Brownian motion. If the local volatilityfunction σ : [0, T ]× R+ −→ R+ is Lipschitz, i.e. satisfies the conditions(a) |σ(t, x)− σ(t, y)| ≤ K|x− y|, ∀t ∈ [0, T ], K constant,(b) t 7→ σ(t, x) is right continuous with left limits, ∀x ∈ R+,then the SDE (1.86) has a unique strong solution (cf. Protter 2004, TheoremV.6), for which

St = E

·∫0

σ(u, Su)dWu

t

= exp

t∫0

σ(u, Su)dWu −12

⟨ ·∫0

σ(u, Su)dWu

⟩t

= exp

t∫0

σ(u, Su)dWu −12

t∫0

σ2(u, Su)du

.

Therefore, assuming the canonical setting (Jacod and Shiryaev 2003, p.154), these models fit in the general exponential semimartingale frameworkwith driving process H = (Ht)0≤t≤T and triplet T(H|P ) = (B,C, ν) where

B = −12

·∫0

σ2(u, eHu)du

C =

·∫0

σ2(u, eHu)du

ν ≡ 0,

and, of course, S = eH ∈ Mloc(P ). Now, S = eH ∈ M(P ) holds if, for ex-ample, Novikov’s condition is satisfied, cf. Revuz and Yor (1999, PropositionVIII.(1.15)); then applying Theorem 1.5, we get that T(H ′|P ′) = (B′, C ′, ν ′),where

B′ = −B − C = −12

·∫0

σ2(u, eHu)du

C ′ =

·∫0

σ2(u, eHu)du

ν ′ ≡ 0.

Page 36: Applications of semimartingales and L´evy processes in ...webdoc.sub.gwdg.de/ebook/dissts/Freiburg/Papapantoleon2006.pdf · gales as driving processes. Another one, was to consider

28 1. ON THE DUALITY PRINCIPLE: SEMIMARTINGALE SETTING

1.5. The call-put duality in option pricing

Let S = (St)0≤t≤T be the price process of a financial asset and fT =fT (S) the payoff of an option on this asset. Here fT (S) = fT (St, 0≤ t≤T )is an FS

T -measurable functional, where FST = σ(St, 0≤ t≤ T ). In order to

simplify the notation we assume that the current interest rate is zero; fordetailed 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 (2005b).

As is well known, in a complete market, where the martingale measureP is unique, the rational (or arbitrage-free) price of the option is given byEfT (= EP fT ). In incomplete markets one has to choose an equivalent mar-tingale measure. In this work we do not discuss the problem of the choice ofa reasonable martingale measure, for example, in the sense of minimizationof a distance (L2-distance, Hellinger distance, entropic distance, etc.) fromthe “physical” measure or in the sense of constructing the simplest possi-ble measure (e.g. Esscher transformation). The practitioners’ point of viewis that the choice of this measure is the result of a calibration to marketprices of plain vanilla options. We will assume that the initial measure Pis a martingale measure and all our calculations of EP fT will be done withrespect to this measure P . In the case of an incomplete market this optionprice EP fT could be called a quasi rational option price.

A. European call and put options. In case of a standard call optionthe payoff function is

fT = (ST −K)+, K > 0, (1.87)

whereas for a put option it is

fT = (K − ST )+, K > 0. (1.88)

The corresponding option prices are given by the formulae

CT (S;K) = E[(ST −K)+] (1.89)

andPT (K;S) = E[(K − ST )+] (1.90)

where E is the expectation operator with respect to the martingale measureP . From (1.89) for S = eH we get

CT (S;K) = E[ST

fTST

]= E′

[ fTST

]= E′[(1−KS′T )+]

= KE′[( 1K− S′T

)+]= KE′[(K ′ − S′T )+] (1.91)

where K ′ = 1K . Comparing here the right hand side with (1.90) we obtain

the following result.

Theorem 1.18. For standard call and put options the option prices sat-isfy the following duality relations:

1K

CT (S;K) = P′T (K ′;S′)

and1K

PT (K;S) = C′T (S′;K ′)

Page 37: Applications of semimartingales and L´evy processes in ...webdoc.sub.gwdg.de/ebook/dissts/Freiburg/Papapantoleon2006.pdf · gales as driving processes. Another one, was to consider

1.5. THE CALL-PUT DUALITY IN OPTION PRICING 29

where K ′ = 1K . P′T (K ′;S′) and C′

T (S′,K ′) are the corresponding prices forput and call options with S′ as underlying price process, computed with re-spect to the dual measure P ′.

Corollary 1.19. Call and put prices in markets (S, P ) and (S′, P ′)which satisfy the duality relation, are connected by the following “call-callparity”

CT (S;K) = KC′T (S′;K ′) + 1−K

and the following “put-put parity”

PT (K;S) = KP′T (K ′;S′) +K − 1.

Proof. From the identity (ST −K)+ = (K − ST )+ + ST −K we get,taking expectations with respect to the measure P , the well-known call-putparity :

CT (S;K) = PT (K;S) + 1−K.

The result follows from the duality relations in Theorem 1.18.

B. American call and put options. The general theory of pricing ofAmerican options (see, for example, Shiryaev 1999, Chapters VI and VIII)states that, for payoff functions described by the process e−λtft, 0 ≤ t ≤ T ,λ ≥ 0, the price VT (S) of the American option is given by the formula

VT (S) = supτ∈MT

E[e−λτfτ ], (1.92)

where MT is the class of stopping times τ such that 0 ≤ τ ≤ T .For a standard call option fτ = (Sτ−K)+ and for a standard put option

fτ = (K − Sτ )+, where K > 0 is a constant strike.Denote

CT (S;K) = supτ∈MT

E[e−λτ (Sτ −K)+] (1.93)

andPT (K;S) = sup

τ∈MT

E[e−λτ (K − Sτ )+]. (1.94)

Similarly to the case of European options, we get for fτ = (Sτ −K)+

CT (S;K) = supτ∈MT

E[e−λτfτ

STST

]= sup

τ∈MT

E′[e−λτ

fτST

]= sup

τ∈MT

E′[e−λτfτS′T

]= sup

τ∈MT

E′[e−λτfτE′(S′T |Fτ )

]= sup

τ∈MT

E′[e−λτfτS′τ

]= sup

τ∈MT

E′[e−λτ (Sτ −K)+S′τ

]= sup

τ∈MT

E′[e−λτ (1−KS′τ )

+]

= K supτ∈MT

E′[e−λτ (K ′ − S′τ )

+]

= K P′T (K ′;S′),

Page 38: Applications of semimartingales and L´evy processes in ...webdoc.sub.gwdg.de/ebook/dissts/Freiburg/Papapantoleon2006.pdf · gales as driving processes. Another one, was to consider

30 1. ON THE DUALITY PRINCIPLE: SEMIMARTINGALE SETTING

Option type\payoff Asian LookbackFixed Strike call (ΣT −K)+ (ST −K)+

Fixed Strike put (K − ΣT )+ (K − ST )+

Floating Strike call (ST − ΣT )+ (ST − ST )+

Floating Strike put (ΣT − ST )+ (ST − ST )+

Table 1.1. Types of payoffs for Asian and lookback options

where again K ′ = 1K . Thus, similarly to the statements in Theorem 1.18 we

have for American options the following duality relations:

1K

CT (S;K) = P′T (K ′;S′)

and also1K

PT (K;S) = C′T (S′;K ′).

C. Lookback options. Let S = (St)0≤t≤T and S = (St)0≤t≤T denotethe supremum and infimum processes of the asset price process, that is

St = sup0≤u≤t

Su and St = inf0≤u≤t

Su.

There exist two types of lookback options traded in the market: floating andfixed strike options. In the case of the floating strike option, the supremum(resp. infimum) plays the role of the strike. The different variants of thelookback option are summarized in Table 1.1.

A variant of floating strike lookback options are partial lookback options,with payoff

(ST − αST )+ and (βST − ST )+

for the call and put option respectively. Here, α ∈ [1,∞) and β ∈ (0, 1]denote the degree of partiality. The incentive behind trading partial lookbackoptions is that “classical” lookback options are too expensive relative to theirEuropean plain vanilla counterparts, since S ≤ S ≤ S a.s., which makesthem unattractive for investors.

Suppose S ∈ M(P ), then for a floating strike, or partial, lookback calloption we get

CT

(S;α inf S

)= E

[(ST − α inf

t≤TSt)+]

= E

[ST

(1− α inft≤T St

ST

)+]= E′

[(1− αeinft≤T Ht−HT

)+]= E′

[(1− αeH

′T−supt≤T H

′t

)+]= αE′

[( 1α− eH

′T−supt≤T H

′t

)+]. (1.95)

Page 39: Applications of semimartingales and L´evy processes in ...webdoc.sub.gwdg.de/ebook/dissts/Freiburg/Papapantoleon2006.pdf · gales as driving processes. Another one, was to consider

1.5. THE CALL-PUT DUALITY IN OPTION PRICING 31

In order to further simplify the last expression, let us assume that the processH ′ = (H ′

t)0≤t≤T satisfies the following reflection principle:

Law(

supt≤T

H ′t −H ′

T |P ′)

= Law(− inft≤T

H ′t|P ′

). (1.96)

This property holds, of course, if the process H ′ is a Levy process withrespect to P ′ (see e.g. Kyprianou 2006, Lemma 3.5). Combining (1.95) and(1.96) we get that

CT

(S;α inf S

)= E′

[( 1α− einft≤T H

′t

)+]= E′

[( 1α− inft≤T

S′t

)+]= P′T

( 1α

; inf S′). (1.97)

Similarly, assuming the following reflection principle

Law(H ′T − inf

t≤TH ′t|P ′

)= Law

(supt≤T

H ′t|P ′

)which again holds for Levy processes (Kyprianou 2006, Lemma 3.5) we get

PT(β supS;S

)= C′

T

(supS′;

). (1.98)

Concluding, we have the following result.

Theorem 1.20. Let H be a Levy process. The calculation of the prices offloating strike lookback call and put options CT (S;α inf S) and PT (β supS;S),(α ≥ 1, 0 < β ≤ 1), can be reduced via the duality relations

CT

(S;α inf S

)= P′T

( 1α

; inf S′)

and

PT(β supS;S

)= C′

T

(supS′;

)to the calculation of the prices of fixed strike lookback put and call optionsP′T ( 1

α ; inf S′) and C′T (supS′; 1

β ) respectively.

D. Asian options. Define ΣT to be the arithmetic average of the assetprice process S during the time interval [0, T ]. In case the price process iscontinuously monitored, then ΣT = 1

T

∫ T0 Sudu, while in the case of discrete

monitoring we have ΣT = 1N

∑Ni=1 STi , where 0 = T0 < T1 < · · · < TN = T .

Similarly to lookback options, there exist floating and fixed strike Asianoptions traded in the market. In the floating strike case, the average playsthe role of the strike. The different variants of the Asian option are againsummarized in Table 1.1. Note that for Asian options there exists a put-callparity relationship, which can be derived by the elementary equality

(ΣT − ST −K)+ − (K + ST − ΣT )+ = ΣT − ST −K.

Page 40: Applications of semimartingales and L´evy processes in ...webdoc.sub.gwdg.de/ebook/dissts/Freiburg/Papapantoleon2006.pdf · gales as driving processes. Another one, was to consider

32 1. ON THE DUALITY PRINCIPLE: SEMIMARTINGALE SETTING

Suppose that S ∈ M(P ) and consider the price of the floating strikeAsian call option; then

CT

(S;

1T

∫S)

= E

[(ST −

1T

T∫0

Stdt

)+]= E

[ST

(1− 1

T

T∫0

StST

dt

)+]

= E′

[(1− 1

T

T∫0

S′TS′t

dt

)+]= E′

[(1− 1

T

T∫0

eH′T−H

′tdt

)+]

= E′

[(1− 1

T

T∫0

eH′T−H

′(T−u)−du

)+]. (1.99)

If H ′ is a P ′-Levy process, then the following duality property holds (see e.g.Kyprianou 2006, Lemma 3.4)

Law(H ′T −H ′

(T−t)−; 0 ≤ t < T |P ′) = Law(H ′t; 0 ≤ t < T |P ′). (1.100)

From (1.99) and (1.100) we conclude

CT

(S;

1T

∫S)

= E′

[(1− 1

T

T∫0

eH′udu

)+]

= E′

[(1− 1

T

T∫0

S′udu

)+]= P′T

(1;

1T

∫S′). (1.101)

Similarly,

PT( 1T

∫S;S

)= C′

T

( 1T

∫S′; 1

). (1.102)

Therefore, we have the following result.

Theorem 1.21. Let H be a Levy process. Then, calculating the pricesof floating strike Asian call and put options CT (S; 1

T

∫S) and PT ( 1

T

∫S;S),

can be reduced via the duality relations

CT

(S;

1T

∫S)

= P′T(1;

1T

∫S′)

and

PT( 1T

∫S;S

)= C′

T

( 1T

∫S′; 1

)to the calculation of the prices of fixed strike Asian put and call optionsP′T (1; 1

T

∫S′) and C′

T ( 1T

∫S′; 1) respectively.

Remark 1.22. The duality relationships of Theorem 1.21 remain true ifwe replace the arithmetic average ΣT by the geometric or harmonic average,i.e. when the averaging is of the form

ΓT =( N∏i=1

STi

) 1N

or AT =N∑Ni=1

1STi

. (1.103)

Page 41: Applications of semimartingales and L´evy processes in ...webdoc.sub.gwdg.de/ebook/dissts/Freiburg/Papapantoleon2006.pdf · gales as driving processes. Another one, was to consider

1.5. THE CALL-PUT DUALITY IN OPTION PRICING 33

Proof. Indeed, using (1.100) we have that

CT

(S; Γ

)= E

[(ST −

( N∏i=1

STi

) 1N

)+]= E′

[(1−

( N∏i=1

STi

ST

) 1N

)+]

= E′

[(1−

( N∏i=1

eH′T−H

′Ti

) 1N

)+]

= E′

[(1−

( N∏k=1

eH′T−H

′T−Tk

) 1N

)+]

= E′

[(1−

( N∏k=1

eH′Tk

) 1N

)+]= P′T

(1; Γ′

),

where Γ′ denotes the geometric average corresponding to S′. Similarly, weget for the harmonic average:

CT

(S;A

)= E

[(ST −

N∑Ni=1

1STi

)+]= E′

[(1− N

ST∑N

i=11STi

)+]

= E′

[(1− N∑N

i=1

S′TiS′T

)+]= E′

[(1− N∑N

i=11

eH′

T−H′

Ti

)+]

= E′

[(1− N∑N

k=11

eH′

Tk

)+]= P′T

(1;A′

),

where now A′ is the harmonic average corresponding to S′. The proofs ofthe dualities between floating strike put and fixed strike call options followalong the same lines.

E. Forward-start options. The payoff of a forward-start call optionis (ST − St)+, where t ∈ [0, T ) is some prespecified future date; similarly,the payoff of the forward-start put option is (St−ST )+. Note that both theforward start call and the put option are at-the-money at time t, when thestrike is activated.

Suppose S ∈ M(P ). Therefore, for the price of the forward-start calloption we get that

Ct,T (S;S) = E[(ST − St

)+] = E[ST

(ST − St)+

ST

]= E′

[(1− St

ST

)+]= E′

[(1−

S′TS′t

)+]= E′[(1− eH

′T−H

′t)+]

= E′[(1− eH′T−H

′(T−u)−

)+],

where u = T − t. Appealing once again to the duality property (1.100),we obtain the following relation between a forward start call option and a

Page 42: Applications of semimartingales and L´evy processes in ...webdoc.sub.gwdg.de/ebook/dissts/Freiburg/Papapantoleon2006.pdf · gales as driving processes. Another one, was to consider

34 1. ON THE DUALITY PRINCIPLE: SEMIMARTINGALE SETTING

European plain vanilla put option

Ct,T (S;S) = E′[(1− eH′T−t)+]

= P′T−t(1;S′). (1.104)

Similarly, we get a relationship between a forward-start put option and aEuropean plain vanilla call option

Pt,T (S;S) = C′T−t(S

′; 1). (1.105)

Therefore, we have the following result.

Theorem 1.23. Let H be a Levy process. Then, calculating the prices offorward-start call and put options Ct,T (S;S) and Pt,T (S;S), can be reducedvia the duality relations

Ct,T (S;S) = P′T−t(1;S′)

and

Pt,T (S;S) = C′T−t(S

′; 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.

Page 43: Applications of semimartingales and L´evy processes in ...webdoc.sub.gwdg.de/ebook/dissts/Freiburg/Papapantoleon2006.pdf · gales as driving processes. Another one, was to consider

CHAPTER 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 dualityprinciple and its applications in option pricing. In Chapter 1 we concentratedon 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 payofffunction, see Definition 2.21, like power options.

The driving process is chosen to be a time-inhomogeneous Levy process,although most of the results can equally well be proved for general semi-martingales. Nevertheless, the class of time-inhomogeneous Levy processesconstitutes a very convenient class for financial modeling, since it providesenough flexibility to model the empirically observed behavior in financialmarkets, while allowing the fast pricing of the most liquidly traded assets.

It is well known that the efforts to calibrate standard Gaussian modelsto the empirically observed volatility surfaces very often do not producesatisfactory results. This phenomenon is not restricted to data from equitymarkets, but it is observed in interest rate and foreign exchange marketsas well. There are two basic aspects to which the classical models cannotrespond appropriately: the underlying distribution is not flexible enoughto capture the implied volatilities either across different strikes or acrossdifferent maturities. The first phenomenon is the so-called volatility smileand the second one the term structure of smiles; together they lead to thenon-flat implied volatility surface, a typical example of which can be seen inFigure 2.1. One way to improve the calibration results is to use stochasticvolatility models; we refer to Psychoyios, Skiadopoulos, and Alexakis (2003)for a thorough review of the various stochastic volatility approaches.

A fundamentally different approach is to replace the driving process.Levy processes offer a large variety of distributions that are capable of fittingthe return distributions in the real world and the volatility smiles in therisk-neutral world. Nevertheless, they cannot capture the term structure ofsmiles adequately. In order to take care of the change of the smile acrossmaturities, one has to go a step further and consider time-inhomogeneousLevy processes – also called additive processes – as the driving processes.For term structure models this approach was introduced in Eberlein et al.(2005) and further investigated in Eberlein and Kluge (2006a), where capand swaption volatilities were calibrated quite successfully.

35

Page 44: Applications of semimartingales and L´evy processes in ...webdoc.sub.gwdg.de/ebook/dissts/Freiburg/Papapantoleon2006.pdf · gales as driving processes. Another one, was to consider

36 2. ON THE DUALITY PRINCIPLE II: MULTIDIMENSIONAL PIIAC

1020

3040

5060

7080

90 1 2 3 4 5 6 7 8 9 10

10

10.5

11

11.5

12

12.5

13

13.5

14

maturitydelta (%) or strike

impl

ied

vol (

%)

Figure 2.1. Implied volatility surface of vanilla options onthe Euro/Dollar rate; date: 5 November 2001. Data availableat http://www.mathfinance.de/FF/sampleinputdata.txt

We should also point out two more aspects of our modeling approach.Firstly, since we are interested in options on several assets, we consider amulti-dimensional time-inhomogeneous Levy 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 Levyprocess allows us to capture all the phenomena we are interested in, hencewe do not need to employ a “multi-factor” model. Of course, given thata Levy process can have an infinite number of jumps on each finite timeinterval, hence an infinite number of assets should be employed to hedgea contingent claim perfectly, it is rather simplistic to call a Levy-drivenmodel a “one-factor” model. On the contrary, a non-trivial (i.e. not simplyGaussian) Levy process is already a high dimensional object.

Secondly, the theory is developed for general time-inhomogeneous Levyprocesses. Nevertheless, when we calibrate the model to empirical data, itis normally sufficient to split the trading horizon in two or three pieces(e.g. short, middle and long term) and employ a Levy process for each ofthese pieces. Hence, the driving motion is a “piecewise” Levy process. As aresult, the number of parameters used in the calibration is an input to thecalibration routine and not an output, depending on the data.

The chapter is organized as follows: in section 2.2 we present a detailedaccount of time-inhomogeneous Levy processes, and in section 2.3 we de-scribe the asset price model. In section 2.4 we describe a method for ex-ploring the duality principle in option pricing. The next section 2.5 containsduality relationships between options with α-homogeneous payoff functionsand in the final section 2.6 we derive duality results for options dependingon several assets.

Page 45: Applications of semimartingales and L´evy processes in ...webdoc.sub.gwdg.de/ebook/dissts/Freiburg/Papapantoleon2006.pdf · gales as driving processes. Another one, was to consider

2.2. TIME-INHOMOGENEOUS LEVY PROCESSES 37

2.2. Time-inhomogeneous Levy processes

1. Let Rd denote the d-dimensional Euclidean space. The Euclideanscalar product between two vectors u, v ∈ Rd is denoted by 〈u, v〉 or u>v,where u> denotes the transpose of the vector (or matrix) u. The Euclideannorm is denoted by | · |, ei denotes the unit vector where the i -th entry is 1and all others zero, i.e. ei = (0, . . . , 1, . . . , 0)> and 1 denotes the vector withall entries equal to 1, i.e. 1 = (1, . . . , 1)>.

The inner product is extended from real to complex numbers as follows:for u = (uk)1≤k≤d and v = (vk)1≤k≤d in Cd, set 〈u, v〉 :=

∑dk=1 ukvk; there-

fore we do not use the Hermitian inner product∑d

k=1 ukvk. Moreover, wedenote iv := (ivk)1≤k≤d and <v := (<vk)1≤k≤d ∈ Rd.

Let Md(R) denote the space of real d × d matrices and let ‖ · ‖ denotethe norm on Md(R) induced by the Euclidean norm on Rd. In addition, letMnd(R) denote the space of real n × d matrices and similarly ‖ · ‖ denotesthe induced norm on this space. Note that we could equally well work withany vector norm on Rd and the norms induced by, or consistent with, it onMd(R) and Mnd(R).

Define the set D := x ∈ Rd : |x| > 1, hence Dc is the unit ballin Rd. The function h = h(x) denotes a truncation function, where thecanonical choice is h(x) = x1|x|≤1 = x1Dc(x). We assume that h satisfiesthe antisymmetry property h(−x) = −h(x).

2. Let B = (Ω,F , (Ft)0≤t≤T , P ) be a complete stochastic basis and de-note by M(P ), resp. Mloc(P ) the class of martingales, resp. local martin-gales, on this stochastic basis. Throughout this chapter, we will work withthe following definition of a time-inhomogeneous Levy process.

Definition 2.1. A time-inhomogeneous Levy process, is an adapted,cadlag Rd-valued stochastic process L = (Lt)0≤t≤T with L0 = 0 a.s., suchthat the following conditions hold:

(D1): L has independent increments, i.e. Lt − Ls is independent ofFs, 0 ≤ s < t ≤ T ,

(D2): the law of Lt, for all t ∈ [0, T ], is described by the characteristicfunction

E[ei〈u,Lt〉

]= exp

t∫0

(i〈u, bs〉 −

12〈u, csu〉

+∫Rd

(ei〈u,x〉 − 1− i〈u, h(x)〉)λs(dx))

ds, (2.1)

where bt ∈ Rd, ct is a symmetric non-negative definite d×d matrix and λt is aLevy measure on Rd, i.e. it satisfies λt(0) = 0 and

∫Rd(|x|2∧1)λt(dx) <∞

for all t ∈ [0, T ]. Moreover, Assumption (AC) holds.

Assumption (AC). The triplets (bt, ct, λt) satisfyT∫

0

(|bt|+ ‖ct‖+

∫Rd

(1 ∧ |x|2)λt(dx))

dt <∞.

Page 46: Applications of semimartingales and L´evy processes in ...webdoc.sub.gwdg.de/ebook/dissts/Freiburg/Papapantoleon2006.pdf · gales as driving processes. Another one, was to consider

38 2. ON THE DUALITY PRINCIPLE II: MULTIDIMENSIONAL PIIAC

Applications in mathematical finance postulate that the asset price pro-cess is a martingale with respect to a risk-neutral measure; this naturallyleads to the existence of exponential moments for the driving process. There-fore we will often use the following assumption.

Assumption (EM). There exists a constant M > 1, such that the Levymeasures λt satisfy

T∫0

∫D

exp〈u, x〉λt(dx)dt <∞, ∀u ∈ [−M,M ]d.

Moreover, without loss of generality, we assume∫D exp〈u, x〉λt(dx) <∞ for

all t ∈ [0, T ] and u ∈ [−M,M ]d.

Remark 2.2. A time-inhomogeneous Levy process L = (Lt)0≤t≤T isan additive process, i.e. a cadlag, stochastically continuous process withindependent increments and L0 = 0 a.s. (Sato 1999, Definition 1.6).

3. A time-inhomogeneous Levy process that satisfies Assumption (AC)is a semimartingale on the stochastic basis (Ω,F ,F, P ). Indeed, it is a pro-cess with independent increments and absolutely continuous characteristics(in the sequel abbreviated PIIAC); cf. Lemmata 1.4 and 1.5 in Kluge (2005).The canonical representation of L = (Lt)0≤t≤T (cf. Jacod and Shiryaev 2003,II.2.34 and Eberlein et al. 2005) is

Lt =

t∫0

bsds+

t∫0

c1/2s dWs +

t∫0

∫Rd

h(x)d(µL − ν) +

t∫0

∫Rd

(x− h(x))dµL,

where c1/2 is a measurable version of the square root of c, W = (Wt)0≤t≤T isa P -standard Brownian motion on Rd, µL is the random measure of jumpsof the process L and ν(dt,dx) = λt(dx)dt is the P -compensator of the jumpmeasure µL. The triplet of predictable or semimartingale characteristics ofL with respect to P , T(L|P ) = (B,C, ν), is

Bt =

t∫0

bsds, Ct =

t∫0

csds, ν([0, t]×A) =

t∫0

∫A

λs(dx)ds, (2.2)

where A ∈ B(Rd). The triplet (b, c, λ) is called the triplet of differentiable orlocal characteristics of L.

A semimartingale with absolutely continuous characteristics admits nofixed times of discontinuity, therefore it is a quasi-left-continuous process.Consequently, the cumulant process of the time-inhomogeneous Levy processL = (Lt)0≤t≤T , defined as

K(u) = 〈u,B〉+12〈u,Cu〉+ (e〈u,x〉 − 1− 〈u, h(x)〉) ∗ ν, (2.3)

is continuous, hence E(K(u)) = eK(u) and never vanishes (cf. PropositionII.2.9 and Theorem III.7.4 in Jacod and Shiryaev 2003). In addition, we have

Page 47: Applications of semimartingales and L´evy processes in ...webdoc.sub.gwdg.de/ebook/dissts/Freiburg/Papapantoleon2006.pdf · gales as driving processes. Another one, was to consider

2.2. TIME-INHOMOGENEOUS LEVY PROCESSES 39

that, for all t ∈ [0, T ] and all u ∈ Rd

E[ei〈u,Lt〉

]= eK(iu)t and

ei〈u,L〉

eK(iu)∈Mloc(P ). (2.4)

Note that since the increments of the process L are independent, the cumu-lant and the predictable characteristics are deterministic processes.

Remark 2.3. Assumption (EM) renders the process L = (Lt)0≤t≤T aspecial semimartingale. Therefore, the canonical representation resumes theform (Jacod and Shiryaev 2003, II.2.38)

Lt =

t∫0

bsds+

t∫0

c1/2s dWs +

t∫0

∫Rd

x(µL − ν)(ds,dx), (2.5)

where B − B = (x − h(x)) ∗ ν. Of course, the characteristic function (2.1)and the cumulant (2.3) are modified accordingly, omitting the use of a trun-cation function and replacing b by b. In the sequel, whenever we work with atime-inhomogeneous Levy process that is a special semimartingale, we willsuppress the notation B and b and write B and b instead.

Remark 2.4. In addition, subject to Assumption (EM) every componentLi = (Li

t)0≤t≤T , i ∈ 1 . . . , d, of the time-inhomogeneous Levy process L =(L1, . . . , Ld)> becomes an exponentially special semimartingale (cf. Kallsenand Shiryaev 2002a, 2.12, 2.13).

4. An important consequence of the independent increments propertyis that the triplet of semimartingale characteristics T(L|P ) determines thelaw of the random variables generating the process L. This is the subject ofthe next result.

Lemma 2.5. The distribution of Lt, for a fixed t ∈ [0, T ], is infinitelydivisible with Levy triplet (b, c, λ), where

b :=

t∫0

bsds, c :=

t∫0

csds, λ(dx) :=

t∫0

λs(dx)ds. (2.6)

(The integrals should be understood componentwise.)

Proof. Firstly, we immediately have that b ∈ Rd and c is a symmetricnon-negative definite d×d matrix. Secondly, an application of the monotoneconvergence theorem yields that λ is a Borel measure on Rd and for anyintegrable function f we have∫

Rd

f(x)λ(dx) =

t∫0

∫Rd

f(x)λs(dx)ds. (2.7)

Therefore, using Assumption (AC) we get that∫

Rd(1 ∧ |x|2)λ(dx) <∞ andλ(0) = 0, i.e. λ is a Levy measure. Now, it suffices to notice that we can

Page 48: Applications of semimartingales and L´evy processes in ...webdoc.sub.gwdg.de/ebook/dissts/Freiburg/Papapantoleon2006.pdf · gales as driving processes. Another one, was to consider

40 2. ON THE DUALITY PRINCIPLE II: MULTIDIMENSIONAL PIIAC

rewrite (2.1) as

E[ei〈u,Lt〉

]= exp

i〈u, b〉 − 12〈u, cu〉+

∫Rd

(ei〈u,x〉 − 1− i〈u, h(x)〉)λ(dx)

,

(2.8)

and the result follows from the Levy–Khintchine formula.

5. The finiteness of the g-moment of the random variable Lt, for a PI-IAC L = (Lt)0≤t≤T and a submultiplicative function g, is related to anintegrability property of its compensator measure ν. For the notions of theg-moment and submultiplicative function, we refer to Definitions 25.1 and25.2 in Sato (1999).

Lemma 2.6 (g-Moment). Let g be a submultiplicative, locally bounded,measurable function on Rd. Then, the following statements are equivalent

(a)∫ t

0

∫Dg(x)λs(dx)ds <∞, for any t ∈ [0, T ]

(b) E[g(Lt)

]<∞, for any t ∈ [0, T ].

Proof. We argue along the lines of Kluge (2005, Lemma 1.6). Assumethat condition (a) holds and consider the Levy process L = (Lt)0≤t≤T such

that L1d= Lt, for some fixed t ∈ [0, T ]. Then, the Levy triplet of L, (b, c, λ),

is given by Lemma 2.5 and we have that∫D

g(x)λ(dx) =

t∫0

∫D

g(x)λs(dx)ds <∞. (2.9)

Applying Theorem 25.3 in Sato (1999), we get that E[g(Lt)

]<∞, or equiv-

alently E[g(L1)

]<∞, which immediately yields E

[g(Lt)

]<∞.

Conversely, assume that condition (b) holds and consider again the Levyprocess L = (Lt)0≤t≤T as above. By definition, E

[g(L1)

]= E

[g(Lt)

]< ∞,

which yields that E[g(Lt)

]<∞. Applying Theorem 25.3 in Sato (1999), we

conclude thatt∫

0

∫D

g(x)λs(dx)ds =∫D

g(x)λ(dx) <∞. (2.10)

The result is proved.

Consequently, since g(x) = exp〈u, x〉 is a submultiplicative function, weimmediately get the following result concerning Assumption (EM).

Corollary 2.7. The following statements are equivalent

(1)∫ T

0

∫D

exp〈u, x〉ν(dt,dx) <∞, ∀u ∈ [−M,M ]d

(2) E[exp〈u, LT 〉

]<∞, ∀u ∈ [−M,M ]d.

Page 49: Applications of semimartingales and L´evy processes in ...webdoc.sub.gwdg.de/ebook/dissts/Freiburg/Papapantoleon2006.pdf · gales as driving processes. Another one, was to consider

2.2. TIME-INHOMOGENEOUS LEVY PROCESSES 41

The following result, which is a straightforward generalization of Theo-rem 25.17 in Sato (1999), provides the existence of the characteristic functionof the random variable Lt on some strip in the complex plane.

Lemma 2.8. Let L = (Lt)0≤t≤T be an Rd-valued time-inhomogeneousLevy process with triplet of predictable characteristics (B,C, ν), that satisfiesAssumption (EM). For a fixed t ∈ [0, T ] and z ∈ Cd with <z ∈ [−M,M ]d,we have that

θs(z) = 〈z, bs〉+12〈z, csz〉+

∫Rd

(e〈z,x〉 − 1− 〈z, x〉)λs(dx) (2.11)

is well defined for all s ∈ [0, t], E∣∣e〈z,Lt〉

∣∣ <∞ and

E[e〈z,Lt〉

]= exp

t∫0

θs(z)ds. (2.12)

Proof. Firstly, as in the proof of Lemma 2.6, consider the Levy processL = (Lt)0≤t≤T such that L1

d= Lt, for some fixed t ∈ [0, T ]. Then we havethat

t∫0

∫D

∣∣e〈z,x〉∣∣λs(dx)ds =∫D

∣∣e〈z,x〉∣∣λ(dx) =∫D

e〈<z,x〉λ(dx) <∞, (2.13)

therefore E∣∣e〈z,Lt〉

∣∣ < ∞. Secondly, using Assumption (EM) and Theorem25.17(iii) in Sato (1999), we deduce that θs in (2.11) is well defined andfinite for all s ∈ [0, t]. Thirdly, using (2.13), Theorem 25.17 in Sato (1999)and Lemma 2.5, we can define for the Levy triplet (b, c, λ) of L

K(z) = 〈z, b〉+12〈z, cz〉+

∫Rd

(e〈z,x〉 − 1− 〈z, x〉)λ(dx)

and

E[e〈z,bL1〉

]= expK(z).

Finally, using Lemma 2.5 once again, we conclude that K(z) =∫ t0 θs(z)ds,

therefore E[e〈z,Lt〉

]= exp

∫ t0 θs(z)ds and the result is proved.

Remark 2.9. The function θs in (2.11) is called the cumulant associatedwith the infinitely divisible distribution characterized by the Levy triplet(bs, cs, λs), s ∈ [0, T ]. Moreover, the cumulant K of the process L, definedby (2.3), and the cumulant associated with the differential characteristics ofL, defined by (2.11), are related via

K(u) =∫ ·

0θs(u)ds. (2.14)

Page 50: Applications of semimartingales and L´evy processes in ...webdoc.sub.gwdg.de/ebook/dissts/Freiburg/Papapantoleon2006.pdf · gales as driving processes. Another one, was to consider

42 2. ON THE DUALITY PRINCIPLE II: MULTIDIMENSIONAL PIIAC

6. We will later consider linear transformations and projections of time-inhomogeneous Levy processes. The following result describes the law andthe predictable characteristics under such a transformation. Analogous re-sults for Levy processes can be found in Sato (1999, Proposition 11.10).

Proposition 2.10. Let L = (Lt)0≤t≤T be a time-inhomogeneous Levyprocess on Rd with predictable characteristics (B,C, ν). Let U be a real-valued n×d matrix (U ∈Mnd(R)). Then, UL = (ULt)0≤t≤T is an Rn-valuedtime-inhomogeneous Levy process with characteristics (BU , CU , νU ), where

bUs = Ubs +∫Rd

(h(Ux)− Uh(x))λs(dx)

cUs = UcsU> (2.15)

λUs (E) = λs(x ∈ Rd : Ux ∈ E), E ∈ B(Rn\0).

Here h(x) is a truncation function on Rn.

Proof. We clearly have that UL is an Rn-valued adapted, cadlag pro-cess, UL0 = 0 a.s. and the linearity of the transformation preserves theindependence of the increments. Regarding the law, we have for any z ∈ Rn

E[ei〈z,ULt〉

]= E

[ei〈U

>z,Lt〉]

= exp

t∫0

(i〈U>z, bs〉 −

12〈U>z, csU

>z〉

+∫Rd

(ei〈U>z,x〉 − 1− i〈U>z, h(x)〉)λs(dx)

)ds

= exp

t∫0

(i〈z, bUs 〉 −

12〈z, UcsU>z〉

+∫

Rn

(ei〈z,y〉 − 1− i〈z, h(y)〉)λUs (dy))

ds,

where bU is given by (2.15).Now, for ease of notation we consider h and h as the canonical truncation

functions. Since bs and λs satisfy Assumption (AC), and from the definitionof the induced norm we get that |Ux| ≤ ‖U‖|x| for all U ∈ Mnd(R) andx ∈ Rd, we can conclude that

T∫0

|bUs |ds ≤T∫

0

(|Ubs|+

∣∣ ∫Rd

(h(Ux)− Uh(x))λs(dx)∣∣)ds

≤ ‖U‖T∫

0

|bs|ds+

T∫0

∫Rd

|h(Ux)− Uh(x)|λs(dx)ds <∞

Page 51: Applications of semimartingales and L´evy processes in ...webdoc.sub.gwdg.de/ebook/dissts/Freiburg/Papapantoleon2006.pdf · gales as driving processes. Another one, was to consider

2.2. TIME-INHOMOGENEOUS LEVY PROCESSES 43

because

T∫0

∫Rd

|h(Ux)−Uh(x)|λs(dx)ds

≤T∫

0

∫Rd

|Ux||1|Ux|≤1−1|x|≤1|λs(dx)ds

=

T∫0

∫Rd

|Ux||1|Ux|≤1<|x| − 1|x|≤1<|Ux||λs(dx)ds

≤T∫

0

∫|Ux|≤1<|x|

|Ux|λs(dx)ds+

T∫0

∫|x|≤1<|Ux|

|Ux|λs(dx)ds

≤T∫

0

∫|x|>1

λs(dx)ds+ ‖U‖T∫

0

∫|x|≤1<‖U‖|x|

|x|λs(dx)ds

≤T∫

0

∫|x|>1

λs(dx)ds+ ‖U‖2

T∫0

∫ 1‖U‖<|x|≤1

|x|2λs(dx)ds <∞

Similarly, we have that

T∫0

‖cUs ‖ds =

T∫0

‖UcsU>‖ds ≤ ‖U‖‖U>‖T∫

0

‖cs‖ds <∞

and

T∫0

∫Rn

(|y|2 ∧ 1)λUs (dy)ds =

T∫0

∫Rd

(|Ux|2 ∧ 1)λs(dx)ds

≤ (‖U‖2 ∨ 1)

T∫0

∫Rd

(|x|2 ∧ 1)λs(dx)ds <∞.

Therefore, the triplets (bUt , cUt , λ

Ut ) satisfy Assumption (AC) and the state-

ment is proved.

Remark 2.11. Assume that the process L = (Lt)0≤t≤T is a special semi-martingale and consider a matrix U ∈ Mnd(R). Then the process UL =

Page 52: Applications of semimartingales and L´evy processes in ...webdoc.sub.gwdg.de/ebook/dissts/Freiburg/Papapantoleon2006.pdf · gales as driving processes. Another one, was to consider

44 2. ON THE DUALITY PRINCIPLE II: MULTIDIMENSIONAL PIIAC

(ULt)0≤t≤T is also a special semimartingale, since

T∫0

∫|y|>1

|y|λUs (dy)ds =

T∫0

∫|Ux|>1

|Ux|λs(dx)ds ≤ ‖U‖T∫

0

∫|Ux|>1

|x|λs(dx)ds

≤ ‖U‖T∫

0

∫‖U‖|x|>1

|x|λs(dx)ds (2.16)

≤ ‖U‖T∫

0

∫|x|>1

|x|λs(dx)ds

+ ‖U‖2

T∫0

∫ 1‖U‖<|x|≤1

|x|2λs(dx)ds <∞, (2.17)

where we have implicitly assumed that ‖U‖ ≥ 1; otherwise, we can concludealready from (2.16). Therefore, we can omit the use of truncation functionsin (2.15) and we have that BU = UB; compare with Remark 2.3.

Remark 2.12. Assume that the process L = (Lt)0≤t≤T satisfies As-sumption (EM) and consider a matrix U ∈ Mnd(R), a vector v ∈ Rn, andthe process UL = (ULt)0≤t≤T . If U>v ∈ [−M,M ]d, then

E[e〈v,ULT 〉] = E[e〈U>v,LT 〉] <∞

which follows directly from Assumption (EM) and Corollary 2.7. Therefore,UL has finite exponential moment if U>1 ∈ [−M,M ]d.

Remark 2.13. 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 normto consider is the max-column-sum norm ‖U‖1, defined by

‖U‖1 = maxj=1,...,d

n∑i=1

|Uij |.

Then, UL has finite exponential moment if

‖U‖1 ≤M.

7. We can describe the triplet of semimartingale characteristics of thedual of a 1-dimensional time-inhomogeneous Levy process in terms of thetriplet of the original process. In Chapter 1 we proved an analogous relation-ship for general semimartingales; cf. Theorem 1.5. However, since the tripletof a time-inhomogeneous Levy process determines the law, the relationshipbetween the triplets implies a relationship between the laws of the originaland the dual process.

Lemma 2.14. Let L = (Lt)0≤t≤T be a time-inhomogeneous Levy pro-cess with characteristic triplet (B,C, ν). Then L? := −L is again a time-inhomogeneous Levy process with triplet T(−L|P ) = T(L?|P ) = (B?, C?, ν?),

Page 53: Applications of semimartingales and L´evy processes in ...webdoc.sub.gwdg.de/ebook/dissts/Freiburg/Papapantoleon2006.pdf · gales as driving processes. Another one, was to consider

2.2. TIME-INHOMOGENEOUS LEVY PROCESSES 45

where

B? = −BC? = C (2.18)

1A(x) ∗ ν? = 1A(−x) ∗ ν, A ∈ B(R\0).

Proof. From the Levy–Khintchine representation we have that

ϕLt(u) = E[eiuLt

]= exp

t∫0

[ibsu−

cs2u2 +

∫R

(eiux − 1− iuh(x))λs(dx)]ds.

We get immediately

ϕ−Lt(u) = ϕLt(−u)

= exp

t∫0

[ibs(−u)−

cs2u2 +

∫R

(ei(−u)x − 1− i(−u)h(x))λs(dx)]ds

= exp

t∫0

[i(−bs)u−

cs2u2 +

∫R

(eiu(−x) − 1− iuh(−x))λs(dx)]ds.

Then, b?t = −bt, c?t = ct, and∫

1A(x)λ?s(dx) =∫

1A(−x)λs(dx), and theyclearly satisfy Assumption (AC). Since L? is a process with independentincrements and L?0 = 0 a.s., we can conclude it is a time-inhomogeneous Levyprocess and has characteristics B?

t =∫ t0 b

?sds = −Bt, C?t =

∫ t0 c

?sds = Ct and∫

1A(x)ν?(dt,dx) =∫

1A(x)λ?t (dx)dt =∫

1A(−x)ν(dt,dx).

8. Remark 2.15. The PIIACs L1, . . . , Ld are independent if and onlyif the matrices ct are diagonal and the Levy measures λt are supported onthe union of the coordinate axes. This follows similarly to Exercise 12.10 inSato (1999). Describing the dependence is a more difficult task; the bookof Muller and Stoyan (2002) consists of a comprehensive exposition of vari-ous dependence concepts and their applications. Tankov (2003) and Kallsenand Tankov (2006) introduced the notion of a Levy copula to describe thedependence of the components of multidimensional Levy processes.

Remark 2.16. If the triplet of local characteristics (bt, ct, λt) is not time-dependent, then the time-inhomogeneous Levy process L (PIIAC) becomes a(homogeneous) Levy process, i.e. a process with independent and stationaryincrements (PIIS). In that case, the distribution of L is described by theLevy triplet (b, c, λ), where λ is the Levy measure and the compensator ofµL becomes a product measure of the form ν = λ⊗ λ\, where λ\ denotes theLebesgue measure. In that case, equation (2.1) takes the form

E[ei〈u,Lt〉

]= etψ(u) (2.19)

where

ψ(u) = i〈u, b〉 − 12〈u, cu〉+

∫Rd

(ei〈u,x〉 − 1− i〈u, h(x)〉)λ(dx), (2.20)

which is called the characteristic exponent of the Levy process L.

Page 54: Applications of semimartingales and L´evy processes in ...webdoc.sub.gwdg.de/ebook/dissts/Freiburg/Papapantoleon2006.pdf · gales as driving processes. Another one, was to consider

46 2. ON THE DUALITY PRINCIPLE II: MULTIDIMENSIONAL PIIAC

2.3. Asset price model

1. The asset price process is modeled as follows: each component S i ofthe vector of asset price processes S = (S1, . . . , Sd)> is an exponential time-inhomogeneous Levy process, i.e. a stochastic process with representation

S it = S i

0 expLit, 0 ≤ t ≤ T ; (2.21)

here the superscript i refers to the i -th coordinate, i ∈ 1, . . . , d. Thedriving process L = (Lt)0≤t≤T is an Rd-valued time-inhomogeneous Levyprocess that satisfies Assumption (EM), with canonical decomposition

Lt =

t∫0

bsds+

t∫0

c1/2s dWs +

t∫0

∫Rd

x(µL − ν)(ds,dx). (2.22)

We assume that P is a risk-neutral (or, martingale) measure, i.e. theasset prices have mean rate of return µi , r − δi , where r is the risk-freeinterest rate and δi is the dividend yield of the i -th asset. Therefore, theauxiliary price processes Si = (Si

t)0≤t≤T , where Sit := eδ

i tS it, i ∈ 1, . . . , d,

once discounted at the rate r, become P -martingales. Consequently, thedrift characteristic B of the driving process L is completely determined bythe other two characteristics (C, ν), the rate of return of the asset and therisk-free interest rate.

2. Assume for the moment that r = δi ≡ 0 and S i0 ≡ 1 for all i ∈

1, . . . , d. Then, Si = S i = eLi

and we wish to determine the drift char-acteristic Bi in this special case, such that eL

i ∈ Mloc(P ). Using TheoremII.2.42, Corollary II.2.48 and section III.7a in Jacod and Shiryaev (2003),we know that for all predictable, integrable with respect to L processes ϑ(i.e. ϑ ∈ L(L)), such that ϑ · L is exponentially special, the following hold:

eϑ·L − eϑ·L ·K(u) ∈Mloc(P ) (2.23)

and

eϑ·L

G(ϑ)∈Mloc(P ), (2.24)

where K is the cumulant process of L, cf. (2.3) and Remarks 2.3 and 2.9,i.e. K(ϑ) =

∫ ·0 θs(ϑ)ds, with

θs(ϑ) = 〈ϑ, bs〉+12〈ϑ, csϑ〉+

∫Rd

(e〈ϑ,x〉 − 1− 〈ϑ, x〉)λs(dx), (2.25)

and

G(ϑ) = E(K(ϑ)) = eK(ϑ). (2.26)

As was already mentioned, L is a quasi-left-continuous process and the cu-mulant process K is continuous; moreover, K(ϑ) is deterministic if ϑ isdeterministic (see also Remarks III.7.15 in Jacod and Shiryaev 2003).

Page 55: Applications of semimartingales and L´evy processes in ...webdoc.sub.gwdg.de/ebook/dissts/Freiburg/Papapantoleon2006.pdf · gales as driving processes. Another one, was to consider

2.3. ASSET PRICE MODEL 47

Since Li = 〈ei , L〉, ei ∈ (−M,M)d ⊂ Rd and subject to Assumption(EM) each coordinate process Li of L is an exponentially special semimartin-gale, we get from (2.24) that

e〈ei ,L〉

G(ei )=

eLi

eK(ei )∈Mloc(P ). (2.27)

Now, from (2.27) we immediately get that

eLi ∈Mloc(P ) ⇔ K(ei ) = 0; (2.28)

the “if” part being obvious, the “only if” part follows from the uniquenessof the multiplicative decomposition of a special semimartingale (cf. Jacod1979, VI.2a and Theoreme (6.19)). Therefore, from (2.27) and (2.28) weconclude that

S i ∈Mloc(P ) ⇔ 〈ei , B〉+12〈ei , Cei 〉+

(e〈ei ,x〉 − 1− 〈ei , x〉

)∗ ν = 0.

(2.29)

Finally, using that every exponential PIIAC that is a local martingale isindeed a martingale (cf. Eberlein, Jacod, and Raible 2005, pp. 79-80), weconclude that, for all i ∈ 1, . . . , d

S i = eLi ∈M(P ) ⇔ Bi +

12C ii +

(ex

i − 1− xi) ∗ ν = 0; (2.30)

compare with (1.29).

Remark 2.17. Another way to derive the condition above, is to extractthe i -th coordinate process from L and then apply the results of Chapter1. Indeed, using Proposition 2.10 for U = e>i , we get that Li has localcharacteristics (bi , cii , λi ), where λi (E) = λ(x ∈ Rd : xi ∈ E), E ∈ B(R).Then, applying (1.29) we immediately get (2.30).

Remark 2.18. Note that since each asset is driven by one componentof the multidimensional time-inhomogeneous Levy process, the off-diagonalelements of the matrix C and the contribution of the jump measure notconcentrated on the coordinate axes, do not participate in the martingalecondition (2.30). Nevertheless, they describe the dependence between theasset price processes S i and S j , j 6= i . See also Example 2.29.

3. Returning to the general case, where r ≥ 0, δi ≥ 0 and S i0 ≥ 0,

i ∈ 1, . . . , d, it is immediately clear from the previous considerations thatthe i -th component of the drift vector B must have the form

Bit =

t∫0

(r − δi )ds− 12

t∫0

ciisds−

t∫0

∫Rd

(exi − 1− xi )ν(ds,dx) (2.31)

and then Si discounted is a P -martingale, for all i ∈ 1, . . . , d.

4. Markets modeled by exponential time-inhomogeneous Levy processesare incomplete and there exists a large class of equivalent martingale (or, riskneutral) measures. Eberlein and Jacod (1997) provide a complete character-ization of the class of equivalent martingale measures for exponential Levy

Page 56: Applications of semimartingales and L´evy processes in ...webdoc.sub.gwdg.de/ebook/dissts/Freiburg/Papapantoleon2006.pdf · gales as driving processes. Another one, was to consider

48 2. ON THE DUALITY PRINCIPLE II: MULTIDIMENSIONAL PIIAC

models; this was later extended to general semimartingales in Gushchin andMordecki (2002).

In this work, we do not dive into the theory of choosing a martingalemeasure, we rather assume that the choice has already taken place. We referto Eberlein and Keller (1995) and Kallsen and Shiryaev (2002a) for theEsscher transform, Frittelli (2000) and Fujiwara and Miyahara (2003) forthe minimal entropy martingale measure and Bellini and Frittelli (2002) forminimax martingale measures, to mention just a small part of the literatureon this subject. A unifying exposition – in terms of f -divergences – of thedifferent methods for selecting an equivalent martingale measure can befound in Goll and Ruschendorf (2001).

Alternatively, one can consider the choice of the martingale measureas the result of a calibration to the smile of the vanilla options market.Hakala and Wystup (2002) describe the calibration procedure in detail; werefer to Cont and Tankov (2004, 2006) and Belomestny and Reiß (2006) fornumerically stable calibration methods for Levy driven models.

Remark 2.19. In the above setting, we can easily incorporate dynamictime-dependent interest rates and dividend yields; the drift term (2.31)would have a similar form, taking rs and δs into account.

Remark 2.20. 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 pointof view though, it is not too restrictive to assume (EM), since all examplesof Levy models we are interested in, e.g. the Generalized Hyperbolic model(cf. Eberlein and Prause 2002), the CGMY model (cf. Carr et al. 2002) orthe Meixner model (cf. Schoutens 2002), possess moments of all order andmoment generating functions.

2.4. General description of the method

In Chapter 1 we developed the mathematical tools required to study theduality principle in an abstract semimartingale framework. Here, we aim atexploring the duality principle in the more practically-oriented frameworkdescribed in sections 2.2 and 2.3: when interest rates and dividend yieldsare present and the driving process is a time-inhomogeneous Levy process.In addition, we want to derive put-call dualities for options with payofffunctions homogeneous of degree higher than one and for options on severalunderlying assets. We begin by defining an α-homogeneous payoff functionand then describe the method we will use to explore dualities; it is basedon the choice of a suitable numeraire and the subsequent change of theprobability measure, pioneered by Geman et al. (1995).

Definition 2.21. A payoff function f : R+ × R+ → R+ is called α-homogeneous if f is a homogeneous function of degree α ≥ 1, that is forc, x, y ∈ R+, holds

f(cx, cy) = cαf(x, y).

In order to derive duality relationships in this framework, the discountedasset price process corrected for dividends serves as the numeraire, in case

Page 57: Applications of semimartingales and L´evy processes in ...webdoc.sub.gwdg.de/ebook/dissts/Freiburg/Papapantoleon2006.pdf · gales as driving processes. Another one, was to consider

2.4. GENERAL DESCRIPTION OF THE METHOD 49

the option payoff is homogeneous of degree one. Using this numeraire, eval-uated at the time of maturity, as the Radon–Nikodym derivative, we form anew measure. Under this new measure, the numeraire asset is riskless whileall other assets, including the savings account, are now risky. As a result,the new “riskless rate” is the dividend yield of the numeraire asset.

In case the payoff function is α-homogeneous for α > 1, we will haveto modify the asset price process appropriately, so that it serves as thenumeraire. Consequently, the asset price dynamics under the new measurewill depend on α as well. For the sake of simplicity, we assume here thatα = 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 priceprocess(es):

(P1): L = L1 is a 1-dimensional PIIAC, L2 = k is constant andS1 = S1

0 expL1, S2 = expL2 = K;(P2): L = (L1, L2) is a 2-dimensional PIIAC and S i = S i

0 expLi ,i = 1, 2.

According to the general arbitrage pricing theory, the value of an optionon assets S1, S2 with payoff function f is equal to its discounted expectedpayoff under an equivalent martingale measure. Throughout this chapter,we will assume that options start at time 0 and mature at T ; therefore wehave

V = e−rTE[f(S1T , S

2T

)]. (2.32)

We choose, without loss of generality, asset S1 as the numeraire andexpress the value of the option in terms of this numeraire, which yields

V = e−rTE[S1T f

(1,S2T

S1T

)]= S1

0e−δ1TE

[e−rTS1

T

e−δ1TS10

f

(1,S2T

S1T

)]. (2.33)

Now, we can define on (Ω,F , (Ft)0≤t≤T ) a new probability measure P ′ viathe Radon–Nikodym derivative

dP ′

dP=

e−rTS1T

e−δ1TS10

=: ZT , (2.34)

and EZT = 1. The valuation problem, under the measure P ′, becomes

V = S10e−δ

1TE′[f(1, S1,2

T

)](2.35)

where we have defined the process S1,2 = (S1,2t )0≤t≤T with S1,2

t = S2t

S1t.

Since the discounted auxiliary process S1 is a P -martingale, we deduced(P ′|Ft)d(P |Ft)

= Zt, 0 ≤ t ≤ T

and because Z > 0 P -a.s., we get that P ′ ∼ P . Therefore, we can applyGirsanov’s theorem for semimartingales, that allows us to determine thedynamics of S1,2 under P ′. If S1,2, discounted at the new “riskless rate” δ1,is a P ′-martingale, then we have transformed the original valuation probleminto a simpler one.

Page 58: Applications of semimartingales and L´evy processes in ...webdoc.sub.gwdg.de/ebook/dissts/Freiburg/Papapantoleon2006.pdf · gales as driving processes. Another one, was to consider

50 2. ON THE DUALITY PRINCIPLE II: MULTIDIMENSIONAL PIIAC

2.5. Options with α-homogeneous payoff functions

1. The setting we will work in, is that of (P1): L = (Lt)0≤t≤T is thedriving real-valued time-inhomogeneous Levy process, with triplet of pre-dictable characteristics (B,C, ν), S1 = S0 expL = S and L2 = k, such thatS2 = ek = K is the strike price of the option.

In accordance with the standard notation in mathematical finance, wewill denote by σ2, instead of c, the local diffusion characteristic, which cor-responds to the volatility in the Black–Scholes model. Therefore, the char-acteristic C in (2.2) has the form C =

∫ ·0 σ

2sds.

2. The main aim of this section is to prove a duality relationship betweenpower options. The payoff of the power call and put option respectively is[

(ST −K)+]α and

[(K − ST )+

]αwhere α ∈ N is the power index. We introduce the following notation for thevalue of a power call option with strike K and power index α

C(S0,K, α; r, δ, C, ν) = e−rTE[(ST −K)+

]αwhere the asset price process is modeled as an exponential PIIAC accordingto (2.21), (2.22) and (2.31). Similarly, for a power put option we set

P(S0,K, α; r, δ, C, ν) = e−rTE[(K − ST )+

]α.

If the power index equals one, then we have a standard European (plainvanilla) 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. LetD+ = D ∩ R+ and D− = D ∩ R−.

Assumption (M). The Levy measures λt satisfyT∫

0

∫D−

|x|λt(dx)dt <∞ and

T∫0

∫D+

xeαxλt(dx)dt <∞.

Theorem 2.22. Assume that the asset price process evolves as an ex-ponential time-inhomogeneous Levy process according to (2.21), (2.22) and(2.31) and assumption (M) is in force. Then, we can relate the power calland put option via the following duality:

C(S0,K, α; r, δ, C, ν

)= KαSα0 CαTP

( 1S0,K, α; δ, r, C, ν

)(2.36)

where

CαT = exp((α−1)(δ−r)T +

α

2(1−α)CT + (eαx−1−αeαx+αe(α−1)x) ∗ νT

),

K = K−1e−C?T , C?T is given by (2.47) and 1A(x) ∗ ν = 1A(−x)eαx ∗ ν.

Proof. We observe that, since the payoff function is α-homogeneousfor α > 1, the discounted asset price process corrected for dividends cannotserve as a numeraire; that is because(

[e−rtSt]α)

0≤t≤T=([e(δ−r)tSt]α

)0≤t≤T

=(Sα0 eα(δ−r)t+αLt

)0≤t≤T

Page 59: Applications of semimartingales and L´evy processes in ...webdoc.sub.gwdg.de/ebook/dissts/Freiburg/Papapantoleon2006.pdf · gales as driving processes. Another one, was to consider

2.5. OPTIONS WITH α-HOMOGENEOUS PAYOFF FUNCTIONS 51

is not a P -martingale, unless α = 1.Let us denote by Lα = (Lαt )0≤t≤T the martingale part of the exponent,

that is

Lαt =

t∫0

ασsdWs +

t∫0

∫R

αx(µL − ν)(ds,dx).

Since Lα is an exponentially special semimartingale, using Lemma 2.15 andTheorem 2.18 in Kallsen and Shiryaev (2002a), we deduce that its expo-nential compensator, denoted by CLα = (CLαt )0≤t≤T , exists and has theform

CLαt =12

t∫0

α2σ2sds+

t∫0

∫R

(eαx − 1− αx)ν(ds,dx).

Then, by definition of the exponential compensator and using Eberlein et al.(2005, pp. 79-80), we conclude that exp(Lα − CLα) ∈M.

Now, the price of the power call option can be expressed as follows:

C = e−rTE[(ST −K)+

]α= e−δTSα0E

[e−rTSαTK

α

e−δTSα0

[(K−1 − S−1

T )+]α]

= e−δTSα0KαE

[exp

((δ − r)T + αBT + CLαT

)× exp

(LαT − CLαT

) [(K−1 − S−1

T )+]α ]

= e−δTSα0KαCTE

[exp

(LαT − CLαT

) [(K−1 − S−1

T )+]α ] (2.37)

where, using (2.31) and (2.2), we have that

log CT := (δ − r)T + αBT + CLαT

= (α− 1)(r − δ)T +α(α− 1)

2

T∫0

σ2sds

+

T∫0

∫R

(eαx − αex + α− 1)ν(ds,dx). (2.38)

We define on (Ω,F , (Ft)0≤t≤T ) a new probability measure P ′, via theRadon–Nikodym derivative

dP ′

dP= exp

(LαT − CLαT

)= ZT (2.39)

and the valuation problem (2.37) becomes

C = e−δTSα0KαCTE

′ [(K ′ − S′T )+]α (2.40)

where K ′ := 1K and S′ := 1

S .Since the measures P and P ′ are related via the density process Z =

(Zt)0≤t≤T , which is a positive P -martingale with Z0 = 1, we immediately

Page 60: Applications of semimartingales and L´evy processes in ...webdoc.sub.gwdg.de/ebook/dissts/Freiburg/Papapantoleon2006.pdf · gales as driving processes. Another one, was to consider

52 2. ON THE DUALITY PRINCIPLE II: MULTIDIMENSIONAL PIIAC

deduce that P ′ ∼ P . The density process can be represented in the “usual”form

Zt = E

[dP ′

dP

∣∣∣∣Ft] = exp(Lαt − CLαt

)= exp

( t∫0

ασsdWs +

t∫0

∫R

αx(µL − ν)(ds,dx)

− 12

t∫0

α2σ2sds−

t∫0

∫R

(eαx − 1− αx)ν(ds,dx)

), (2.41)

or as the stochastic exponential of a suitable time-inhomogeneous Levy pro-cess Xα, i.e. Z = E(Xα), where

Xα =

·∫0

ασsdWs +

·∫0

∫R

(eαx − 1

)(µL − ν)(ds,dx);

cf. Kallsen and Shiryaev (2002a, Lemma 2.6).An application of Girsanov’s theorem for semimartingales, cf. Jacod and

Shiryaev (2003, Theorem III.3.24), yields that the triplet of predictable char-acteristics of L under P ′ is T(L|P ′) = (B′, C ′, ν ′), for

B′ = B + βα · C + x(Y α − 1) ∗ νC ′ = C (2.42)ν ′ = Y α · ν,

where we can take the following versions of βα and Y α:

βα ≡ α and Y α = eαx. (2.43)

Indeed, as in part (a) of the proof of Theorem 1.5 and using Appendix B,we have that

〈Zc, Lc〉 =

⟨ ·∫0

Zs−ασsdWs,

·∫0

σsdWs

⟩=

·∫0

Zs−ασ2sds,

therefore, we conclude that βα ≡ α. Moreover, we can choose Y α = eαx,because for any non-negative P-measurable function U = U(ω; t, x), we get

MPµL(eαxU) = E

[ T∫0

∫R

eαxU(ω; t, x)µL(ω; dt,dx)]

= E

[ ∑0≤t≤T

eα∆Lt(ω)U(ω; t,∆Lt(ω))1∆Lt(ω) 6=0

]

= E

[ T∫0

∫R

Zt(ω)Zt−(ω)

U(ω; t, x)µL(ω; dt,dx)]

= MPµL

( ZZ−

U),

Page 61: Applications of semimartingales and L´evy processes in ...webdoc.sub.gwdg.de/ebook/dissts/Freiburg/Papapantoleon2006.pdf · gales as driving processes. Another one, was to consider

2.5. OPTIONS WITH α-HOMOGENEOUS PAYOFF FUNCTIONS 53

since ZZ−

1∆Z 6=0 = exp(α∆L).Now, using Theorem II.4.15 in Jacod and Shiryaev (2003), we deduce

that a time-inhomogeneous Levy process remains a time-inhomogeneousLevy process under the measure P ′, because the P ′-characteristics of L aredeterministic and satisfy Assumption (AC).

In addition, the P ′-canonical decomposition of L is

Lt =

t∫0

b′sds+

t∫0

σsdW ′s +

t∫0

∫R

x(µL − ν ′)(ds,dx) (2.44)

where

B′t =

t∫0

b′sds = (r − δ)t+(α− 1

2

) t∫0

σ2sds

+

t∫0

∫R

(e−αx − e(1−α)x + x)ν ′(ds,dx). (2.45)

Here, W ′ = W −∫ ·0 ασsds is a P ′-Brownian motion and ν ′ = Y αν is the P ′

compensator of the jumps of L.Define the dual process of L, L? := −L; its triplet of predictable char-

acteristics T(L?|P ′) = (B?, C?, ν?), is given by Lemma 2.14 – in terms ofT(L|P ′). The canonical decomposition of L? is

L?t = −t∫

0

b′sds+

t∫0

σsdW ′s +

t∫0

∫R

x(µL? − ν?)(ds,dx), (2.46)

where µL?

is the random measure associated with the jumps of L?, i.e.1A(x) ∗ µL?

= 1A(−x) ∗ µL.We have that S′ = 1

S = 1S0

e−L = 1S0

eL?

and from equations (2.46) and(2.45), we can easily deduce that

(e(r−δ)tS′t

)0≤t≤T is not a P ′-martingale for

α 6= 1. Nevertheless, we can easily construct a martingale, and then takeadvantage of the fact that the compensating terms of a time-inhomogeneousLevy process are deterministic.

Indeed, adding the appropriate terms, we can re-write L? as L? = C?+L,where

C? = (1− α)

·∫0

σ2sds−

·∫0

∫R

(e−αx − e(1−α)x + 1− e−x)ν ′(ds,dx) (2.47)

and L = (Lt)0≤t≤T is a time-inhomogeneous Levy process with triplet ofpredictable characteristics T(L|P ′) = (B? − C?, C, ν?). Define S = S−1

0 ebLand then

(e(r−δ)tSt

)0≤t≤T ∈M(P ′).

Page 62: Applications of semimartingales and L´evy processes in ...webdoc.sub.gwdg.de/ebook/dissts/Freiburg/Papapantoleon2006.pdf · gales as driving processes. Another one, was to consider

54 2. ON THE DUALITY PRINCIPLE II: MULTIDIMENSIONAL PIIAC

Therefore, we can conclude the proof, since

C = e−δTSα0KαCTE

′ [(K ′ − S′T )+]α

= e−δTSα0KαCTE

′[(K ′ − eC?

T ST )+]α

= e−δTSα0KαCαTE′

[(K− ST )+

]α,

where K := K ′e−C?T = K−1e−C?

T and CαT := CT eαC?T .

Setting α = 1 in the previous Theorem, we immediately get a dualityrelationship between European (plain vanilla) call and put options; comparewith Theorem 1.18.

Corollary 2.23. Assume that the asset price evolves as an exponentialtime-inhomogeneous Levy process and assumption (M) is in force. Then, wecan relate European call and put options via the following duality:

C(S0,K; r, δ, C, ν

)= KS0P

( 1S0,

1K

; δ, r, C, ν)

(2.48)

where 1A(x) ∗ ν = 1A(−x)ex ∗ ν, A ∈ B(R\0).

Remark 2.24. It is interesting to point out that a different duality re-lating European (and American) call and put options, in the Black-Scholesmodel, was derived by Peskir and Shiryaev (2002). They use the mathemat-ical concept of negative volatility and their main result states that

C(ST ,K;σ) = P(−ST ,−K;−σ). (2.49)

See also the discussion – and the corresponding cartoon! – in Haug (2002).

3. In this framework, we can easily derive duality relationships betweenself-quanto and European plain vanilla options. This result is, of course, aspecial case of Theorem 2.30, that will be proved later. Nevertheless, wegive a short proof, since it simplifies considerably due to the driving processbeing 1-dimensional.

The payoff of the self-quanto call and put option is

ST (ST −K)+ and ST (K − ST )+

respectively. Introduce the following notation for the value of the self-quantocall option

QC(S0,K; r, δ, C, ν) = e−rTE[ST (ST −K)+

]and similarly, for the self-quanto put option we set

QP(S0,K; r, δ, C, ν) = e−rTE[ST (K − ST )+

].

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 Levy measures λt satisfyT∫

0

∫D−

|x|λt(dx)dt <∞ and

T∫0

∫D+

e2xλt(dx)dt <∞.

Page 63: Applications of semimartingales and L´evy processes in ...webdoc.sub.gwdg.de/ebook/dissts/Freiburg/Papapantoleon2006.pdf · gales as driving processes. Another one, was to consider

2.5. OPTIONS WITH α-HOMOGENEOUS PAYOFF FUNCTIONS 55

Theorem 2.25. Assume that the asset price evolves as an exponentialtime-inhomogeneous Levy process and assumption (M′) is in force. We canrelate the self-quanto and European plain vanilla call and put options via thefollowing dualities:

QC(S0,K; r, δ, C, ν) = S0C?TC(S0,K; δ, r, C, fν) (2.50)

QP(S0,K; r, δ, C, ν) = S0C?TP(S0,K; δ, r, C, fν) (2.51)

where C?T = eC?T , C?T is given by (2.53), K = Ke−C

?T and f(x) = ex.

Proof. Obviously, we choose asset S as the numeraire, define a newprobability measure P ′ via the Radon–Nikodym derivative in (2.34), andthe original valuation problem becomes

QC = e−δTS0E′ [(ST −K)+

]. (2.52)

We want to calculate the P ′-characteristic triplet of L. Arguing as inthe proof of Theorem 2.22, the density process Z = (Zt)0≤t≤T has the form(2.41) for α = 1. Hence, the tuple of predictable processes (β, Y ) associatedwith the process L and the measure P ′ is

β ≡ 1 and Y = ex.

Therefore, similarly to (2.44) and (2.45), L has the P ′-canonical decompo-sition

Lt =

t∫0

b′sds+

t∫0

σsdW ′s +

t∫0

∫R

x(µL − ν ′)(ds,dx)

where

b′s = r − δ +σ2s

2+∫R

(e−x − 1 + x)λ′s(dx).

Here, W ′ = W −∫ ·0 σsds is a P ′-Brownian motion and ν ′ = Y ν is the P ′-

compensator of µL. Consequently, T(L|P ′) = (B′, C, ν ′), which are describedabove.

Notice that(e(r−δ)teLt

)0≤t≤T is not a P ′-martingale; however, we can

define L? = (L?t )0≤t≤T where

L?t :=

t∫0

σsdW ′s +

t∫0

∫R

x(µL − ν ′)(ds,dx)

+ (δ − r)t−t∫

0

σ2s

2ds−

t∫0

∫R

(ex − 1− x)ν ′(ds,dx)

and then(e(r−δ)teL

?t)0≤t≤T ∈M(P ′). Next, we can express L as L = L?+C?,

where

C?T = 2(r − δ)T +

T∫0

σ2sds+

T∫0

∫R

(ex + e−x − 2)ν ′(ds,dx). (2.53)

Page 64: Applications of semimartingales and L´evy processes in ...webdoc.sub.gwdg.de/ebook/dissts/Freiburg/Papapantoleon2006.pdf · gales as driving processes. Another one, was to consider

56 2. ON THE DUALITY PRINCIPLE II: MULTIDIMENSIONAL PIIAC

Now, by re-arranging the terms in (2.52), using again the fact that thecompensating terms are deterministic, the result follows, since

QC = e−δTS0E′[(S0eLT −K

)+]= e−δTS0E

′[(S0eL

?T eC

?T −K

)+]= e−δTS0eC

?TE′

[(S0eL

?T −Ke−C

?T)+]

= e−δTS0eC?TE′

[(S0eL

?T −K

)+], (2.54)

where K := Ke−C?T . The case of the quanto put option is similar.

2.6. Options on several assets

1. The aim in this section is to derive duality relationships betweenoptions involving two, or more, assets – such as swaps and quanto options –and European plain vanilla options. Therefore, similarly to the case for exoticoptions, the duality principle allows us to reduce a problem involving two,or more, random variables and joint distributions to a problem involving asingle random variable and distribution. The latter problem can be treatedusing 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 of-ten referred to as “Margrabe” option. Schroder (1999) worked in a gen-eral semimartingale setting, but provided explicit expressions only in theBlack–Scholes framework. Fajardo and Mordecki (2006b) considered (ho-mogeneous) Levy processes as the driving processes.

The setting we will work in is that of (P2): L = (L1, . . . , Ld)> is the driv-ing Rd-valued time-inhomogeneous Levy process with triplet of predictablecharacteristics T(L|P ) = (B,C, ν). The vector of asset price processes isS = (S1, . . . , Sd)>, where we set for convenience

S it = S i

0 exp((r − δi )t+ Li

t

), i = 1, . . . , d. (2.55)

The canonical decomposition of the time-inhomogeneous Levy process L =(Lt)0≤t≤T , which satisfies Assumption (EM), is

Lt =

t∫0

bsds+

t∫0

c1/2s dWs +

t∫0

∫Rd

x(µL − ν)(ds,dx) (2.56)

and the i -th component of the (modified) drift characteristic is set to be

Bit = −1

2

t∫0

ciisds−

t∫0

∫Rd

(exi − 1− xi )ν(ds,dx); (2.57)

compare with (2.31).

Page 65: Applications of semimartingales and L´evy processes in ...webdoc.sub.gwdg.de/ebook/dissts/Freiburg/Papapantoleon2006.pdf · gales as driving processes. Another one, was to consider

2.6. OPTIONS ON SEVERAL ASSETS 57

2. The following result is the key tool required to derive the dualityrelationships in this setting. It provides the characteristic triplet of a 1-dimensional time-inhomogeneous Levy process – defined as the scalar prod-uct of a vector with the d-dimensional Levy process L – in terms of thetriplet of L under an equivalent change of probability measure. The newmeasure is defined via an Esscher transformation, cf. Kallsen and Shiryaev(2002a).

Theorem 2.26. Let L = (Lt)0≤t≤T be an Rd-valued time-inhomogeneousLevy process that satisfies Assumption (EM), with triplet of predictable char-acteristics T(L|P ) = (B,C, ν). Let u, v be vectors in Rd such that v ∈(−M,M)d and u + v ∈ [−M,M ]d. Define the measure P ′ via the Radon–Nikodym derivative

dP ′

dP=

e〈v,LT 〉

E[e〈v,LT 〉].

Then, the process Lu = (Lut )0≤t≤T , where Lut := 〈u, Lt〉, is a 1-dimensionaltime-inhomogeneous Levy process with triplet of predictable characteristicsT(Lu|P ′) = (Bu, Cu, νu) with

bus = 〈u, bs〉+ 〈u, csv〉+∫

Rd

〈u, x〉(e〈v,x〉 − 1

)λs(dx)

cus = 〈u, csu〉 (2.58)

λus (E) = λ′s(x ∈ Rd : 〈u, x〉 ∈ E), E ∈ B(R\0),

where λ′s is a measure defined by

λ′s(A) =∫A

e〈v,x〉λs(dx), A ∈ B(Rd\0).

Proof. There are several equivalent ways to prove the above result,which are represented in the following diagram:

T(L|P ′)

(b)

(U)

))SSSSSSSSSSSSSS

T(L|P )(e)

E′[eiz〈u,L〉]

//

(G)

(a)

55lllllllllllll

(c)

(U) ))RRRRRRRRRRRRRT(Lu|P ′)

T(Lu|P )

(d)

(G)

55kkkkkkkkkkkkkk

(2.59)

where(G) // means that we use Girsanov’s theorem to calculate the right

side triplet from the left side one and(U) // means that we use Proposition

2.10 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 Eberleinand Papapantoleon (2005b) and will not be pursued further here, since theother methods are more general. We will prove the result using steps (a)and (b), with the proof using steps (c) and (d) being analogous.

Page 66: Applications of semimartingales and L´evy processes in ...webdoc.sub.gwdg.de/ebook/dissts/Freiburg/Papapantoleon2006.pdf · gales as driving processes. Another one, was to consider

58 2. ON THE DUALITY PRINCIPLE II: MULTIDIMENSIONAL PIIAC

Step 1: (a) T(L|P )(G) // T(L|P ′)

The process 〈v, L〉 is a 1-dimensional time-inhomogeneous Levy processon (Ω,F ,F, P ), see Proposition 2.10, a special and exponentially specialsemimartingale since v ∈ (−M,M)d, cf. Remark 2.11, and

E[e〈v,Lt〉] = E(K(v))t = eK(v)t , 0 ≤ t ≤ T . (2.60)

The process Z = (Zt)0≤t≤T defined by Z := e〈v,L〉−K(v) satisfies Z > 0a.s., Z ∈M(P ) and EZT = 1, therefore the probability measure P ′ definedon (Ω,F , (F)0≤t≤T ) via dP ′ = ZTdP is equivalent to P (P ′ ∼ P ) and thedensity is given by

Zt =d(P ′|Ft)d(P |Ft)

= e〈v,Lt〉−K(v)t , 0 ≤ t ≤ T . (2.61)

Moreover, using Theorem 2.19 in Kallsen and Shiryaev (2002a), we canexpress Z as follows:

Z = E(v> · Lc + (ev

>x − 1) ∗ (µL − ν)). (2.62)

Now, an application of Girsanov’s theorem for semimartingales (Jacodand Shiryaev 2003, Theorem III.3.24), yields that T(L|P ′) = (B′, C ′, ν ′) is

B′i = Bi + (Cβ)i + xi (Y − 1) ∗ ν, i ∈ 1, . . . , dC ′ = C (2.63)

ν ′ = Y · ν

where we can take the following versions of β and Y :

β = v and Y = e〈v,x〉. (2.64)

Indeed, using (2.62), we have that

Zc = Z− · (v>Lc) = (Z−v)> · Lc

from which we immediately get that

〈Zc, Li ,c〉 =

·∫0

d∑j=1

cijs v

jZs−ds;

see also Lemma 2.20 in Kallsen and Shiryaev (2002a). Similarly, for theverification of Y we can either rely on Lemma 2.20 in Kallsen and Shiryaev(2002a), or calculate explicitly, for any P-measurable function U = U(ω; t, x)

Page 67: Applications of semimartingales and L´evy processes in ...webdoc.sub.gwdg.de/ebook/dissts/Freiburg/Papapantoleon2006.pdf · gales as driving processes. Another one, was to consider

2.6. OPTIONS ON SEVERAL ASSETS 59

on Ω× [0, T ]× Rd, that

MPµL

(e〈v,x〉U

)= E

[ T∫0

∫Rd

e〈v,x〉U(ω; t, x)µL(ω; dt,dx)]

= E

[ ∑0≤t≤T

e〈v,∆Lt(ω)〉U(ω; t,∆Lt(ω))1∆Lt(ω) 6=0

]

= E

[ ∑0≤t≤T

e〈v,Lt(ω)〉−K(v)t

e〈v,Lt−(ω)〉−K(v)t−U(ω; t,∆Lt(ω))1∆Lt(ω) 6=0

]

= E

[ T∫0

∫Rd

Zt(ω)Zt−(ω)

U(ω; t, x)µL(ω; dt,dx)]

= MPµL

( ZZ−

U). (2.65)

In addition, since there exists a deterministic version of the P ′-characteristicsof L we conclude, using Theorem II.4.15 in Jacod and Shiryaev (2003), thatL is a semimartingale with independent increments on (Ω,F ,F, P ′).

Regarding the differential characteristics of L under P ′, we have that

T∫0

|b′s|ds ≤T∫

0

(|bs|+ |csv|+

∣∣ ∫Rd

x(ev>x − 1)λs(dx)

∣∣)ds≤

T∫0

(|bs|+ |v|‖cs‖+

∫Rd

|x(ev>x − 1)|λs(dx))ds <∞

because (bs, cs, λs) satisfy Assumption (AC), λs satisfies Assumption (EM)and v ∈ (−M,M)d. Similarly, we have that

T∫0

‖c′s‖ds =

T∫0

‖cs‖ds <∞

and

T∫0

∫Rd

(|x|2 ∧ |x|)λ′s(dx)ds =

T∫0

∫Rd

(|x|2 ∧ |x|)e〈v,x〉λs(dx)ds

≤KT∫

0

∫Dc

|x|2λs(dx)ds+T∫

0

∫D

|x|e〈v,x〉λs(dx)ds <∞

where K is a positive constant. Therefore, the differential characteristics sat-isfy Assumption (AC), and we can conclude that L is a P ′-time-inhomogene-ous Levy process and a special semimartingale.

Page 68: Applications of semimartingales and L´evy processes in ...webdoc.sub.gwdg.de/ebook/dissts/Freiburg/Papapantoleon2006.pdf · gales as driving processes. Another one, was to consider

60 2. ON THE DUALITY PRINCIPLE II: MULTIDIMENSIONAL PIIAC

Step 2: (b) T(L|P ′)(U) // T(Lu|P ′)

Applying Proposition 2.10 to the time-inhomogeneous Levy process Lunder P ′ and the vector U = u>, we have that Lu = 〈u, L〉 is also a P ′-time-inhomogeneous Levy process with triplet of predictable characteristicsT(Lu|P ′) = (Bu, Cu, νu), where

bus = u>b′s = u>bs + u>csv + u>∫Rd

x(ev>x − 1)λs(dx)

cus = u>csu

λus (E) = λ′s(x ∈ Rd : u>x ∈ E), E ∈ B(R\0).

Note that Lu is a special semimartingale, which follows directly from theanalogous property of L under P ′, cf. Remark 2.11. Moreover, since u+ v ∈[−M,M ]d we can deduce that Lu is an exponentially special semimartingale,and the statement is proved.

3. The payoff of a swap option, also coined a “Margrabe” option oroption to exchange one asset for another, is(

S1T − S2

T

)+and we denote its value by

M(S10 , S

20 ; r, δ, C, ν) = e−rTE

[(S1T − S2

T

)+].

The payoff of the quanto call and put option respectively, is

S1T

(S2T −K

)+ and S1T

(K − S2

T

)+and we will use the following notation for the value of the quanto call option

QC(S10 , S

20 ,K; r, δ, C, ν) = e−rTE

[S1T

(S2T −K

)+]and similarly for the quanto put option

QP(S10 , S

20 ,K; r, δ, C, ν) = e−rTE

[S1T

(K − S2

T

)+].

The different variants of the quanto option traded in Foreign Exchange mar-kets are explained in detail in Musiela and Rutkowski (2005).

The payoff of a digital (cash-or-nothing) and a correlation digital optionrespectively, is

1ST>K and S1T 1S2

T>K.

Hence, the holder of a correlation digital option receives one unit of thepayment asset (S1) at expiration, if the measurement asset (S2) ends upin the money. Of course, this is a generalization of the (standard) digitalasset-or-nothing option, where the holder receives one unit of the asset if itends up in the money. We denote the value of the digital option by

D(S0,K; r, δ, C, ν) = e−rTE[1ST>K

]

Page 69: Applications of semimartingales and L´evy processes in ...webdoc.sub.gwdg.de/ebook/dissts/Freiburg/Papapantoleon2006.pdf · gales as driving processes. Another one, was to consider

2.6. OPTIONS ON SEVERAL ASSETS 61

and the value of the correlation digital option by

CD(S10 , S

20 ,K; r, δ, C, ν) = e−rTE

[S1T 1S2

T>K

].

Remark 2.27. Notice that in the case of the digital option r, δ, C andν correspond to a 1-dimensional driving process, while for all other optionsto a 2-dimensional one.

Theorem 2.28. Assume that the asset price processes evolve as expo-nential time-inhomogeneous Levy processes according to (2.21), (2.22) and(2.31), and assumption (EM) is in force. Then we can relate the value of aswap and a plain vanilla option via the following duality:

M(S1

0 , S20 ; r, δ, C, ν

)= C P

(S2

0/S10 ,K; δ1, C ′, ν ′

)(2.66)

where K = eδ2T , C = S1

0e−δ2T and the characteristics (C ′, ν ′) are given by

Theorem 2.26 for v = (1, 0)> and u = (−1, 1)>.

Proof. We will use, without loss of generality, asset S1 as the numeraireasset; if we used asset S2 instead, then we would get a duality relationshipwith a call option. The value of the swap, or “Margrabe”, option is

M = e−rTE[(S1T − S2

T

)+]= e−δ

1TS10E

[e−rTS1

T

e−δ1TS10

(1−

S2T

S1T

)+]

= e−δ1TS1

0E

[eL

1T

(1−

S2T

S1T

)+]

= e−δ1TS1

0E

[e〈v,LT 〉

(1−

S2T

S1T

)+], (2.67)

where we have used (2.55) and v = (1, 0)>. Moreover, from (2.56) and (2.57),it is immediately obvious that e〈v,L〉 ∈M(P ) (cf. also (2.24)–(2.28)).

Define a new measure P ′ via the Radon–Nikodym derivative

dP ′

dP= e〈v,LT 〉

and the valuation problem takes the form

M = e−δ1TS1

0E′

[(1−

S2T

S1T

)+]

where, using (2.55) again, we get that

S2t

S1t

=S2

0

S10

e(r−δ2)t

e(r−δ1)t

eL2t

eL1t

=S2

0

S10

e(δ1−δ2)te〈u,Lt〉, 0 ≤ t ≤ T (2.68)

where u = (−1, 1)>.Now, applying Proposition III.3.8 in Jacod and Shiryaev (2003) and

(2.56)–(2.57), we obtain that

e〈u,L〉 ∈M(P ′) since e〈u,L〉e〈v,L〉 = eL2 ∈M(P ).

Page 70: Applications of semimartingales and L´evy processes in ...webdoc.sub.gwdg.de/ebook/dissts/Freiburg/Papapantoleon2006.pdf · gales as driving processes. Another one, was to consider

62 2. ON THE DUALITY PRINCIPLE II: MULTIDIMENSIONAL PIIAC

Therefore, we define the process S′ = (S′t)0≤t≤T with S′t = S20/S

10eδ

1te〈u,Lt〉,which once discounted at the new “risk-free rate” δ1 becomes a P ′-martingale.Then, we have that

M = e−δ1TS1

0E′[(

1− eδ2TS′T

)+]

= e−δ1TS1

0e−δ2TE′

[(eδ

2T − S′T

)+]

and the proof is completed.

Example 2.29 (Margrabe 1978). Consider two assets, S1 and S2, wherethe dynamics of each asset are

S it = exp(rt+ Li

t), i = 1, 2, 0 ≤ t ≤ T , (2.69)

where Li , i = 1, 2, is a Brownian motion with drift. In other words, thecharacteristics of L = (L1, L2) are

C =(

σ21 ρσ1σ2

ρσ1σ2 σ22

)and ν ≡ 0,

where σ1, σ2 ∈ R+ and ρ ∈ [−1, 1] is the correlation coefficient of the Brown-ian 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 characteristichas the form

B = −12

(C11

C22

)= −1

2

(σ2

1

σ22

).

The price of the option to exchange asset S1 for asset S2, according toTheorem 2.28, is equal to the price of a put option with strike 1, on an assetS′ with characteristics (C ′, ν ′) described by Theorem 2.26 for v = (1, 0)>

and u = (−1, 1)>. Hence, we get that

C ′ = u>Cu =(−1 1

)( σ21 ρσ1σ2

ρσ1σ2 σ22

)(−11

)= σ2

1 + σ22 − 2ρσ1σ2

and ν ′ ≡ 0. Therefore we have recovered the original result of Margrabe, cf.Margrabe (1978, p. 179), as a special case in our setting. Moreover, we havethat the drift term B′ of S′ has the form

B′ = u>B + u>Cv

= −12(−1 1

)(σ21

σ22

)+(−1 1

)( σ21 ρσ1σ2

ρσ1σ2 σ22

)(10

)= −1

2(σ2

1 + σ22 − 2ρσ1σ2) = −1

2C ′,

as was expected, since S′ discounted is a P ′-martingale.

Theorem 2.30. Assume that the asset price processes evolve as expo-nential time-inhomogeneous Levy processes according to (2.21), (2.22) and(2.31), and assumption (EM) is in force. Then we can relate the value of aquanto and a European call option via the following duality:

QC(S1

0 , S20 ,K; r, δ, C, ν

)= X C

(S2

0 ,K; δ1, C ′, ν ′)

(2.70)

Page 71: Applications of semimartingales and L´evy processes in ...webdoc.sub.gwdg.de/ebook/dissts/Freiburg/Papapantoleon2006.pdf · gales as driving processes. Another one, was to consider

2.6. OPTIONS ON SEVERAL ASSETS 63

where K = Ke(δ1+δ2−r)T e−K(1)T , X = S10e(r−δ1−δ2)T eK(1)T and the charac-

teristics (C ′, ν ′) are given by Theorem 2.26 for v = (1, 0)> and u = (0, 1)>.

Proof. Obviously, we will use asset S1 as the numeraire asset, andsimilarly to the proof of Theorem 2.28, we have

QC = e−rTE[S1T

(S2T −K

)+]= e−δ

1TS10 E

[eL

1T(S2T −K

)+]= e−δ

1TS10 E

[e〈v,LT 〉

(S2T −K

)+], (2.71)

where v = (1, 0)> and e〈v,L〉 ∈M(P ).Define a measure P ′ via the Radon–Nikodym derivative

dP ′

dP= e〈v,LT 〉

and the valuation problem takes the form

QC = eδ1TS1

0E′[(S2T −K

)+].

Notice that, using (2.55) again, we have that

S2t = S2

0e(r−δ2)teL2t = S2

0e(r−δ2)te〈u,Lt〉, 0 ≤ t ≤ T (2.72)

where u = (0, 1)>. An application of Proposition III.3.8 in Jacod andShiryaev (2003) and (2.56)–(2.57), yield that

e〈u,L〉 /∈M(P ′) since e〈u,L〉e〈v,L〉 = e〈1,L〉 /∈M(P ).

Nevertheless, since e〈1,L〉 is an exponentially special semimartingale, we getfrom (2.24) and (2.26) that

e〈1,L〉

eK(1)∈M(P ).

Hence, using Proposition III.3.8 in Jacod and Shiryaev (2003) again, we getthat

e〈u,L〉

eK(1)∈M(P ′) since

e〈u,L〉

eK(1)e〈v,L〉 =

e〈1,L〉

eK(1)∈M(P ).

Therefore, we define the process S′ = (S′t)0≤t≤T with

S′t = S20eδ

1te〈u,Lt〉e−K(1)t ,

which once discounted at the new “risk-free rate” δ1 becomes a P ′-martingale.Now, using again that the exponential compensator is a deterministic pro-cess, we can conclude that

QC = e−δ1TS1

0E′[(

e(r−δ1−δ2)T eK(1)TS′T −K)+]

= e−δ1TXE′

[(S′T −K

)+]where X := S1

0e(r−δ1−δ2)T eK(1)T and K := Ke(δ1+δ2−r)T e−K(1)T .

Page 72: Applications of semimartingales and L´evy processes in ...webdoc.sub.gwdg.de/ebook/dissts/Freiburg/Papapantoleon2006.pdf · gales as driving processes. Another one, was to consider

64 2. ON THE DUALITY PRINCIPLE II: MULTIDIMENSIONAL PIIAC

Theorem 2.31. Assume that the asset price processes evolve as expo-nential time-inhomogeneous Levy processes according to (2.21), (2.22) and(2.31), and assumption (EM) is in force. Then we can relate the value of acorrelation digital and a digital option via the following duality:

CD(S1

0 , S20 ,K; r, δ, C, ν

)= S1

0 D(S2

0 ,K; δ1, C ′, ν ′)

(2.73)

where K = Ke(δ1+δ2−r)T e−K(1)T and the characteristics (C ′, ν ′) are given byTheorem 2.26 for v = (1, 0)> and u = (0, 1)>.

Proof. The proof follows along similar lines to the proof of Theorem2.30 and is therefore omitted.

4. As a final application, we will treat an option that depends on threeassets which will be called, for obvious reasons, a quanto-swap option. Thepayoff of the quanto-swap option is

S1T (S2

T − S3T )+

and can be interpreted as a swap option struck in a foreign currency. Let usdenote its value by

QM(S10 , S

20 , S

30 ; r, δ, C, ν) = e−rTE

[S1T

(S2T − S3

T

)+].

Theorem 2.32. Assume that the asset price processes evolve as expo-nential time-inhomogeneous Levy processes according to (2.21), (2.22) and(2.31), and assumption (EM) is in force. Then we can relate the value of aquanto-swap and a plain vanilla put option via the following duality:

QM(S1

0 , S20 , S

30 ; r, δ, C, ν

)= Z P

(S3

0/S20 ,K; δ1 + δ2, C ′, ν ′

)(2.74)

where K = e(δ1+δ3)T eK(v)T−K(w)T , Z = S10S

20e(r−δ1−δ3)T eK(w)T , w = (1, 0, 1)>

and the characteristics (C ′, ν ′) are given by Theorem 2.26 for v = (1, 1, 0)>

and u = (0,−1, 1)>.

Proof. Instead of changing measure once using S1 as the numeraireand then once more using either S2 or S3, we will combine S1 and S2 (orS3) directly. We have that

QM = e−rTE[S1T

(S2T − S3

T

)+]= e−δ

1TS10e−δ

2TS20erT E

[eL

1T eL

2T

(1−

S3T

S2T

)+]

= e(r−δ1−δ2)TS10S

20 E

[e〈v,LT 〉

(1−

S3T

S2T

)+], (2.75)

where v = (1, 1, 0)>. Note immediately that e〈v,L〉 /∈ M(P ), but sincev ∈ (−M,M)3 we conclude that e〈v,L〉 is an exponentially special semi-martingale. Hence, we define a measure P ′ on (Ω,F , (Ft)0≤t≤T ) via theRadon–Nikodym derivative

dP ′

dP=

e〈v,LT 〉

eK(v)T

Page 73: Applications of semimartingales and L´evy processes in ...webdoc.sub.gwdg.de/ebook/dissts/Freiburg/Papapantoleon2006.pdf · gales as driving processes. Another one, was to consider

2.6. OPTIONS ON SEVERAL ASSETS 65

and the valuation problem takes the form

QM = e(r−δ1−δ2)TS10S

20 E

[eK(v)T

(1−

S3T

S2T

)+].

Now, using (2.55) again, we get that

S3t

S2t

=S3

0

S20

e(r−δ3)t

e(r−δ2)t

eL3t

eL2t

=S3

0

S20

e(δ2−δ3)te〈u,Lt〉, 0 ≤ t ≤ T (2.76)

where u = (0,−1, 1)>. Moreover, denote by w := v+u = (1, 0, 1)> and then,using Proposition III.3.8 in Jacod and Shiryaev (2003), we get that

e〈u,L〉

eK(w)−K(v)∈M(P ′) since

e〈u,L〉

eK(w)−K(v)

e〈v,L〉

eK(v)=

e〈w,L〉

eK(w)∈M(P ).

Therefore, we define the process S′ = (S′t)0≤t≤T with

S′t =S3

0

S20

e(δ1+δ2)te〈u,Lt〉eK(v)t−K(w)t ,

which once discounted at the new “risk-free rate” δ1 + δ2 becomes a P ′-martingale. Now, using again that the exponential compensator is a deter-ministic process, we can conclude that

QM = e−(δ1+δ2)T S10S

20erTE′

[(eK(v)T − S′T e−(δ1+δ3)T eK(w)T

)+]

= e−(δ1+δ2)TZE′[(

K− S′T)+]

where Z := S10S

20e(r−δ1−δ3)T eK(w)T and K := e(δ1+δ3)T eK(v)T e−K(w)T .

Page 74: Applications of semimartingales and L´evy processes in ...webdoc.sub.gwdg.de/ebook/dissts/Freiburg/Papapantoleon2006.pdf · gales as driving processes. Another one, was to consider
Page 75: Applications of semimartingales and L´evy processes in ...webdoc.sub.gwdg.de/ebook/dissts/Freiburg/Papapantoleon2006.pdf · gales as driving processes. Another one, was to consider

CHAPTER 3

Valuation of exotic derivatives in Levy models

3.1. Introduction

The purpose of this chapter is twofold: on the one hand we derive gen-eral formulae for the valuation of vanilla and exotic options on single ormultiple assets; on the other hand, as the title already dictates, we are par-ticularly interested in the valuation of exotic options on assets driven byLevy processes.

The first part of this chapter is devoted to the derivation of generalvaluation formulae. We consider the framework of Chapter 1, where theasset is driven by a general semimartingale and aim at pricing vanilla andexotic options with arbitrary payoff functions. The options we consider areEuropean style, in the sense that they cannot be exercised or terminatedbefore maturity. Other than that, the variable that determines the payoffcan 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 thatthe characteristic function of this variable is known. The payoff function isalso an arbitrary function, subject to some mild integrability conditions; theexamples include all the commonly traded contracts in financial markets,such as standard call and put options, digital options and double digitaloptions.

These formulae allow the fast pricing of European plain vanilla options,and hence the calibration to market data, for a large variety of drivingprocesses, such as Levy processes and stochastic volatility models drivenby Levy processes. When considering the valuation of exotic options, thesituation becomes much more delicate since, in most cases, the characteristicfunction is not known in advance.

Therefore, in the second part of this chapter we concentrate on derivingthe characteristic function of the supremum of a Levy process. This willallow us to price exotic path-dependent options on assets driven by Levyprocesses, such as one-touch and lookback options. Indeed, making use ofclassical results from fluctuation theory, more specifically of the celebratedWiener–Hopf factorization of a Levy process, we present an expression forthe characteristic function of the supremum and subsequently apply it tovaluation problems. Of course, we can similarly derive the characteristicfunction of the infimum of a Levy process; this result would also have inter-esting applications in credit risk. However, for the sake of brevity, we haveshied away from doing so.

It should be pointed out that Levy processes are becoming standardtools in financial modeling, since they can describe the observed reality infinancial markets in a quite accurate way. The recent volume of Kyprianou

67

Page 76: Applications of semimartingales and L´evy processes in ...webdoc.sub.gwdg.de/ebook/dissts/Freiburg/Papapantoleon2006.pdf · gales as driving processes. Another one, was to consider

68 3. EXOTIC DERIVATIVES IN LEVY MODELS

et al. (2005) contains an up-to-date account of applications of Levy pro-cesses in finance. For more general considerations about Levy processes onecould refer to the books of Bertoin (1996), Sato (1999), and Applebaum(2004). The books of Schoutens (2003) and Cont and Tankov (2003) discussLevy processes with particular focus on their applications in mathematicalfinance.

This chapter is organized as follows: in section 3.2 we present the val-uation formulae, while in section 3.3 some basic facts on Levy processesand their fluctuations are presented and the characteristic function of thesupremum of a Levy process is obtained. In section 3.4 we present someexamples 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 andmulti-asset options

3.2. Option valuation: general formulae

1. Let B = (Ω,F ,F, P ) be a stochastic basis where F = FT and F =(Ft)0≤t≤T . We model the price process of a financial asset, e.g. a stock or anFX rate, as an exponential semimartingale S = (St)0≤t≤T , i.e. a stochasticprocess with representation

St = eHt , 0 ≤ t ≤ T (3.1)

(shortly: S = eH), where H = (Ht)0≤t≤T is a semimartingale with H0 = 0.Every semimartingale H = (Ht)0≤t≤T admits a canonical representation

H = H0 +B +Hc + h(x) ∗ (µ− ν) + (x− h(x)) ∗ µ, (3.2)

where B = (Bt)0≤t≤T is a predictable process of bounded variation, Hc =(Hc

t )0≤t≤T is the continuous martingale part of H, h = h(x) is a truncationfunction, µ = µ(ω; ds,dx) is the random measure of jumps of H and ν =ν(ω; ds,dx) is the predictable compensator of µ with respect to P .

The continuous martingale part Hc of H has predictable quadratic char-acteristic 〈Hc〉 which will be denoted by C = (Ct)0≤t≤T . For the processesB, C, and the measure ν we use the notation

T(H|P ) = (B,C, ν)

which is called the triplet of predictable characteristics of the semimartingaleH with respect to the probability measure P .

Let M(P ), resp. Mloc(P ), denote the class of all martingales, resp. localmartingales, 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 1x>1ex ∗ ν has bounded variation.

Subject to this assumption, we can deduce that (see Chapter 1 or Eber-lein, Papapantoleon, and Shiryaev (2006) for details)

S = eH ∈Mloc(P ) ⇔ B +C

2+ (ex − 1− h(x)) ∗ ν = 0. (3.3)

Throughout this work, we assume that P is an (equivalent) martingale mea-sure for the asset S. We do not elaborate on how this measure has been cho-sen; it can be the outcome of the minimization of a distance (e.g. L2-distance,

Page 77: Applications of semimartingales and L´evy processes in ...webdoc.sub.gwdg.de/ebook/dissts/Freiburg/Papapantoleon2006.pdf · gales as driving processes. Another one, was to consider

3.2. OPTION VALUATION: GENERAL FORMULAE 69

Hellinger distance, entropic distance, etc.) to the “physical” measure or theresult of a calibration to market data of vanilla options.

Remark 3.1. The assumption that the process 1x>1ex∗ν has boundedvariation, is equivalent by Kallsen and Shiryaev (2002a, Lemma 2.13) to theassumption that the semimartingale H is exponentially special, i.e., the priceprocess S = eH is a special semimartingale.

2. Let X = (Xt)0≤t≤T be any stochastic process on the given stochasticbasis. We denote by X = (Xt)0≤t≤T and X = (Xt)0≤t≤T , where

Xt = sup0≤u≤t

Xu and Xt = inf0≤u≤t

Xu

the supremum and infimum processes ofX respectively. Notice that since theexponential function is monotonically increasing, the supremum processesof S and H are related via

ST = sup0≤t≤T

St = sup0≤t≤T

(eHt)

= esup0≤t≤T Ht = eHT . (3.4)

Similarly, the infimum processes of S and H are related via

ST = eHT . (3.5)

3. We derive and prove a general option pricing formula, analogous toRaible’s option pricing formula (cf. Raible 2000, Chapter 3), which does notrequire the existence of a Lebesgue density for the driving process. The proofrelies on the Fourier transform of the payoff function, generalizing the idea ofBorovkov and Novikov (2002). A similar representation of the payoff functionis used for hedging purposes by Hubalek, Kallsen, and Krawczyk (2006) andCerny (2007). Of course, our method also has similarities and generalizes themethod of Carr and Madan (1999). Note that Carr and Madan (1999) andRaible (2000) consider the (Fourier) transform of the option price insteadof the payoff function.

Moreover, we would like to tackle plain vanilla options, such as Europeancall and put options, and exotic path-dependent options, such as lookbackand one-touch options, in a unified framework.

Therefore, assume we want to price an option on an asset S, whereS = eH , with payoff f(XT ), where f(XT ) = f(Ht, 0 ≤ t ≤ T ) is an FT -measurable functional and FT = FS

T = σ(St, 0 ≤ t ≤ T ); in other words,XT is a random variable that depends on the distribution of H, or somefunctional of H, at maturity T . The functionals we consider are “Europeanstyle”, in the sense that the option writer or holder do not have the right toexercise or terminate the option before maturity. Nevertheless, the payofffunctional can depend on the whole history of the price process and not juston 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 supre-

mum/infimum of the asset price process or an average of the asset priceprocess. This will always be denoted by X (i.e. X = H or X = H orX = H, etc.);

(b) the payoff function: it is an arbitrary function f : R → R+, for examplef(x) = (ex −K)+ or f(x) = 1ex>B, for K,B ∈ R+.

Page 78: Applications of semimartingales and L´evy processes in ...webdoc.sub.gwdg.de/ebook/dissts/Freiburg/Papapantoleon2006.pdf · gales as driving processes. Another one, was to consider

70 3. EXOTIC DERIVATIVES IN LEVY MODELS

We want to derive a valuation formula for an option with an arbitrarypayoff function, and impose the following conditions.

(R1): Assume that∫

R e−Rxf(x)dx <∞ for all R ∈ I1 ⊂ R.(R2): Assume that MXT

(z), the moment generating function of XT ,exists for all z ∈ I2 ⊂ R.

(R3): Assume that I1 ∩ I2 6= ∅.

Theorem 3.2. Assume that the asset price process is modeled as anexponential semimartingale process according to (3.1)–(3.3) and conditions(R1)–(R3) are in force. Then, the price Vf (X) of an option on S = (St)0≤t≤Twith payoff function f = f(XT ) is given by

Vf (X) =12π

∫R

ϕXT(−u− iR)Ff (u+ iR)du, (3.6)

where ϕXTdenotes the extended characteristic function of XT and Ff de-

notes the Fourier transform of f .

Proof. According to the first fundamental theorem of asset pricing –see Delbaen and Schachermayer (1994, 1998) and references therein – theprice of an integrable contingent claim is equal to its discounted expectedpayoff with respect to a martingale measure. Without loss of generality, weassume that interest rates are equal to zero.

Introduce the dampened payoff function g(x) = e−Rxf(x), for an R ∈I1 ∩ I2. Since S = eH ∈Mloc(P ), we have

Vf (X) = E [f(XT )] =∫Ω

f(XT )dP =∫Ω

eRXT g(XT )dP

=∫R

eRxg(x)PXT(dx). (3.7)

Under assumption (R1), we have that

‖g‖L1(R) =∫R

|g(x)|dx <∞, (3.8)

hence g ∈ L1(R) and the Fourier transform of g

Fg(u) =∫R

eiuxg(x)dx

is well defined for every u ∈ R. Using Theorem 9.6 in Rudin (1987), theFourier transform is continuous and

‖Fg‖L∞(R) ≤ ‖g‖L1(R). (3.9)

Applying Theorem 3.8 in Rudin (1987), we get that

‖Fg‖L1(R) ≤ ‖Fg‖L∞(R)

and since g ∈ L1(R) we conclude, using (3.9) and (3.8), that Fg ∈ L1(R).Therefore, the prerequisites of the Inversion Theorem (cf. Rudin 1987, The-orem 9.11) are satisfied and the Fourier transform of g can be inverted; we

Page 79: Applications of semimartingales and L´evy processes in ...webdoc.sub.gwdg.de/ebook/dissts/Freiburg/Papapantoleon2006.pdf · gales as driving processes. Another one, was to consider

3.2. OPTION VALUATION: GENERAL FORMULAE 71

get

g(x) =12π

∫R

e−ixuFg(u)du. (3.10)

Now, returning to the valuation problem (3.7) we get that

Vf (X) =∫R

eRxg(x)PXT(dx)

=∫R

eRx(

12π

∫R

e−ixuFg(u)du

)PXT

(dx)

=12π

∫R

(∫R

ei(−u−iR)xPXT(dx)

)Fg(u)du

=12π

∫R

ϕXT(−u− iR)Ff (u+ iR)du, (3.11)

where for the third equality we have applied the Fubini theorem and for thelast equality we have

Fg(u) =∫R

eiuxg(x)dx =∫R

eiuxe−Rxf(x)dx

=∫R

ei(u+iR)xf(x)dx

= Ff (u+ iR). (3.12)

Finally, the application of Fubini’s theorem is justified as follows; definethe function

F (x, u) = eRxe−iuxFg(u),

for any x, u ∈ R. Then∫R

∫R

|F (x, u)|duPXT(dx) =

∫R

∫R

eRx|e−iux||Fg(u)|duPXT(dx)

≤∫R

∫R

eRx · 1 · |Fg(u)|duPXT(dx)

=∫R

(∫R

|Fg(u)|du)

eRxPXT(dx)

≤ K

∫R

eRxPXT(dx)

= KMXT(R) <∞,

where for the second inequality we used that Fg ∈ L1(R) and the finitenessof MXT

(R) for R ∈ I1 ∩ I2 follows from Assumptions (R2) and (R3). Hence,we can conclude that F ∈ L1(λ\⊗ PXT

). Theorem 3.2 is proved.

Page 80: Applications of semimartingales and L´evy processes in ...webdoc.sub.gwdg.de/ebook/dissts/Freiburg/Papapantoleon2006.pdf · gales as driving processes. Another one, was to consider

72 3. EXOTIC DERIVATIVES IN LEVY MODELS

Remark 3.3. If the moment generating function of a random variableexists in some interval I2 ⊂ R, then using standard arguments from complexanalysis, one can extend the characteristic function to complex argumentswith imaginary part in J2, where J2 = −I2 (where for a set [a, b] ⊂ R wedenote by −[a, b] := [−b,−a]).

Remark 3.4. The valuation formulae of Theorem 3.2 bears, as was men-tioned already before, great similarities to Raible’s option pricing formula;compare with Theorem 3.2 in Raible (2000). In addition, the prerequisitesof Theorem 3.2 in Raible (2000) are the same as (R1)–(R3), requiring ad-ditionally the existence of a Lebesgue density for the random variable XT

(cf. p. 63 in Raible 2000). The latter is needed to establish the convolutionrepresentation of the option price, which we do not use. Let us also pointout the following; the proof of the result, that the Laplace transform of aconvolution equals the product of the Laplace transforms of the convolutionfactors, which is crucial in Raible’s work, relies essentially on the Fubinitheorem.

4. Next, we would like to establish some valuation formulae for optionsthat depend on two functionals of the driving process. Examples of suchoptions are barrier options, with payoff

(ST −K)+1ST>B, K,B ∈ R+

and slide-in or corridor options, with payoff

(ST −K)+N∑i=1

1L<STi<H, H,K,L ∈ R+.

It should be pointed out that these options cannot be treated with theduality methods developed in Chapters 1 and 2.

The setting is the familiar one: S = eH is the asset price process, whereH is a semimartingale with canonical decomposition (3.2), that satisfiesAssumption (ES) and (3.3). We also consider two payoff functions f and g,which are arbitrary mappings from R to R+, and two functionals X and Yof the driving process H (e.g. X = H and Y = H). Moreover, we impose thefollowing condition on the payoff functions and the underlying processes.

(T1): Assume that∫

R e−R1xf(x)dx <∞ for all R1 ∈ I1 ⊂ R.(T2): Assume that

∫R e−R2xg(x)dx <∞ for all R2 ∈ I2 ⊂ R.

(T3): Assume that MXT ,YT(u, v), the moment generating function

of the random vector (XT , YT ), exists for all u ∈ I3 ⊂ R and allv ∈ I4 ⊂ R.

(T4): Assume that I1 ∩ I3 6= ∅ and I2 ∩ I4 6= ∅.Theorem 3.5. Assume that the asset price process is modeled as an

exponential semimartingale process according to (3.1)–(3.3) and conditions(T1)–(T4) are in force. Then, the price Vf,g(X,Y ) of an option on S =(St)0≤t≤T with payoff function f(XT )g(YT ) is given by

Vf,g(X,Y ) =1

4π2

∫R

∫R

ϕXT ,YT(−u− iR1,−v − iR2)

× Fg(v + iR2)Ff (u+ iR1)dvdu, (3.13)

Page 81: Applications of semimartingales and L´evy processes in ...webdoc.sub.gwdg.de/ebook/dissts/Freiburg/Papapantoleon2006.pdf · gales as driving processes. Another one, was to consider

3.2. OPTION VALUATION: GENERAL FORMULAE 73

where ϕXT ,YTdenotes the extended characteristic function of the random

vector (XT , YT ).

Proof. Introduce the dampened payoff function g′(x) = e−R2xg(x) foran R2 ∈ I2 ∩ I4. Define the random variable Y := g(YT )

E[g(YT )] and let us notethat E[Y] = 1; in addition, we trivially have that g(YT ) = eR2YT g′(YT ).Moreover, we denote by Π := E[g(YT )]; it can evaluated using Theorem 3.2and therefore is finite. Now, we have that

Vf,g(X,Y ) = E [f(XT )g(YT )]

= ΠE[f(XT )

g(YT )E[g(YT )]

]= ΠE′ [f(XT )] (3.14)

where we have defined the measure P ′ on (Ω,F , (Ft)0≤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 XT withrespect to P ′, denoted by ϕ′XT

. Using that dP ′ = YdP , conditions (T1) and(T2), (3.10), (3.12) and Fubini’s theorem, we get that, for all u ∈ R

ϕ′XT(u) = E′ [eiuXT

]=∫Ω

eiuXT dP ′

=∫Ω

eiuXTg(YT )

E[g(YT )]dP

=1Π

∫R×R

eiuxeR2yg′(y)PXT ,YT(dx,dy)

=1Π

∫R×R

eiuxeR2y

(12π

∫R

e−ivyFg′(v)dv

)PXT ,YT

(dx,dy)

=1Π

12π

∫R

( ∫R×R

ei(ux−vy−iR2y)PXT ,YT(dx,dy)

)Fg(v + iR2)dv

=1Π

12π

∫R

ϕXT ,YT(u,−v − iR2)Fg(v + iR2)dv. (3.15)

Therefore, returning to the valuation problem (3.14) and using (3.6),(3.15) and Assumptions (T2)–(T4), we can conclude that

Vf,g(X,Y ) = ΠE′ [f(XT )]

= Π12π

∫R

ϕ′XT(−u− iR1)Ff (u+ iR1)du

=1

4π2

∫R

∫R

ϕXT ,YT(−u− iR1,−v − iR2)

× Fg(v + iR2)dvFf (u+ iR1)du, (3.16)

where R1 ∈ I1 ∩ I3.

Page 82: Applications of semimartingales and L´evy processes in ...webdoc.sub.gwdg.de/ebook/dissts/Freiburg/Papapantoleon2006.pdf · gales as driving processes. Another one, was to consider

74 3. EXOTIC DERIVATIVES IN LEVY MODELS

Finally, the application of the Fubini theorem is justified as follows:define the function

G(x, y, v) = eiuxe(R2−iv)yFg′(v)

for x, y, v ∈ R and R2 ∈ I2 ∩ I4. Then, we have that∫∫∫G(x, y, v)dvPXT ,YT

(dx,dy)=∫∫∫

|eiuxe(R2−iv)yFg′(v)|dvPXT ,YT(dx,dy)

≤∫∫

eR2y

(∫|Fg′(v)|dv

)PXT ,YT

(dx,dy)

≤K∫∫

eR2yPXT ,YT(dx,dy) <∞

where K is a positive constant; hence, G ∈ L1(λ\⊗ PXT ,YT). The statement

is proved.

Remark 3.6. Theorem 3.5 states that the knowledge of the joint charac-teristic function is sufficient for the valuation of options that depend on twofunctionals of the driving process. Unfortunately, such joint characteristicfunctions are not readily available for most of the interesting cases.

3.3. Levy processes and their fluctuations

The focal point of this section is to exploit results from fluctuation theoryfor Levy processes to derive the characteristic function of the supremum ofa Levy process. The characteristic function of the infimum can be derivedanalogously.

1. Let L = (Lt)0≤t≤T be a Levy process on the stochastic basis B =(Ω,F ,F, P ), i.e. L is a semimartingale with independent and stationaryincrements (PIIS). We denote the triplet of predictable characteristics of Lwith respect to the probability measure P by T(L|P ) = (B,C, ν); moreover,we denote the triplet of local characteristics of L by (b, c, λ) and using Jacodand Shiryaev (2003, II.4.20), the two triplets are related via

Bt(ω) = bt, Ct(ω) = ct, ν(ω; dt,dx) = dt λ(dx).

The triplet of predictable characteristics of a semimartingale with inde-pendent increments determines the law of the random variables generatingthe process. More specifically, for a Levy process we know from the Levy–Khintchine formula that

E[eiuLt

]= exp

t(iub+u2

2c+

∫R

(eiux − 1− iuh(x))λ(dx))

for all t ∈ [0, T ] and all u ∈ R. In the sequel we will assume that the followingcondition is in force.

Assumption (EM). There exists a constant M > 1 such that∫|x|>1

euxλ(dx) <∞, ∀u ∈ [−M,M ].

Page 83: Applications of semimartingales and L´evy processes in ...webdoc.sub.gwdg.de/ebook/dissts/Freiburg/Papapantoleon2006.pdf · gales as driving processes. Another one, was to consider

3.3. LEVY PROCESSES AND THEIR FLUCTUATIONS 75

It is immediately clear that assumption (EM) is a mild strengthening ofassumption (ES). Moreover, subject to (EM) L is a special and exponentiallyspecial semimartingale and the use of the truncation function will be omitted(cf. also Remark 2.3).

Applying Theorem 25.3 in Sato (1999), we have that

E[euLt

]<∞, ∀u ∈ [−M,M ], ∀t ∈ [0, T ].

Moreover, applying Lemma 12 in Keller (1997), we have that the character-istic function of Lt is holomorphic in the horizontal strip z ∈ C : −M <=z < M and can be represented as a Fourier integral in the complex plane;hence, for z ∈ C with −M < =z < M we have that

ϕLt(z) = E[eizLt

]=∫R

eizxPLt(dx).

Therefore, assumption (R2) of Theorem 3.2 is satisfied with I2 = [−M,M ].We model the asset price process S = (St)0≤t≤T as an exponential Levy

process, that is as a stochastic process with representation

St = S0 expLt, 0 ≤ t ≤ T (3.17)

where, subject to Assumption (EM), L is a special semimartingale withcanonical decomposition (Jacod and Shiryaev 2003, II.2.38)

Lt = bt+√cWt +

t∫0

∫R

x(µL − νL)(ds,dx). (3.18)

Using Assumption (EM) and (3.3) we have that

S ∈Mloc(P ) ⇔ b+c

2+∫R

(ex − 1− x)ν(dx) = 0 (3.19)

and by Lemma 4.4 in Kallsen (2000), we can even conclude that S ∈M(P ).Let us denote the supremum and infimum processes of L = (Lt)0≤t≤T

by L = (Lt)0≤t≤T and L = (Lt)0≤t≤T respectively and, as in the previoussection, we have that S = eL and S = eL.

2. We establish some results regarding the existence and analyticity ofthe characteristic function of the supremum of a Levy process. The nexttheorem will be useful for our considerations.

Theorem 3.7. Let L = (Lt)0≤t≤T be a Levy process and define

L∗t = sup

0≤u≤t|Lu|, 0 ≤ t ≤ T .

Let g(r) be a submultiplicative, non-negative, continuous function on [0,∞),increasing to ∞ as r →∞. Then, the following are equivalent:

(a) E[g(L∗t )] <∞, ∀t ∈ [0, T ],(b) E[g(|Lt|)] <∞, ∀t ∈ [0, T ].

Proof. See Theorem 25.18, pp. 166–167 in Sato (1999).

Page 84: Applications of semimartingales and L´evy processes in ...webdoc.sub.gwdg.de/ebook/dissts/Freiburg/Papapantoleon2006.pdf · gales as driving processes. Another one, was to consider

76 3. EXOTIC DERIVATIVES IN LEVY MODELS

Making use of the assumption on exponential moments (EM), we imme-diately have that

E[euLt

]<∞, ∀u ∈ [−M,M ], ∀t ∈ [0, T ].

In particular, we get that

E[eMLt

]<∞, and (3.20)

E[e−MLt

]<∞, ∀t ∈ [0, T ]. (3.21)

Therefore, we can easily conclude that

E[eM |Lt|] =

∫R

eM |x|PLt(dx)

=∫

(−∞,0)

e−MxPLt(dx) +∫

(0,∞)

eMxPLt(dx) <∞ (3.22)

for all t ∈ [0, T ].

Lemma 3.8. Let L = (Lt)0≤t≤T be a Levy process that satisfies assump-tion (EM). Then, the moment generating function of Lt is well defined forall u ∈ (−∞,M ], for all t ∈ [0, T ].

Proof. Consider the function g(x) = eMx, x ∈ [0,∞). It is a submul-tiplicative, non-negative, continuous function on [0,∞), which increases to∞ as x → ∞. Hence, it satisfies the prerequisites of Theorem 3.7. Usingequation (3.22), we have that condition (b) of Theorem 3.7 is satisfied forthis choice of g, i.e.

E[eM |Lt|] <∞, ∀t ∈ [0, T ].

Applying Theorem 3.7, we get that

E[eML

∗t]<∞, ∀t ∈ [0, T ]. (3.23)

Now, it suffices to notice that

L∗t = sup

0≤u≤t|Lu| ≥ sup

0≤u≤tLu = Lt

and since M > 0, we have that MLt ≤ML∗t . Since the exponential function

is monotone increasing, we have that eMLt ≤ eML∗t , therefore

E[eMLt

]≤ E

[eML

∗t]<∞. (3.24)

Moreover, we trivially have that for any M ′ > 0 and any t ∈ [0, T ]

E[e−M

′Lt]≤ E[1] <∞, (3.25)

because L is an increasing process starting at zero, i.e. the measure of Lt issupported on [0,∞).

Lemma 3.9. Let L = (Lt)0≤t≤T be a Levy process that satisfies assump-tion (EM). Then, the characteristic function ϕLt

of Lt is holomorphic in the

Page 85: Applications of semimartingales and L´evy processes in ...webdoc.sub.gwdg.de/ebook/dissts/Freiburg/Papapantoleon2006.pdf · gales as driving processes. Another one, was to consider

3.3. LEVY PROCESSES AND THEIR FLUCTUATIONS 77

half plane z ∈ C : −M < =z < ∞ and can be represented as a Fourierintegral in the complex domain

ϕLt(z) = E

[eizLt

]=∫R

eizxPLt(dx).

Proof. We will follow the proof of Lemma 12 in Keller (1997). Usingassumption (EM) and Lemma 3.8, we have that the moment generatingfunction MLt

of Lt exists for u ∈ (−∞,M ]. Hence, MLtis a real analytic

function and can be expanded in a power series around zero, cf. Gut (1995,Remark III.3.4), that is

MLt(u) =

∞∑n=0

un

n!E[(Lt)n],

where the radius of convergence is −M ′ < u < M . Using the ExtensionTheorem for power series, we obtain that MLt

is holomorphic within z ∈C : −M ′ < =z < M, hence the above function can be applied to complexarguments. Taking ϕLt

(u) = MLt(iu) for −M < u < M ′, we have that ϕLt

is analytic and thus can be extended to a holomorphic function in z ∈C : −M < =z < M ′. Now, applying Theorem 7.1.1 in Lukacs (1970), therequired result is proved.

3. The next step is to establish a relationship between the characteristicfunction of the supremum at a fixed and at an independent and exponentiallydistributed time. Independent exponential times play a fundamental role influctuation theory for Levy processes.

Let θ denote an exponentially distributed random variable with param-eter q > 0, independent of L = (Lt)0≤t≤T . We denote by Lθ the supremumprocess of L sampled at an independent and exponentially distributed time,that is

Lθ = sup0≤u≤θ

Lu.

Lemma 3.10. Let L = (Lt)0≤t≤T be a Levy process that satisfies assump-tion (EM) and consider β ∈ C with <β ∈ [−M,∞). The Laplace transformsof Lt, t ∈ [0, T ] and Lθ, θ ∼ Exp(q) are related via

E[e−βLθ

]= q

∞∫0

e−qtE[e−βLt

]dt. (3.26)

Moreover, the Laplace transform of Lθ is finite for β ∈ C with <β ∈[−M,∞).

Proof. It suffices to notice that an application of Fubini’s theoremyields

E[e−βLθ

]= E

[ ∞∫0

qe−qte−βLtdt]

= q

∞∫0

e−qtE[e−βLt

]dt

and the finiteness of E[e−βLθ ] is a direct consequence of Fubini’s theorem(see e.g. Corollary 13.9 in Schilling 2005).

Page 86: Applications of semimartingales and L´evy processes in ...webdoc.sub.gwdg.de/ebook/dissts/Freiburg/Papapantoleon2006.pdf · gales as driving processes. Another one, was to consider

78 3. EXOTIC DERIVATIVES IN LEVY MODELS

The application of Fubini’s theorem is justified as follows. Define thefunction

H(t, x) = qe−qte−βx

for t > 0, q > 0 and β = γ + ic ∈ C with <β = γ ∈ [−M,∞). Then∞∫0

∞∫0

|H(t, x)|dtPLt(dx) =

∞∫0

∞∫0

qe−qt|e−βx|dtPLt(dx)

≤∞∫0

∞∫0

qe−qte−γxdtPLt(dx)

=

∞∫0

q

( ∞∫0

e−qtdt)

e−γxPLt(dx)

=

∞∫0

e−γxPLt(dx) <∞,

using Lemma 3.8. Hence, H ∈ L1(λ\⊗ PLt) and the result is proved.

Remark 3.11. For an alternative proof of the existence of moments ofthe supremum process, we refer to Kyprianou and Surya (2005, Lemma 5).

4. Next, we present some facts from fluctuation theory for Levy pro-cesses that will be useful in deriving the characteristic function of the supre-mum of a Levy process. Fluctuation identities for Levy processes orig-inate from analogous results for random walks, based on combinatorialmethods, see e.g. Spitzer (1964) or Feller (1971). Greenwood and Pitman(1980a,1980b) proved those results for random walks and Levy processesusing excursion theory. Our presentation relies on the beautiful recent bookKyprianou (2006).

The most celebrated result concerning fluctuation identities for Levyprocesses is the so-called Wiener–Hopf factorization, which serves as a com-mon reference to a multitude of statements regarding the distributional de-composition of the excursions of a Levy process sampled at an independentand exponentially distributed time. A statement often referred to as theWiener–Hopf factorization, relates the characteristic function of the supre-mum, the infimum and the Levy process itself, sampled at an independentand exponentially distributed time, and reads as follows:

E[eizLθ

]= E

[eizLθ

]E[eizLθ

]or equivalently,

q

q − ψ(z)= ϕ+

q (z)ϕ−q (z), z ∈ R

where ψ denotes the characteristic exponent of L and ϕ+q , ϕ−q are the so-

called Wiener–Hopf factors. In the sequel, we will make use of the followingform of the Wiener–Hopf factorization.

Page 87: Applications of semimartingales and L´evy processes in ...webdoc.sub.gwdg.de/ebook/dissts/Freiburg/Papapantoleon2006.pdf · gales as driving processes. Another one, was to consider

3.3. LEVY PROCESSES AND THEIR FLUCTUATIONS 79

Theorem 3.12 (Wiener–Hopf factorization). Let L be a Levy process(but not a compound Poisson process). The Laplace transform of Lθ at anindependent and exponentially distributed time θ can be identified from theWiener–Hopf factorization of L via

E[e−βLθ

]=κ(q, 0)κ(q, β)

. (3.27)

The Laplace exponent of the ascending ladder process κ(α, β), for α ≥ 0, β ≥0 and k > 0, is given by

κ(α, β) = k exp

( ∞∫0

∞∫0

(e−t − e−αt−βx)1tPLt(dx)dt

). (3.28)

Moreover, κ(α, β) can be analytically extended to α, β ∈ C with <a ≥ 0 and<β ≥ −M .

Proof. The first part follows directly from Theorem 6.16 (ii) in Kypri-anou (2006). The second part follows from Theorem 6.16 (iii) in Kyprianou(2006) and Lemma 3.10.

5. The final step is to invert the Laplace transform in Theorem 3.12 torecover the characteristic function of Lt. This is done in Theorem 3.14, afteran auxiliary lemma.

Lemma 3.13. The map t 7→ E[e−βLt

]is continuous for all β ∈ C with

<β ∈ [−M,∞).

Proof. Since the Levy process L is stochastically continuous and L isan increasing process, we get that Ls ↑ Lt a.s. as s→ t. We first consider realarguments and have to distinguish two cases. If β > 0 then e−βLs → e−βLt

a.s. as s→ t and we have that |e−βLs | ≤ 1 a.s. because Ls ≥ 0 a.s. Applyingthe dominated convergence theorem, we get that

E[e−βLs

]→ E

[e−βLt

]as s→ t.

If −M ≤ β ≤ 0 then e−βLs ↑ e−βLt a.s. as s→ t. Because the random vari-ables e−βLs are positive, we can apply the monotone convergence theoremto conclude

E[e−βLs

]→ E

[e−βLt

]as s→ t.

The proof for complex arguments is analogous, considering the real andimaginary parts separately.

Theorem 3.14. Let L = (Lt)0≤t≤T be a Levy process (but not a com-pound Poisson process). The Laplace transform of Lt at a fixed time t,t ∈ [0, T ], is given by

E[e−βLt

]=

12π

∫R

et(Y+iv)

Y + iv

κ(Y + iv, 0)κ(Y + iv, β)

dv, (3.29)

for Y > 0. Moreover, the Laplace transform can be extended to the complexplane for β ∈ C with <β ∈ [−M,∞).

Page 88: Applications of semimartingales and L´evy processes in ...webdoc.sub.gwdg.de/ebook/dissts/Freiburg/Papapantoleon2006.pdf · gales as driving processes. Another one, was to consider

80 3. EXOTIC DERIVATIVES IN LEVY MODELS

Proof. Combining equations (3.26) and (3.27), we get that∞∫0

e−qtE[e−βLt

]dt =

1q

κ(q, 0)κ(q, β)

. (3.30)

In addition, the Laplace transform on the right hand side is convergent onthe half plane <q > 0. Therefore, applying Satz 4.5.2 in Doetsch (1950), wehave that this Laplace transform can be inverted and

E[e−βLt

]=

12π

Y+i∞∫Y−i∞

etz

z

κ(z, 0)κ(z, β)

dz

=12π

∫R

et(Y+iv)

Y + iv

κ(Y + iv, 0)κ(Y + iv, β)

dv, (3.31)

a.e. for Y > 0. Now, using the continuity of the map t 7→ E[e−βLt

], cf.

Lemma 3.13, the result follows.

3.4. Examples of payoff functions

Almost every payoff function commonly used in mathematical financesatisfies assumption (R1). Here we list the most representative examplestogether with their Fourier transform and the range of definition I1.

Example 3.15 (Call and put option). The payoff of a call option isf(x) = (ex −K)+, K ∈ R+. Let z ∈ C with =z ∈ (1,∞), then the Fouriertransform of the payoff function of the call option is

Ff (z) =∫R

eizx(ex −K)+dx =∫R

eizx(ex −K)1x>lnKdx

=

∞∫lnK

e(1+iz)xdx−K

∞∫lnK

eizxdx

= − 11 + iz

e(1+iz) lnK +K

izeiz lnK

= −K1+iz 11 + iz

+KizK

iz

=K1+iz

iz(1 + iz).

Therefore,

Ff (u+ iR) =K1+iu−R

(iu−R)(1 + iu−R), R ∈ I1 = (1,∞). (3.32)

Similarly, for a put option, where f(x) = (K − ex)+, K ∈ R+, we have that

Ff (u+ iR) =K1+iu−R

(iu−R)(1 + iu−R), R ∈ I1 = (−∞, 0). (3.33)

Page 89: Applications of semimartingales and L´evy processes in ...webdoc.sub.gwdg.de/ebook/dissts/Freiburg/Papapantoleon2006.pdf · gales as driving processes. Another one, was to consider

3.4. EXAMPLES OF PAYOFF FUNCTIONS 81

Example 3.16 (Digital option). The payoff of a digital call option is1ex>B, B ∈ R+. Let z ∈ C with =z ∈ (0,∞), then the Fourier transformof the payoff function of the digital call option is

Ff (z) =∫R

eizx1x>lnBdx =

∞∫lnB

eizxdx = − 1iz

eiz lnB

= −Biz 1iz.

Therefore,

Ff (u+ iR) = −Biu−R

iu−R, R ∈ I1 = (0,∞). (3.34)

Similarly, for a digital put option, where f(x) = 1ex<B, B ∈ R+, we havethat

Ff (u+ iR) =Biu−R

iu−R, R ∈ I1 = (−∞, 0). (3.35)

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 theasset-or-nothing digital call option is f(x) = ex1ex>B, B ∈ R+, and theFourier transform, for z ∈ C with =z ∈ (1,∞), similarly to the case of the“standard” digital call, is

Ff (z) =∫R

eizxex1x>lnBdx = −B1+iz 11 + iz

.

Therefore,

Ff (u+ iR) = − B1+iu−R

1 + iu−R, R ∈ I1 = (1,∞). (3.36)

Similarly, for an asset-or-nothing digital put option, where f(x) = ex1ex<B,B ∈ R+, we have that

Ff (u+ iR) =B1+iu−R

1 + iu−R, R ∈ I1 = (−∞, 1). (3.37)

Example 3.17 (Double digital option). The payoff of a double digitalcall option is 1B<ex<B, B,B ∈ R+. Let z ∈ C with =z ∈ (−∞, 0)∪ (0,∞),then the Fourier transform of the double digital call option is

Ff (z) =∫R

eizx1B<ex<Bdx =

lnB∫lnB

eizxdx

=1iz

eiz lnB − 1iz

eiz lnB

=1iz

(Biz −Biz

).

Therefore,

Ff (u+ iR) =1

iu−R

(Biu−R −Biu−R

), R ∈ I1 = R\0. (3.38)

Page 90: Applications of semimartingales and L´evy processes in ...webdoc.sub.gwdg.de/ebook/dissts/Freiburg/Papapantoleon2006.pdf · gales as driving processes. Another one, was to consider

82 3. EXOTIC DERIVATIVES IN LEVY MODELS

Example 3.18 (Self-quanto option). The payoff of a self-quanto calloption is f(x) = ex(ex −K)+, K ∈ R+. Let z ∈ C with =z ∈ (2,∞), thenthe Fourier transform of the payoff function of the self-quanto call option,similarly to the case of the “standard” call option, is

Ff (z) =∫R

eizxex(ex −K)1x>lnKdx

=

∞∫lnK

e(2+iz)xdx−K

∞∫lnK

e(1+iz)xdx

= −K2+iz 12 + iz

+K1+iz K

1 + iz

=K2+iz

(1 + iz)(2 + iz).

Therefore,

Ff (u+ iR) =K2+iu−R

(1 + iu−R)(2 + iu−R), R ∈ I1 = (2,∞). (3.39)

Similarly, for a self-quanto put option, where f(x) = (K − ex)+, K ∈ R+,we have that

Ff (u+ iR) =K2+iu−R

(1 + iu−R)(2 + iu−R), R ∈ I1 = (−∞, 1). (3.40)

3.5. Applications

A. European options. Assume we are interested in pricing a Europeanplain vanilla option on the asset S = eH , e.g. a call or put option or adigital option. Then, it is sufficient to know the characteristic function ofthe random variable XT ≡ HT , and HT must possess a moment generatingfunction in [0, 1] ⊂ I2 ⊆ R. Note that the restriction [0, 1] ⊂ I2 stems fromno-arbitrage considerations and is directly related to Assumption (ES).

Examples of options that can be treated include plain vanilla call andput options, with payoffs (ST −K)+ and (K−ST )+, digital cash-or-nothingand asset-or-nothing options, with payoffs 1ST>B and ST 1ST>B, doubledigital options, with payoff 1B<ST<B, and self-quanto options. The priceof the option follows by simply using Theorem 3.2 with the given ϕHT

andthe Fourier transform of the corresponding payoff function from section 3.4;therefore, we do not elaborate further. Below we describe some characteristicexamples of models used in mathematical finance.

Example 3.19 (Black–Scholes model). In the standard Black and Sc-holes (1973) model, the driving process H = (Ht)0≤t≤T follows a Brownianmotion with Law(H1|P ) = Normal(µ, σ2), σ > 0. The characteristic func-tion of Ht, t ∈ [0, T ], is

ϕHt(u) = exp(iuµt− u2σ2

2t

), (3.41)

and the moment generating function exists for R ∈ I2 = R.

Page 91: Applications of semimartingales and L´evy processes in ...webdoc.sub.gwdg.de/ebook/dissts/Freiburg/Papapantoleon2006.pdf · gales as driving processes. Another one, was to consider

3.5. APPLICATIONS 83

Example 3.20 (Levy models). In exponential (time-inhomogeneous)Levy models, the driving process is a (time-inhomogeneous) Levy processwith triplet of semimartingale characteristics (B,C, ν). The triplet of charac-teristics determines the law of the random variables generating the process.Indeed, using the Levy–Khintchine formula we have that, for all t ∈ [0, T ]

E[eiuLt

]= exp

t∫0

(iubs + u2 cs

2+∫R

(eiux − 1− iuh(x))λs(dx))ds. (3.42)

Moreover, subject to a moment condition such as Assumption (EM), wecan extend the characteristic function to the complex plane, with R ∈ I2 =[−M,M ]. Therefore, the methods developed here are particularly suitablefor the class of Levy models.

Examples of parametric models are:

Example 3.20.1 (Generalized hyperbolic model). Let H = (Ht)0≤t≤Tbe a generalized hyperbolic process with Law(H1|P ) = GH(λ, α, β, δ, µ), cf.Eberlein (2001, p. 321) or Eberlein and Prause (2002). The characteristicfunction of H1 is

ϕH1(u) = eiuµ(

α2 − β2

α2 − (β + iu)2

)λ2 Kλ

(δ√α2 − (β + iu)2

)Kλ

(δ√α2 − β2

) , (3.43)

where Kλ denotes the Bessel function of the third kind with index λ (cf.Abramowitz and Stegun 1968) and the moment generating function existsfor R ∈ I2 = (−α− β, α− β).

The class of generalized hyperbolic distributions is not closed under con-volution, hence the law or the characteristic function of the random variableHt is not known in closed form; of course, using the properties of Levyprocesses, we have that

ϕHt(u) = (ϕH1(u))t .

The only exception is the class of normal inverse Gaussian distributions,where λ = −1

2 (cf. Barndorff-Nielsen 1998). In that case, Law(Ht|P ) =NIG(α, β, δt, µt) and the characteristic function resumes the form

ϕHt(u) = eiuµtexp(δt

√α2 − β2)

exp(δt√α2 − (β + iu)2)

. (3.44)

Example 3.20.2 (CGMY model). LetH = (Ht)0≤t≤T be a CGMY Levyprocess, cf. Carr, Geman, Madan, and Yor (2002); another name for thisprocess is (generalized) tempered stable process (see e.g. Cont and Tankov2003). The characteristic function of Ht, t ∈ [0, T ], is

ϕHt(u) = exp(tCΓ(−Y )

[(M − iu)Y + (G+ iu)Y −MY −GY

])(3.45)

where C,G,M > 0 and Y < 2, and the moment generating function existsfor R ∈ I2 ⊂ R, depending on the parameters C,G,M, Y . For example, ifY = 0 then we recover the bilateral gamma distribution (Kuchler and Tappe2006) and the moment generating function exists for R ∈ I2 = (−G,M).

Page 92: Applications of semimartingales and L´evy processes in ...webdoc.sub.gwdg.de/ebook/dissts/Freiburg/Papapantoleon2006.pdf · gales as driving processes. Another one, was to consider

84 3. EXOTIC DERIVATIVES IN LEVY MODELS

Example 3.20.3 (Meixner model). Let H = (Ht)0≤t≤T be a Meixnerprocess with Law(H1|P ) = Meixner(α, β, δ), α > 0, −π < β < π, δ > 0,cf. Schoutens and Teugels (1998) and Schoutens (2002). The characteristicfunction Ht, t ∈ [0, T ], is

ϕHt(u) =

(cos β2

cosh αu−iβ2

)2δt

(3.46)

and the moment generating function exists for R ∈ I2 ⊂ R, depending onthe parameters α, β, δ.

Example 3.21 (Stochastic volatility Levy models). This class of modelswas proposed by Carr, Geman, Madan, and Yor (2003) and further investi-gated in Schoutens (2003). Let X = (Xt)0≤t≤T be a pure jump Levy processand Y = (Yt)0≤t≤T be an increasing process, independent of X. The processY acts as a stochastic clock measuring activity in business time and has theform

Yt =

t∫0

ysds

where y = (ys)0≤s≤T is a positive process. Carr et al. (2003) consider the CIRprocess as a candidate for y, i.e. the solution of the stochastic differentialequation

dyt = K(η − yt)dt+ λy12t dWt,

where W = (Wt)0≤t≤T is a standard Brownian motion. For other choices ofY see Schoutens (2003, Chapter 7).

The stochastic volatility Levy process is defined by time-changing theLevy process X with the increasing process Y , that is

Ht = XYt , 0 ≤ t ≤ T .

If we denote the characteristic functions of Xt and Yt by ϕXt and ϕYt re-spectively, then the characteristic function of Ht, t ∈ [0, T ], is given by

ϕHt(u) =ϕYt(−iϕXt(u))(

ϕYt(−iϕXt(−i)))iu (3.47)

cf. Schoutens (2003, section 7.3). The moment generating function exists forR ∈ I2 ⊂ R, depending on the choice of X and Y .

Example 3.22 (BNS model). The BNS model incorporates stochasticvolatility and jump effects in the asset price process; it was introduced byBarndorff-Nielsen and Shephard (2001). The driving process satisfies theSDE

dHt = (µ+ βσ2t )dt+ σtdWt + ρdZλt, H0 = 0

where

dσ2t = −λσ2

t dt+ dZλt, σ20 > 0, 0 ≤ t ≤ T .

Here, µ, β ∈ R, λ > 0 and ρ ≤ 0, W = (Wt)0≤t≤T is a standard Brownianmotion and Z = (Zt)0≤t≤T is a subordinator with triplet (0, 0, ν).

Page 93: Applications of semimartingales and L´evy processes in ...webdoc.sub.gwdg.de/ebook/dissts/Freiburg/Papapantoleon2006.pdf · gales as driving processes. Another one, was to consider

3.5. APPLICATIONS 85

The moment generating function of HT was derived in Nicolato andVenardos (2003) and is given by

MHT(u)=exp

zµT + (z2 + 2βz)ε(T )

2σ2

0 +

T∫0

λ( ∫

R+

(ef(s,z)x − 1)ν(dx))ds

,where ε(T ) = 1−e−λT

λ and f(s, z) = ρz + 12(z2 + 2βz)ε(T ). An efficient

numerical algorithm for calculating this function has been developed inKeller-Ressel (2006). Moreover, the moment generating function can be an-alytically extended in the complex domain, to an open strip of the formz ∈ C : <z ∈ (θ−, θ+); see Nicolato and Venardos (2003, p. 450).

B. Lookback options. The results concerning the characteristic func-tion of the supremum of a Levy process, cf. section 3.3, allow us to pricelookback options in models driven by Levy processes. Indeed, assuming thatthe asset price evolves according to (3.17), (3.18) and (3.19), a fixed strikelookback call option with payoff

(ST −K)+

can easily be viewed as a call option with driving process the supremum ofthe underlying Levy processes L. Therefore, combining Theorem 3.2, Theo-rem 3.14 and Example 3.15, we get that

CT (S;K) =12π

∫R

ϕLT(−u− iR)

K1+iu−R

(iu−R)(1 + iu−R)du, (3.48)

where

ϕLT(−u− iR) =

12π

∫R

eT (Y+iv)

Y + iv

κ(Y + iv, 0)κ(Y + iv, iu−R)

dv, (3.49)

for R ∈ (1,M ] and Y > 0. Of course, floating strike lookback options canbe treated by the same formulae, making use of the duality relationships inChapter 1, cf. Theorem 1.20.

C. One-touch options. Similarly, we can treat one-touch options onassets driven by Levy processes. Assuming that the asset price evolves ac-cording to (3.17), (3.18) and (3.19), a one-touch call option with payoff

1ST>B

can easily be priced using the developed methods, i.e. as a digital call optionwhere the driving process is the supremum of the underlying Levy process.Indeed, combining Theorem 3.2, Theorem 3.14 and Example 3.16, we getthat

DCT (S;B) =1

4π2

∫R

∫R

eT (Y+iv)

Y + iv

κ(Y + iv, 0)κ(Y + iv, iu−R)

Biu−R

R− iudvdu, (3.50)

where R ∈ (0,M ] and Y > 0.

Page 94: Applications of semimartingales and L´evy processes in ...webdoc.sub.gwdg.de/ebook/dissts/Freiburg/Papapantoleon2006.pdf · gales as driving processes. Another one, was to consider

86 3. EXOTIC DERIVATIVES IN LEVY MODELS

Remark 3.23. There are a number of articles that address the pricing ofexotic options, such as touch, barrier and lookback options in jump-diffusionand general Levy models. Kou and Wang (2003, 2004) derived formulas forthe value of lookback and barrier options in a jump diffusion model where thejumps are double-exponentially distributed; similar results where also provedby Lipton (2002). Boyarchenko and Levendorskiı (2002) applied methodsfrom potential theory and pseudodifferential operators, while Nguyen-Ngocand Yor (2005) use a probabilistic approach based on excursion theory. Fi-nally, Nguyen-Ngoc (2003) takes a similar probabilistic approach, combinedwith the method of Carr and Madan (1999); nevertheless, the formulae hederives for lookback options are quite more involved than (3.48).

D. Options on two assets. As a final application we consider a two-asset correlation option. These options have the payoff of a vanilla option onthe payment asset, denoted by S1, if the measurement asset, denoted by S2,ends up in the money; thus, a special case of these options is the correlationdigital option, considered in Chapter 2.

In the case of the correlation call option, the payoff is

(S1T −K)+1S2

T>B.

Assume, for simplicity, that the process driving the two assets is anR2-valued time-inhomogeneous Levy process L = (L1, L2)>, that satisfiesAssumption (EM), cf. p. 38. The characteristic function of Lt, t ∈ [0, T ],is described by (2.1) and the dynamics of the asset price processes S1 andS2 are described by (2.21), (2.22) and (2.31). Then, applying Theorem 3.5,Example 3.15 and Example 3.16, we get that

TACT (S1, S2;K,B) =1

4π2

∫R

∫R

ϕLT(−u− iR1,−v − iR2)

× K1+iu−R1

(iu−R1)(1 + iu−R1)Biu−R2

R2 − iudvdu,

(3.51)

where R1 ∈ (1,M ] and R2 ∈ (0,M ].

Page 95: Applications of semimartingales and L´evy processes in ...webdoc.sub.gwdg.de/ebook/dissts/Freiburg/Papapantoleon2006.pdf · gales as driving processes. Another one, was to consider

Part 2

Term structure models

Page 96: Applications of semimartingales and L´evy processes in ...webdoc.sub.gwdg.de/ebook/dissts/Freiburg/Papapantoleon2006.pdf · gales as driving processes. Another one, was to consider

The musical notes are only five in number but their melodiesare so numerous that one cannot hear them all.The primary colors are only five in number but their com-binations are so infinite that one cannot visualize them all.The flavors are only five in number but their blends are sovarious that one cannot taste them all.In battle there are only the normal and extraordinaryforces, but their combinations are limitless; none can com-prehend them all.

Sun Tzu, The Art of War.

Page 97: Applications of semimartingales and L´evy processes in ...webdoc.sub.gwdg.de/ebook/dissts/Freiburg/Papapantoleon2006.pdf · gales as driving processes. Another one, was to consider

CHAPTER 4

Duality and valuation in Levy term structuremodels

4.1. Introduction

The modeling of the term structure of interest rates is a contemporarytopic of research that has attracted the attention of academics and practi-tioners alike. On the one hand, the market of over-the-counter (OTC) inter-est rate derivative products represents the largest part of the OTC tradedderivatives; indeed, out of an estimated total amount of 284,819 billion US$in December 2005, an amount of 215,237 billion US$ corresponds to interestrate contracts (see BIS Quarterly Review, September 2006, p. A103). There-fore, appropriate models for the term structure of interest rates are of vitalinterest for fund managers and derivatives traders operating in these mar-kets. On the other hand, interest rate theory represents a unique challengefor researchers in mathematical finance; contrary to stock markets that con-sist of a finite number of traded assets, bond markets consist of the entireterm structure of interest rates: an infinite dimensional object.

Another difference between stock markets and bond markets is the mod-eling object; in stock markets, the individual traded stocks are subject tomodeling. In bond markets, there are different quantities that can be mod-eled. We start by describing these quantities and outlining some of the mod-els proposed in the literature. Of course, we do not aim at presenting a self-contained introduction to interest rate theory here, there are many excellenttextbooks available for that purpose; let us just refer to Bjork (2004), Brigoand Mercurio (2006), Hunt and Kennedy (2004) and Musiela and Rutkowski(2005).

A zero coupon bond is a financial instrument that pays off one unit ofcurrency at maturity. The time-t price of a zero coupon bond with maturityT (t ≤ T ) is denoted by B(t, 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 pur-poses. Let f(t, T ) denote the instantaneous forward rate at time t for in-vesting money over an infinitesimal time period starting at time T . Instan-taneous forward rates are formally defined by f(t, T ) := − ∂

∂T logB(t, T ).Then, zero coupon bond prices can be deduced from forward rates using therelationship B(t, T ) = exp

(−∫ Tt f(t, u)du

).

The instantaneous forward rate prevailing at time t for immediate bor-rowing or lending is called short or spot rate. It is usually denoted by rtand the following relationship holds: rt = f(t, t). An amount of one unit ofcurrency that is continuously reinvested at the short rate, from time zero

89

Page 98: Applications of semimartingales and L´evy processes in ...webdoc.sub.gwdg.de/ebook/dissts/Freiburg/Papapantoleon2006.pdf · gales as driving processes. Another one, was to consider

90 4. LEVY TERM STRUCTURE MODELS

until time T , yields the amount BMT = exp

( ∫ T0 rudu

). The quantity BM is

called the money market account or discount factor.The LIBOR rate is a rate where interest accrues according to a discrete

grid, the tenor structure. The LIBOR rate for investing at time T for a periodof length δ, is denoted L(T, T ) and is related to zero coupon bond prices viaL(T, T ) := 1

δ

(B(T,T )B(T,T+δ) − 1

). The forward LIBOR rate L(t, T ) is the time-t

LIBOR 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 couponbond with settlement date U .

Summarizing, the prices of zero coupon bonds, the forward LIBOR rateand the forward price are related in the following way:

1 + δL(t, T ) =B(t, T )

B(t, T + δ)= F (t, T, T + δ). (4.1)

The different approaches to modeling the term structure of interest ratescorrespond to different choices of modeling objects from the above mentionedquantities. 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 ratesis that of Heath, Jarrow, and Morton (1992). In the Heath–Jarrow–Morton(henceforth HJM) framework subject to modeling are instantaneous contin-uously compounded forward rates, or equivalently bond prices, which aredriven by a d-dimensional Wiener process. However, data from bond mar-kets do not support the use of the normal distribution. Empirical evidencefor the non-Gaussianity of daily returns from bond market data can befound in Raible (2000, chapter 5); the fit of the normal inverse Gaussiandistribution to the same data is particularly good, supporting the use ofLevy processes for modeling interest rates. Similar evidence appears in therisk-neutral world, i.e. from caplet implied volatility smiles and surfaces; seeEberlein and Kluge (2006a).

The Levy forward rate model was developed in Eberlein and Raible(1999) and extended to time-inhomogeneous Levy processes in Eberlein, Ja-cod, and Raible (2005). In these models, forward rates are driven by a (timeinhomogeneous) Levy process; therefore, the model allows to accurately cap-ture the empirical dynamics of interest rates, while it is still analyticallytractable, so that closed form valuation formulas for liquid derivatives canbe derived. Valuation formulas for caps, floors, swaptions and range noteshave been derived in Eberlein and Kluge (2006a, 2006b). The method relieson a convolution representation of the option value and the use of Laplacetransforms, beautifully described in Raible (2000). Estimation and calibra-tion methods are discussed in Eberlein and Kluge (2006a, 2007).

Eberlein, Jacod, and Raible (2005) provide a complete classification ofall equivalent martingale measures in the Levy forward rate model. Theyalso 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 infinitelymany risk factors, the jumps, and infinitely many hedging instruments, theentire term structure of interest rates, which render the market complete.

Page 99: Applications of semimartingales and L´evy processes in ...webdoc.sub.gwdg.de/ebook/dissts/Freiburg/Papapantoleon2006.pdf · gales as driving processes. Another one, was to consider

4.1. INTRODUCTION 91

There have been several other extensions of the HJM framework pro-posed in the literature. We just mention here Bjork et al. (1997), whereforward rates are driven by a d-dimensional Wiener process and a generalrandom measure, and Ozkan and Schmidt (2005), where forward rates aredriven by an infinite dimensional Levy process.

The main pitfall of the HJM framework is the assumption of continuouslycompounded rates, while in real markets interest accrues according to adiscrete grid, the tenor structure. LIBOR market models, that is, arbitrage-free term structure models on a discrete tenor, were constructed in a seriesof articles by Sandmann et al. (1995), Miltersen et al. (1997), Brace et al.(1997), and Jamshidian (1997). The backward induction construction of thelog-normal LIBOR market model was carried out by Musiela and Rutkowski(1997). In addition, LIBOR market models are consistent with the marketpractice of pricing caps and floors using Black’s formula (Black 1976).

Nevertheless, a familiar phenomenon appears: since the model is drivenby a Brownian motion, it cannot be calibrated accurately to the whole termstructure of volatility smiles. As a remedy, Eberlein and Ozkan (2005) de-veloped a LIBOR model driven by time inhomogeneous Levy processes.Valuation methods for caps and floors using an approximation – alreadyemployed by Schlogl (2002) and other authors – and Raible’s method, werepresented in Eberlein and Ozkan (2005). Kluge (2005) uses other approxi-mations together with Raible’s method to value caps, floors and swaptionsin the Levy LIBOR model. Calibration issues for this model are discussed inEberlein and Kluge (2007); see also Belomestny and Schoenmakers (2006).

The Levy forward price model is a market model based on the forwardprice – rather than the LIBOR rate – and driven by time inhomogeneousLevy processes; it was put forward by Eberlein and Ozkan (2005, pp. 342–343). A detailed construction of the model is presented in Kluge (2005,Chapter 3); there, it is also shown how this model can be embedded in theLevy forward rate model.

Although the LIBOR rate and the forward price differ only by an ad-ditive and a multiplicative constant, see (4.1), the two specifications leadto models with very different qualitative and quantitative behavior. In theLIBOR model, LIBOR rates change by an amount relative to their currentlevel, while in the forward price model changes do not depend on the ac-tual level (cf. Kluge 2005, p. 60). There are authors who claim that modelsbased on the forward process – also coined “arithmetic” or “Bachelier” LI-BOR models – are able to better describe the dynamics of the market than“classical” LIBOR models; see Henrard (2005).

Another advantage of the forward price model is that the driving processremains a time-inhomogeneous Levy process under each forward measure,hence this model is particularly suitable for practical implementation. Thedownside is that negative LIBOR rates can occur, like in an HJM model;this comes as no surprise, since this model can be embedded in the Levyforward rate framework.

The aim of this chapter is two-fold; on the one hand, we derive dualityresults relating caplets and floorlets in term structure models driven by Levyprocesses. On the other hand, we derive valuation formulas for an exotic path

Page 100: Applications of semimartingales and L´evy processes in ...webdoc.sub.gwdg.de/ebook/dissts/Freiburg/Papapantoleon2006.pdf · gales as driving processes. Another one, was to consider

92 4. LEVY TERM STRUCTURE MODELS

dependent interest rate derivative, the option on the composition of LIBORrates.

This chapter is organized as follows: in section 4.2 we briefly describetime-inhomogeneous Levy processes; for a detailed account, we refer to sec-tion 2.2. In section 4.3 we review three different approaches to modelinginterest rates based on Levy processes; namely, an HJM forward rate model,a LIBOR model and a model for forward prices. In section 4.4 we providecaplet-floorlet dualities in each of these models and in section 4.5 we derivevaluation formulas for the option on the composition in the forward rateand forward price models.

4.2. Time-inhomogeneous Levy processes

Let (Ω,F ,F, IP) be a complete stochastic basis, where F = FT ∗ andthe filtration F = (Ft)t∈[0,T ∗] satisfies the usual conditions; we assume thatT ∗ ∈ R+ is a finite time horizon. The driving process L = (Lt)t∈[0,T ∗] is atime-inhomogeneous Levy process, or a process with independent incrementsand absolutely continuous characteristics, in the sequel abbreviated PIIAC.Therefore, according to Definition 2.1, L is an adapted, cadlag, real-valuedstochastic process with independent increments, starting from zero, wherethe law of Lt, t ∈ [0, T ∗], is described by the characteristic function

IE[eiuLt

]= exp

t∫0

(ibsu−

cs2u2 +

∫R

(eiux − 1− iux)λs(dx))

ds, (4.2)

where bt ∈ R, ct ∈ R+ and λt is a Levy measure, i.e. satisfies λt(0) = 0and

∫R(1 ∧ |x|2)λt(dx) < ∞, for all t ∈ [0, T ∗]. In addition, the process L

satisfies Assumptions (AC) and (EM) given below.

Assumption (AC). The triplets (bt, ct, λt) satisfyT ∗∫0

(|bt|+ ct +

∫R

(1 ∧ |x|2)λt(dx))

dt <∞. (4.3)

Assumption (EM). There exist constants M, ε > 0 such that for everyu ∈ [−(1 + ε)M, (1 + ε)M ]

T ∗∫0

∫|x|>1

exp(ux)λt(dx)dt <∞. (4.4)

Moreover, without loss of generality, we assume that∫|x|>1 euxλt(dx) <∞

for all t ∈ [0, T ∗] and u ∈ [−(1 + ε)M, (1 + ε)M ].

These assumptions render the process L = (Lt)0≤t≤T ∗ a special semi-martingale, therefore it has the canonical decomposition (cf. Jacod andShiryaev 2003, II.2.38, and Eberlein et al. 2005)

Lt =

t∫0

bsds+

t∫0

√csdWs +

t∫0

∫R

x(µL − ν)(ds,dx), (4.5)

Page 101: Applications of semimartingales and L´evy processes in ...webdoc.sub.gwdg.de/ebook/dissts/Freiburg/Papapantoleon2006.pdf · gales as driving processes. Another one, was to consider

4.2. TIME-INHOMOGENEOUS LEVY PROCESSES 93

where µL is the random measure of jumps of the process L and W =(Wt)0≤t≤T ∗ is a IP-standard Brownian motion. The triplet of predictable orsemimartingale characteristics of L with respect to the measure P , T(L|P ) =(B,C, ν), is

Bt =

t∫0

bsds, Ct =

t∫0

csds, ν([0, t]×A) =

t∫0

∫A

λs(dx)ds, (4.6)

where A ∈ B(R). The triplet (b, c, λ) represents the local or differentialcharacteristics of L. In addition, the triplet of semimartingale characteristics(B,C, ν) determines the distribution of L, cf. Lemma 2.5.

We denote by θs the cumulant associated with the infinitely divisibledistribution with Levy triplet (bs, cs, λs), i.e. for z ∈ [−(1 + ε)M, (1 + ε)M ]

θs(z) := bsz +cs2z2 +

∫R

(ezx − 1− zx)λs(dx). (4.7)

In addition, we can extend θs to the complex domain C, for z ∈ C with<z ∈ [−(1 + ε)M, (1 + ε)M ] and the characteristic function of Lt can bewritten as

IE[eiuLt

]= exp

t∫0

θs(iu)ds. (4.8)

If L is a (time-homogeneous) Levy process, then (bs, cs, λs) and thus alsoθs do not depend on s. In that case, θ equals the cumulant (log-momentgenerating function) of L1.

Lemma 4.1. Let L = (Lt)0≤t≤T ∗ be a time-inhomogeneous Levy processsatisfying assumption (EM) and f : R+ → C a continuous function suchthat |<(f)| ≤M . Then

IE

[exp

t∫0

f(s)dLs

]= exp

t∫0

θs(f(s)

)ds. (4.9)

(The integrals are to be understood componentwise for real and imaginarypart.)

Proof. The proof is similar to the proof of Lemma 3.1 in Eberlein andRaible (1999); see also Kluge (2005, Proposition 1.9).

Lemma 4.2. Let L = (Lt)0≤t≤T ∗ be a time-inhomogeneous Levy pro-cess with characteristic triplet (B,C, ν), satisfying assumption (EM). ThenL? := −L is again a time-inhomogeneous Levy process satisfying assumption(EM), with triplet (B?, C?, ν?), where

B? = −BC? = C (4.10)

1A(x) ∗ ν? = 1A(−x) ∗ ν, A ∈ B(R\0).

Proof. See Lemma 2.14.

Page 102: Applications of semimartingales and L´evy processes in ...webdoc.sub.gwdg.de/ebook/dissts/Freiburg/Papapantoleon2006.pdf · gales as driving processes. Another one, was to consider

94 4. LEVY TERM STRUCTURE MODELS

4.3. Levy fixed income models

In this section we review three different approaches to modeling the termstructure of interest rates, where the driving process is a time-inhomogeneousLevy process.

4.3.1. The Levy forward rate model. In the Levy forward rateframework for modeling the term structure of interest rates, the dynam-ics of forward rates are specified and the prices of zero coupon bonds arethen deduced. Let T ∗ be a fixed time horizon and assume that for everyT ∈ [0, T ∗], there exists a zero coupon bond maturing at T traded in themarket; in addition, let U ∈ [0, T ∗].

The forward rates are driven by a time-inhomogeneous Levy processL = (Lt)t∈[0,T ∗] on the stochastic basis (Ω,F ,F, IP) with semimartingalecharacteristics (B,C, ν) or local characteristics (b, c, λ). The dynamics ofthe instantaneous continuously compounded forward rates for T ∈ [0, T ∗] isgiven by

f(t, T ) = f(0, T ) +

t∫0

α(s, T )ds−t∫

0

σ(s, T )dLs, 0 ≤ t ≤ T. (4.11)

The initial values f(0, T ) are deterministic, and bounded and measurablein T . In general, α and σ are real-valued stochastic processes defined onΩ× [0, T ∗]× [0, T ∗] that satisfy the following conditions:

(A1): for s > T we have α(ω; s, T ) = 0 and σ(ω; s, T ) = 0.(A2): (ω, s, T ) 7→ α(ω; s, T ), σ(ω; s, T ) are P⊗B([0, T ∗])-measurable.(A3): S(ω) := sups,T≤T ∗(|α(ω; s, T )|+ |σ(ω; s, T )|) <∞.

Then, (4.11) is well defined and we can find a “joint” version of all f(t, T )such that (ω; t, T ) 7→ f(t, T )(ω)1t≤T is O ⊗ B([0, T ∗])-measurable.

Taking the dynamics of the forward rates as the starting point, explicitexpressions for the dynamics of zero coupon bond prices and the moneymarket account can be deduced (cf. Proposition 5.2 in Bjork et al. 1997).From Eberlein and Kluge (2006b, (2.6)), we get that the time-T price of azero coupon bond maturing at time U is

B(T,U) =B(0, U)B(0, T )

exp

T∫0

Σ(s, T, U)dLs −T∫

0

A(s, T, U)ds

, (4.12)

where the following abbreviations are used:

Σ(s, T, U) := Σ(s, U)− Σ(s, T ),

A(s, T, U) := A(s, U)−A(s, T ),

and

A(s, T ) :=

T∫s∧T

α(s, u)du and Σ(s, T ) :=

T∫s∧T

σ(s, u)du.

Page 103: Applications of semimartingales and L´evy processes in ...webdoc.sub.gwdg.de/ebook/dissts/Freiburg/Papapantoleon2006.pdf · gales as driving processes. Another one, was to consider

4.3. LEVY FIXED INCOME MODELS 95

Similarly, using Eberlein and Kluge (2006b, (2.5)), we have for the moneymarket account

BMT =

1B(0, T )

exp

T∫0

A(s, T )ds−T∫

0

Σ(s, T )dLs

. (4.13)

In the sequel we will consider only deterministic volatility structures.Therefore, Σ and A are assumed to be deterministic real-valued functionsdefined on ∆ := (s, T ) ∈ [0, T ∗] × [0, T ∗]; s ≤ T, whose paths are con-tinuously differentiable in the second variable. Moreover, they satisfy thefollowing conditions.

(B1): The volatility structure Σ is continuous in the first argumentand bounded in the following way: for (s, T ) ∈ ∆ we have

0 ≤ Σ(s, T ) ≤M,

where M is the constant from Assumption (EM); for the dual-ity results, we will assume that 0 ≤ Σ(s, T ) ≤ M

2 . Furthermore,Σ(s, T ) 6= 0 for s < T and Σ(T, T ) = 0 for T ∈ [0, T ∗].

(B2): The drift coefficients A(·, T ) are given by

A(s, T ) = θs(Σ(s, T )), (4.14)

where θs is the cumulant associated with the triplet (bs, cs, λs),s ∈ [0, T ].

Remark 4.3. The drift condition (4.14) guarantees that bond pricesdiscounted by the money market account are martingales; hence, IP is amartingale measure. In addition, from Theorem 6.4 in Eberlein et al. (2005),we know that the martingale measure is unique.

4.3.2. The Levy LIBOR model. In the Levy LIBOR model, theforward LIBOR rate is modeled directly. Let 0 = T0 < T1 < T2 < · · · <TN < TN+1 = T ∗ denote a discrete tenor structure where δi = Ti+1 − Ti,i ∈ 0, 1, . . . , N; since the model is constructed via backward induction,we denote by T ∗j := TN+1−j for j ∈ 0, 1, . . . , N + 1 and δ∗j := δN+1−jfor j ∈ 1, . . . , N + 1. Consider a complete stochastic basis (Ω,F ,F, IPT ∗)and let L = (Lt)t∈[0,T ∗] be a time-inhomogeneous Levy process satisfyingAssumption (EM). L has semimartingale characteristics (0, C, νT

∗) or local

characteristics (0, c, λT∗) and its canonical decomposition is

Lt =

t∫0

√csdW T ∗

s +

t∫0

∫R

x(µL − νT∗)(ds,dx), (4.15)

where W T ∗ is a IPT ∗-standard Brownian motion, µL is the random measureassociated with the jumps of L and νT

∗is the IPT ∗-compensator of µL. We

further assume that the following conditions are in force.

(LR1): For any maturity Ti there exists a bounded, continuous, de-terministic function λ(·, Ti) : [0, Ti] → R, which represents the

Page 104: Applications of semimartingales and L´evy processes in ...webdoc.sub.gwdg.de/ebook/dissts/Freiburg/Papapantoleon2006.pdf · gales as driving processes. Another one, was to consider

96 4. LEVY TERM STRUCTURE MODELS

volatility of the forward LIBOR rate process L(·, Ti). Moreover,N∑i=1

∣∣λ(s, Ti)∣∣ ≤M,

for all s ∈ [0, T ∗], where M is the constant from Assumption (EM)and λ(s, Ti) = 0 for all s > Ti.

(LR2): The initial term structure B(0, Ti), 1 ≤ i ≤ N + 1, is strictlypositive and strictly decreasing. Consequently, the initial term struc-ture of forward LIBOR rates is given, for 1 ≤ i ≤ N , by

L(0, Ti) =1δi

(B(0, Ti)

B(0, Ti + δi)− 1).

The construction starts by postulating that the dynamics of the for-ward LIBOR rate with the longest maturity L(·, T ∗1 ) is driven by the time-inhomogeneous Levy process L, and evolve as a martingale under the ter-minal forward measure IPT ∗ . Then, the dynamics of the LIBOR rates forthe preceding maturities are constructed by backward induction; therefore,they are driven by the same process L and evolve as martingales under theirassociated forward measures.

Let us denote by IPT ∗j−1the forward measure associated to the settlement

date T ∗j−1, j ∈ 1, . . . , N + 1. The dynamics of the forward LIBOR rateL(·, T ∗j ), for an arbitrary T ∗j , is given by

L(t, T ∗j ) = L(0, T ∗j ) exp

t∫0

bL(s, T ∗j , T∗j−1)ds+

t∫0

λ(s, T ∗j )dLT ∗j−1s

,

(4.16)

where LT∗j−1 is a special semimartingale with canonical decomposition

LT ∗j−1

t =

t∫0

√csdW

T ∗j−1s +

t∫0

∫R

x(µL − νT∗j−1)(ds,dx). (4.17)

Here W T ∗j−1 is a IPT ∗j−1-standard Brownian motion, and νT

∗j−1 is the IPT ∗j−1

-compensator of µL. The dynamics of an arbitrary LIBOR rate again evolvesas a martingale under its corresponding forward measure; therefore, we spec-ify the drift term of the forward LIBOR process L(·, T ∗j ) as

bL(s, T ∗j , T∗j−1) = −1

2(λ(s, T ∗j ))2cs

−∫R

(eλ(s,T ∗j )x − 1− λ(s, T ∗j )x

)λT ∗j−1s (dx). (4.18)

The forward measure IPT ∗j−1, which is defined on (Ω,F , (Ft)0≤t≤T ∗j−1

), isrelated to the terminal forward measure IPT ∗ via

dIPT ∗j−1

dIPT ∗=

j−1∏k=1

1 + δkL(T ∗j−1, T∗k )

1 + δkL(0, T ∗k )=

B(0, T ∗)B(0, T ∗j−1)

j−1∏k=1

1 + δkL(T ∗j−1, T∗k ).

Page 105: Applications of semimartingales and L´evy processes in ...webdoc.sub.gwdg.de/ebook/dissts/Freiburg/Papapantoleon2006.pdf · gales as driving processes. Another one, was to consider

4.3. LEVY FIXED INCOME MODELS 97

Additionally, W T ∗j−1 is a IPT ∗j−1-Brownian motion which is related to the

IPT ∗-Brownian motion via

WT ∗j−1

t = WT ∗j−2

t −t∫

0

α(s, T ∗j−1, T∗j−2)

√csds = . . .

= W T ∗t −

t∫0

(j−1∑k=1

α(s, T ∗k , T∗k−1)

)√csds,

where

α(t, T ∗k , T∗k−1) =

δkL(t−, T ∗k )1 + δkL(t−, T ∗k )

λ(t, T ∗k ).

Similarly, νT∗j−1 is the IPT ∗j−1

-compensator of µL and is related to the IPT ∗-compensator of µL via

νT∗j−1(ds,dx) = β(s, x, T ∗j−1, T

∗j−2)ν

T ∗j−2(ds,dx) = . . .

=

(j−1∏k=1

β(s, x, T ∗k , T∗k−1)

)νT

∗(ds,dx), (4.19)

where

β(t, x, T ∗k , T∗k−1) =

δkL(t−, T ∗k )1 + δkL(t−, T ∗k )

(eλ(t,T ∗k )x − 1

)+ 1. (4.20)

Remark 4.4. Notice that the process LT∗j−1 , driving the forward LIBOR

rate L(·, T ∗j ), and L = LT∗

have the same martingale parts and differ onlyin the finite variation part (drift). An application of Girsanov’s theorem forsemimartingales yields that the IPT ∗j−1

-finite variation part of L is

·∫0

cs

j−1∑k=1

α(s, T ∗k , T∗k−1)ds+

·∫0

∫R

x

(j−1∏k=1

β(s, x, T ∗k , T∗k−1)− 1

)νT

∗(ds,dx).

Remark 4.5. The process L = LT∗

driving the most distant LIBOR rateL(·, T ∗1 ) is – by assumption – a time-inhomogeneous Levy process. However,this is not the case for any of the processes LT

∗j−1 driving the remaining

LIBOR rates, because the random terms δL(t−,T ∗k )

1+δL(t−,T ∗k ) enter into the compen-

sators νT∗j−1 during the construction; see equations (4.19) and (4.20))

4.3.3. The Levy forward price model. In the Levy forward pricemodel the dynamics of forward prices, i.e. ratios of successive bond prices,are specified. Let 0 = T0 < T1 < T2 < · · · < TN < TN+1 = T ∗ denote adiscrete tenor structure where δi = Ti+1 − Ti, i ∈ 0, 1, . . . , N; the modelis again constructed via backward induction, hence we denote by T ∗j :=TN+1−j for j ∈ 0, 1, . . . , N + 1 and δ∗j := δN+1−j for j ∈ 1, . . . , N + 1.Consider a complete stochastic basis (Ω,F ,F, IPT ∗) and let L = (Lt)t∈[0,T ∗]

be a time-inhomogeneous Levy process satisfying Assumption (EM). L has

Page 106: Applications of semimartingales and L´evy processes in ...webdoc.sub.gwdg.de/ebook/dissts/Freiburg/Papapantoleon2006.pdf · gales as driving processes. Another one, was to consider

98 4. LEVY TERM STRUCTURE MODELS

semimartingale characteristics (0, C, νT∗) or local characteristics (0, c, λT

∗)

and its canonical decomposition is

Lt =

t∫0

√csdW T ∗

s +

t∫0

∫R

x(µL − νT∗)(ds,dx), (4.21)

where W T ∗ is a IPT ∗-standard Brownian motion, µL is the random measureassociated with the jumps of L and νT

∗is the IPT ∗-compensator of µL.

Moreover, we assume that the following conditions are in force.

(FP1): For any maturity Ti there exists a bounded, continuous, de-terministic function λ(·, Ti) : [0, Ti] → R, which represents thevolatility of the forward price process F (·, Ti, Ti + δi). Moreover,we require that the volatility structure satisfies∣∣∣ i∑

k=1

λ(s, Tk)∣∣∣ ≤M, ∀ i ∈ 1, . . . , N,

for all s ∈ [0, T ∗], where M is the constant from Assumption (EM)and λ(s, Ti) = 0 for all s > Ti.

(FP2): The initial term structure B(0, Ti), 1 ≤ i ≤ N + 1 is strictlypositive. Consequently, the initial term structure of forward priceprocesses is given, for 1 ≤ i ≤ N , by

F (0, Ti, Ti + δi) =B(0, Ti)

B(0, Ti + δi).

The construction starts by postulating that the dynamics of the for-ward process with the longest maturity F (·, T ∗1 , T ∗) are driven by the time-inhomogeneous Levy process L, and evolve as a martingale under the termi-nal forward measure IPT ∗ . Then, the dynamics of the forward processes forthe preceding maturities are constructed by backward induction; therefore,they are driven by the same process L and evolve as martingales under theirassociated forward measures.

Let us denote by IPT ∗j−1the forward measure associated with the set-

tlement date T ∗j−1, j ∈ 1, . . . , N + 1. The dynamics of the forward priceprocess F (·, T ∗j , T ∗j−1) is given by

F (t, T ∗j , T∗j−1)=F (0, T ∗j , T

∗j−1) exp

t∫0

b(s, T ∗j , T∗j−1)ds+

t∫0

λ(s, T ∗j )dLT ∗j−1s

where

LT ∗j−1

t =

t∫0

√csdW

T ∗j−1s +

t∫0

∫R

x(µL − νT∗j−1)(ds,dx)

is a time-inhomogeneous Levy process. Here W T ∗j−1 is a IPT ∗j−1-standard

Brownian motion and νT∗j−1 is the IPT ∗j−1

-compensator of µL. The forward

Page 107: Applications of semimartingales and L´evy processes in ...webdoc.sub.gwdg.de/ebook/dissts/Freiburg/Papapantoleon2006.pdf · gales as driving processes. Another one, was to consider

4.4. CAPLET-FLOORLET DUALITY 99

price process evolves as a martingale under its corresponding forward mea-sure, hence, we specify the drift of the forward price process to be

b(s, T ∗j , T∗j−1) = −1

2(λ(s, T ∗j ))2cs

−∫R

(eλ(s,T ∗j )x − 1− λ(s, T ∗j )x

)λT ∗j−1s (dx). (4.22)

The forward measure IPT ∗j−1, which is defined on (Ω,F , (Ft)0≤t≤T ∗j−1

), isrelated to the terminal forward measure IPT ∗ via

dIPT ∗j−1

dIPT ∗=

j−1∏k=1

F (T ∗j−1, T∗k , T

∗k−1)

F (0, T ∗k , T∗k−1)

=B(0, T ∗)B(0, T ∗j−1)

j−1∏k=1

F (T ∗j−1, T∗k , T

∗k−1).

In addition, the IPT ∗j−1-Brownian motion is related to the IPT ∗-Brownian

motion via

WT ∗j−1

t = WT ∗j−2

t −t∫

0

λ(s, T ∗j−1)√csds = . . .

= W T ∗t −

t∫0

(j−1∑k=1

λ(s, T ∗k )

)√csds. (4.23)

Similarly, the IPT ∗j−1-compensator of µL is related to the IPT ∗-compensator

of µL via

νT∗j−1(ds,dx) = exp

(λ(s, T ∗j−1)x

)νT

∗j−2(ds,dx) = . . .

= exp

(x

j−1∑k=1

λ(s, T ∗k )

)νT

∗(ds,dx). (4.24)

Remark 4.6. The process L = LT∗, driving the most distant forward

price, and LT∗j−1 , driving the forward price F (·, T ∗j , T ∗j−1), are both time-

inhomogeneous Levy processes, sharing the same martingale parts and dif-fering only in the finite variation parts. Applying Girsanov’s theorem forsemimartingales yields that the IPT ∗j−1

-finite variation part of L is

·∫0

cs

(j−1∑k=1

λ(s, T ∗k )

)ds+

·∫0

∫R

x

(exp

(x

j−1∑k=1

λ(s, T ∗k ))− 1

)νT

∗(ds,dx).

4.4. Caplet-floorlet duality

The well-known caplet-floorlet parity relates caps and floors of the samestrike and time to maturity; let L(T, T ) denote the LIBOR rate for theperiod [T, T +δ], then the values of a caplet and a floorlet with strike K andpayoff δ(L(T, T )−K)+ and δ(K − L(T, T ))+ respectively, are related via

C0(K,T ) = F0(K,T ) +B(0, T )− (1 + δK)B(0, T + δ),

Page 108: Applications of semimartingales and L´evy processes in ...webdoc.sub.gwdg.de/ebook/dissts/Freiburg/Papapantoleon2006.pdf · gales as driving processes. Another one, was to consider

100 4. LEVY TERM STRUCTURE MODELS

where B(0, T ) denotes the price of a zero coupon bond maturing at T andC0(K,T ) and F0(K,T ) denote the present value of a caplet and a floorletrespectively, with strike rate K maturing at T .

This section aims at providing duality relationships between caplets andfloorlets with different strikes but the same time of maturity and moneyness,in term structure models driven by time-inhomogeneous Levy processes. By‘moneyness’ of a caplet (resp. floorlet), we mean the ratio of the initialforward LIBOR 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 inthe introduction of Chapter 1. Apart from providing better understandingof valuation formulas and simplifying computational work, such results areapplied for statically hedging other – usually exotic – derivatives; see, e.g.Carr et al. (1998).

The proofs are based on the choice of a suitable numeraire and thesubsequent change of the probability measure; this method was pioneeredby Geman et al. (1995). Three different approaches to modeling interest ratesare considered: a Heath–Jarrow–Morton forward rate model, a model for theLIBOR, and a model for the forward price, driven by time inhomogeneousLevy processes, as described in the previous section.

4.4.1. Duality in the Levy forward rate model. In this section wederive a duality relationship between call and put options on zero couponbonds. As a direct corollary of this result, we obtain a duality relating capletsand floorlets in the Levy forward rate model.

For the duality result, we define the constant D via

D := IE

[B(T,U)(BMT

)2]

= IE

[B(0, U)B(0, T ) exp

( T∫0

(Σ(s, U) + Σ(s, T )

)dLs

)

× exp

( T∫0

−(A(s, U) +A(s, T )

)ds

)]

= B(0, U)B(0, T ) exp

( T∫0

θs(Σ(s, T, U)

)ds

)

× exp

( T∫0

−(θs(Σ(s, U)) + θs(Σ(s, T ))

)ds

),

where we used the abbreviation Σ(s, T, U) := Σ(s, U)+Σ(s, T ). In addition,we define a measure IP via the Radon–Nikodym derivative

dIPdIP

:=B(T,U)

D(BMT

)2 =B(T,U)(

BMT

)2 IE[B(T,U)

(BMT )2

] = ZT , (4.25)

Page 109: Applications of semimartingales and L´evy processes in ...webdoc.sub.gwdg.de/ebook/dissts/Freiburg/Papapantoleon2006.pdf · gales as driving processes. Another one, was to consider

4.4. CAPLET-FLOORLET DUALITY 101

noting that IE[B(T,U)

D(BMT )2

]= 1 and that the two measures, IP and IP, are

equivalent since ZT is strictly positive. The density process Z = (Zt)t∈[0,T ]

related to this change of measure is given by the restriction of the Radon–Nikodym derivative to the σ-field Ft, i.e. for t ≤ T , we get

Zt = IE

[B(T,U)

D(BMT

)2 ∣∣∣∣Ft]

= IE

[exp

( T∫0

Σ(s, T, U)dLs −T∫

0

θs(Σ(s, T, U)

)ds

)∣∣∣∣Ft]

= exp

( t∫0

Σ(s, T, U)dLs −t∫

0

θs(Σ(s, T, U)

)ds

).

Remark 4.7. Notice that using (B1), we have that∫ ·0 Σ(s, T )dLs is well

defined. Moreover, from (B1) and (EM) we get that∫ ·0 Σ(s, T )dLs is expo-

nentially special (cf. Kallsen and Shiryaev 2002a, Definition 2.12). ApplyingTheorem 2.18 in Kallsen and Shiryaev (2002a) we have thatexp

[ t∫0

Σ(s, T, U)dLs −t∫

0

θs(Σ(s, T, U)

)ds]

t∈[0,T ]

is a martingale and the last equality follows. Alternatively, this follows fromLemma 4.1 and Assumption (B1).

Now, using the canonical decomposition of L and (4.7), we can rewritethe density process in the “usual” form

Zt = exp

( t∫0

Σ(s, T, U)√csdWs −

t∫0

(Σ(s, T, U)

)2 cs2

ds

+

t∫0

∫R

xΣ(s, T, U)(µL − ν)(ds,dx)

−t∫

0

∫R

(exΣ(s,T,U) − 1− xΣ(s, T, U)

)ν(ds,dx)

).

Equivalently, we can express Z as the stochastic exponential of a suitabletime-inhomogeneous Levy process denoted by X, that is Z = E(X) where

X :=

·∫0

Σ(s, T, U)√csdWs +

·∫0

∫R

(exΣ(s,T,U) − 1

)(µL − ν)(ds,dx);

here, we have applied Lemma 2.6. in Kallsen and Shiryaev (2002a).

Page 110: Applications of semimartingales and L´evy processes in ...webdoc.sub.gwdg.de/ebook/dissts/Freiburg/Papapantoleon2006.pdf · gales as driving processes. Another one, was to consider

102 4. LEVY TERM STRUCTURE MODELS

Proposition 4.8. The triplet of local characteristics of L = (Lt)t∈[0,T ∗]

under the measure IP isbs = bs + βscs +

∫Rx(Y (s, x)− 1)λs(dx)

cs = cs

λs(dx) = Y (s, x)λs(dx),

(4.26)

where the tuple (β, Y ) of predictable processes associated to the process Lunder this change of measure is

βs = Σ(s, T, U) and Y (s, x) = exΣ(s,T,U). (4.27)

Proof. Applying Girsanov’s theorem for semimartingales, cf. TheoremIII.3.24 in Jacod and Shiryaev (2003), we get that the IP-semimartingalecharacteristics of L are

B = B + β · C + x(Y − 1) ∗ νC = C (4.28)ν = Y · ν

using the notation of Chapter 1. Taking into account the relationship be-tween the local and semimartingale characteristics in (4.6), the result isproved.

Now, it remains to verify that we can use the versions of β and Y asdescribed in (4.27), where β = βs(ω) and Y = Y (ω; s, x) are defined by thefollowing formulae (cf. Jacod and Shiryaev 2003, III.3.28):

〈Zc, Lc〉 = (Z−β) · C (4.29)

and

Y = M IPµL

(Z

Z−

∣∣∣P) . (4.30)

In equation (4.30) P = P ⊗ B(R) denotes the σ-field of predictable setson Ω × [0, T ∗] × R and M IP

µL = µL(ω; dt,dx)IP(dω) is the positive measureon (Ω× [0, T ∗]× R,F ⊗ B([0, T ∗])⊗ B(R)) defined by

M IPµL(W ) = IE(W ∗ µL)T ∗ (4.31)

for measurable nonnegative functions W = W (ω; t, x) on Ω×[0, T ∗]×R. Theconditional expectation M IP

µL

(ZZ−

∣∣P) is, by definition, the M IPµL-a.s. unique

P-measurable function Y satisfying

M IPµL

(Z

Z−U

)= M IP

µL(Y U) (4.32)

for all nonnegative P-measurable functions U = U(ω; t, x).An application of Ito’s formula, cf. Appendix B, yields that the contin-

uous martingale part of the density process is

Zc =

·∫0

Zs−dXcs ; (4.33)

Page 111: Applications of semimartingales and L´evy processes in ...webdoc.sub.gwdg.de/ebook/dissts/Freiburg/Papapantoleon2006.pdf · gales as driving processes. Another one, was to consider

4.4. CAPLET-FLOORLET DUALITY 103

using in addition I.4.41 in Jacod and Shiryaev (2003), we get

〈Zc, Lc〉t =

⟨ ·∫0

Zs−dXcs , L

c

⟩t

=

t∫0

Zs−d 〈Xc, Lc〉s

=

t∫0

Zs−d

⟨ ·∫0

Σ(u, T, U)√cudWu,

·∫0

√cudWu

⟩s

=

t∫0

Zs−Σ(s, T, U)csds.

Therefore, we conclude that βs = Σ(s, T, U).Finally, we prove that we can choose Y (s, x) = exΣ(s,T,U). We have

M IPµL

(exΣ(s,T,U)U

)= IE

[ T ∗∫0

∫R

exΣ(t,T,U)U(ω; t, x)µL(ω; dt,dx)]

= IE[ ∑

0≤t≤T ∗e∆Lt(ω)Σ(t,T,U)U(ω; t,∆Lt(ω))1∆Lt(ω) 6=0

]

= IE[ T ∗∫

0

∫R

Zt(ω)Zt−(ω)

U(ω; t, x)µL(ω; dt,dx)]

= M IPµL

(ZsZs−

U

), (4.34)

because Z(ω)Z−(ω)1∆Z(ω) 6=0 = exp(Σ(·, T, U)∆L(ω)).

We denote the value of a call option, with maturity T and strike K, ona bond with maturity U , by

C(B(0, T );

B(0, U)B(0, T )

,K;C, ν)

= IE[

1BMT

(B(T,U)−K)+],

where B(0, T ) is the discount factor associated with the option’s maturitydate T and B(0, U)/B(0, T ) is the initial value of the forward price processB(·, U)/B(·, T ). The dynamics of BM

T and B(T,U) are given by equations(4.13) and (4.12) respectively and the drift terms A are determined by thetwo characteristics of the driving process (C, ν) and the volatility structuresΣ, according to equation (4.14). Similar notation is used for the put optionon a zero coupon bond.

Theorem 4.9. Assume that bond prices are modeled according to theLevy forward rate model. Then, we can relate the value of a call and a put

Page 112: Applications of semimartingales and L´evy processes in ...webdoc.sub.gwdg.de/ebook/dissts/Freiburg/Papapantoleon2006.pdf · gales as driving processes. Another one, was to consider

104 4. LEVY TERM STRUCTURE MODELS

option on a bond via the following duality:

C(B(0, T );

B(0, U)B(0, T )

,K;C, ν)

= P(B(0, T );K,

B(0, U)B(0, T )

;C, ν)

where f(s, x) = exp((Σ(s, U) + Σ(s, T ))x

)and 1A(x) ∗ ν = 1A(−x)f ∗ ν,

A ∈ B(R\0).

Proof. The price of a call option with maturity T and strike K, on abond with maturity U , is given by

C = IE[

1BMT

(B(T,U)−K)+]

= IE[KB(T,U)

BMT

(K−1 −B(T,U)−1)+]

= IE

[B(T,U)

D(BMT

)2KDBMT (K−1 −B(T,U)−1)+

]

and changing measure from IP to IP, we get that

C = IE[KDBM

T (K−1 −B(T,U)−1)+].

This can be re-written as

C = IE

[B(0, T )B(0, U)

DBMT

(B(0, U)B(0, T )

−KB(0, U)B(0, T )

B(T,U)−1

)+]

= IE

[1

BMT

(K − B(T,U)

)+], (4.35)

for (BMT )−1 := B(0,T )

B(0,U)DBMT , K := B(0,U)

B(0,T ) and B(T,U) := K B(0,U)B(0,T )B(T,U)−1.

We will calculate the IP-dynamics of B(T,U). Firstly, by its definitionand (4.12), we get that

B(T,U) = K exp

( T∫0

(Σ(s, T )− Σ(s, U)

)dLs +

T∫0

(A(s, U)−A(s, T )

)ds

).

Keeping in mind that the local characteristics of L under IP are givenby Proposition 4.8, we define the time-inhomogeneous Levy process L :=−L. The local characteristics of L, using Lemma 4.2, are (−b, c, λ), where∫

1A(x)λ(dx) =∫

1A(−x)λ(dx). Then we get

B(T,U) = K exp

( T∫0

(Σ(s, U)− Σ(s, T )

)dLs +

T∫0

(A(s, U)−A(s, T )

)ds

).

Page 113: Applications of semimartingales and L´evy processes in ...webdoc.sub.gwdg.de/ebook/dissts/Freiburg/Papapantoleon2006.pdf · gales as driving processes. Another one, was to consider

4.4. CAPLET-FLOORLET DUALITY 105

Secondly, for the deterministic terms we have

exp

( T∫0

A(s, U)ds

)= IE

[exp

( T∫0

Σ(s, U)dLs)]

= IE

[D(BMT

)2B(T,U)

exp( T∫

0

Σ(s, U)dLs)]

= IE

[exp

( T∫0

−Σ(s, T, U)dLs)

× exp( T∫

0

Σ(s, U)dLs)

exp( T∫

0

θs(Σ(s, T, U)

)ds)]

= IE

[exp

( T∫0

−Σ(s, T )dLs)]

exp( T∫

0

θs(Σ(s, T, U)

)ds)

= IE

[exp

( T∫0

Σ(s, T )dLs)]

exp( T∫

0

θs(Σ(s, T, U)

)ds)

= exp

( T∫0

A(s, T )ds

)exp

( T∫0

θs(Σ(s, T, U)

)ds

),

(4.36)

where A(s, T ) := θs(Σ(s, T )) and θs is the cumulant associated with theLevy triplet (−bs, cs, λs). Similarly, for the other term we have

exp

( T∫0

A(s, T )ds

)= IE

[exp

( T∫0

Σ(s, T )dLs)]

= exp

( T∫0

A(s, U)ds

)exp

( T∫0

θs(Σ(s, T, U)

)ds

).

Therefore, the IP-dynamics of B(T,U) is

B(T,U) = K exp

T∫0

(Σ(s, U)− Σ(s, T )

)dLs +

T∫0

(A(s, T )− A(s, U)

)ds

.

(4.37)

Page 114: Applications of semimartingales and L´evy processes in ...webdoc.sub.gwdg.de/ebook/dissts/Freiburg/Papapantoleon2006.pdf · gales as driving processes. Another one, was to consider

106 4. LEVY TERM STRUCTURE MODELS

Finally, for the term corresponding to the money-market account, wehave

1

BMT

=B(0, T )B(0, U)

B(0, U)B(0, T ) exp

( T∫0

θs(Σ(s, T, U)

)ds

)

× exp

( T∫0

−(θs(Σ(s, U)) + θs(Σ(s, T ))

)ds

)

× 1B(0, T )

exp

( T∫0

θs(Σ(s, T ))ds−T∫

0

Σ(s, T )dLs

)

= B(0, T ) exp

( T∫0

Σ(s, T )dLs

)exp

( T∫0

−A(s, U)ds

)

× exp

( T∫0

θs(Σ(s, T, U)

)ds

)and using equation (4.36), we get

1

BMT

= B(0, T ) exp

( T∫0

Σ(s, T )dLs

)exp

( T∫0

−θs(Σ(s, T, U)

)ds

)

× exp

( T∫0

θs(Σ(s, T, U)

)ds

)exp

( T∫0

−A(s, T )ds

)

= B(0, T ) exp

( T∫0

Σ(s, T )dLs −T∫

0

A(s, T )ds

). (4.38)

In view of equations (4.35), (4.37) and (4.38), the result is proved.

Remark 4.10. Note that B(T,U) discounted by BMT becomes a IP-

martingale.

Remark 4.11. The change of measure from IP to IP is not “structure-preserving” for time-homogeneous processes, e.g. Levy processes. Therefore,even if we had modeled bond prices as exponentials of Levy processes un-der IP, the process driving the bond prices under IP would have been atime-inhomogeneous Levy process; the driving process would remain time-homogeneous only if the jump part vanished or in some pathetic cases (e.g.Σ(·, T ) ≡ 0,∀T ∈ [0, T ∗]). This is obvious from the structure of the functionf in Theorem 4.9. A similar phenomenon does not occur when modelingequities with Levy processes (compare with Theorem 1.18 or Corollary 4.2in Eberlein and Papapantoleon 2005b).

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 directlyrelating the values of caplets and floorlets in the Levy forward rate model.

Page 115: Applications of semimartingales and L´evy processes in ...webdoc.sub.gwdg.de/ebook/dissts/Freiburg/Papapantoleon2006.pdf · gales as driving processes. Another one, was to consider

4.4. CAPLET-FLOORLET DUALITY 107

We denote the value of a floorlet with strike K maturing at time Ti thatsettles in arrears at Ti+1, by

FL(B(0, Ti);L(0, Ti),K;C, ν

)= IE

[1

BMTi+1

δi

(K − L(Ti, Ti)

)+]

= (1 + δiK)IE

[1BMTi

(B(Ti, Ti+1)−K

)+],

where L(0, Ti) = 1δi

( B(0,Ti)B(0,Ti+1) − 1

)is the initial value of the forward LIBOR

rate and the strike K := 1/(1 + δiK). Similar notation is used for a caplet.

Corollary 4.12. Assume that bond prices are modeled according to theLevy forward rate model. Then, we can relate the value of a caplet and afloorlet via the following duality:

FL(B(0, Ti);L(0, Ti),K;C, ν

)= C CL

(B(0, Ti);K,L(0, Ti);C, ν

)where C := 1+δiK

1+δiL(0,Ti), f(s, x) = exp

((Σ(s, Ti) + Σ(s, Ti+1))x

)and 1A(x) ∗

ν = 1A(−x)f ∗ ν, A ∈ B(R\0).

Proof. We simply use the result of Appendix A to express a floorlet asa call option on a zero coupon bond, then apply Theorem 4.9 and then theformula of Appendix A in the other direction, to express a put option on azero coupon bond as a caplet. We get

FL(B(0, Ti);L(0, Ti),K;C, ν

)= K−1C

(B(0, Ti);

B(0, Ti+1)B(0, Ti)

,K;C, ν)

= K−1P(B(0, Ti);K,

B(0, Ti+1)B(0, Ti)

;C, ν)

= CCL(B(0, Ti);K,L(0, Ti);C, ν

).

4.4.2. Duality in the Levy LIBOR model. The aim of this sectionis to provide a duality relationship between caplets and floorlets in the LevyLIBOR model. This result generalizes Theorem 5.1 in Eberlein, Kluge, andPapapantoleon (2006) since we do not approximate the random compensatorby a deterministic one. Instead, we build on the results of Chapter 1 anddeal with the most general case directly.

The payoff of a caplet with strike K, that is settled in arrears at timeT ∗j−1, is δ∗j (L(T ∗j , T

∗j )−K)+; similarly, the payoff of a floorlet with the same

settlement date and strike is δ∗j (K − L(T ∗j , T∗j ))+.

Assuming that LIBOR rates are modeled according to the Levy LIBORmodel, we denote the value of a caplet with strike K, by

C(L(0, T ∗j ),K;C, νT

∗j−1

)= B(0, T ∗j−1) IEIPT∗

j−1

[δ∗j (L(T ∗j , T

∗j )−K)+

],

where L(0, T ∗j ) is the initial value of the forward LIBOR process. Noticethat the drift term is determined by the other two characteristics of thedriving process (C, νT

∗j−1) and the volatility structure λ(·, T ∗j ), according to

Page 116: Applications of semimartingales and L´evy processes in ...webdoc.sub.gwdg.de/ebook/dissts/Freiburg/Papapantoleon2006.pdf · gales as driving processes. Another one, was to consider

108 4. LEVY TERM STRUCTURE MODELS

equation (4.18). Moreover, the discount factor B(0, T ∗j−1) corresponds to thesettlement date T ∗j−1. Similar notation is used for a floorlet.

Theorem 4.13. Let the LIBOR rate be modeled according to the LevyLIBOR model. We can relate the values of caplets and floorlets via the fol-lowing duality

C(L(0, T ∗j ),K;C, νT

∗j−1

)= F

(K,L(0, T ∗j );C, νT

∗j−1

)where f(s, x) = exp(λ(s, T ∗j )x) and 1A(x) ∗ νT

∗j−1 = 1A(−x)f ∗ νT

∗j−1, A ∈

B(R\0).

Proof. From the time-T0 value of a caplet settled at time T ∗j−1, we get

C = B(0, T ∗j−1) IEIPT∗j−1

[δ∗j (L(T ∗j , T

∗j )−K)+

]= B(0, T ∗j−1) IEIPT∗

j−1

[δ∗jKL(T ∗j , T

∗j )(K−1 − L(T ∗j , T

∗j )−1)+

]= B(0, T ∗j−1)KL(0, T ∗j )

× IEIPT∗j−1

[L(T ∗j , T

∗j )

L(0, T ∗j )δ∗j (K

−1 − L(T ∗j , T∗j )−1)+

]. (4.39)

Define the measure IPT ∗j−1on (Ω,F , (Ft)0≤t≤T ∗j−1

) via the Radon–Nikodymderivative

dIPT ∗j−1

dIPT ∗j−1

=L(T ∗j , T

∗j )

L(0, T ∗j )= Z (4.40)

and the valuation problem (4.39), reduces to

C = B(0, T ∗j−1)KL(0, T ∗j ) IEeIPT∗j−1

[δ∗j (K

−1 − L(T ∗j , T∗j )−1)+

]. (4.41)

The density process is given by the restriction of the Radon–Nikodym deriv-ative to the σ-field Ft, and because the forward LIBOR process is a IPT ∗j−1

-martingale, we get

Zt = IEIPT∗j−1

[dIPT ∗j−1

dIPT ∗j−1

∣∣∣∣Ft]

=L(t, T ∗j )L(0, T ∗j )

= exp

( t∫0

λ(s, T ∗j )c1/2s dWT ∗j−1s − 1

2

t∫0

(λ(s, T ∗j )

)2csds

+

t∫0

∫R

xλ(s, T ∗j )(µL − νT∗j−1)(ds,dx)

−t∫

0

∫R

(eλ(s,T ∗j )x − 1− λ(s, T ∗j )x

)νT

∗j−1(ds,dx)

). (4.42)

Page 117: Applications of semimartingales and L´evy processes in ...webdoc.sub.gwdg.de/ebook/dissts/Freiburg/Papapantoleon2006.pdf · gales as driving processes. Another one, was to consider

4.4. CAPLET-FLOORLET DUALITY 109

Equivalently, the density process Z can be expressed as the stochastic ex-ponential of a semimartingale X, that is Z = E(X) where

X :=

·∫0

λ(s, T ∗j )c1/2s dWT ∗j−1s +

·∫0

∫R

(exλ(s,T ∗j ) − 1

)(µL − νT

∗j−1)(ds,dx).

Adapting the results of Proposition 4.8 to the present setting, it followsthat the tuple of predictable processes associated to the process L and thischange of measure is

βs = λ(s, T ∗j ) and Y (s, x) = exλ(s,T ∗j ). (4.43)

Additionally, using Girsanov’s theorem for semimartingales, we immediatelyrecognize W T ∗j−1 = W T ∗j−1 −

∫ ·0 λ(s, T ∗j )c1/2s ds as a IPT ∗j−1

-Brownian motion

and νT∗j−1(dt,dx) = exλ(t,T ∗j )νT

∗j−1(dt,dx) as the IPT ∗j−1

-compensator of µL.

Hence, the IPT ∗j−1-local characteristics of LT

∗j−1 are

bT ∗j−1s = βscs +

∫Rx(Y (s, x)− 1)λ

T ∗j−1s (dx)

cT ∗j−1s = cs

λT ∗j−1s (dx) = Y (s, x)λ

T ∗j−1s (dx)

(4.44)

and the IPT ∗j−1-canonical decomposition of the semimartingale LT

∗j−1 is

LT∗j−1 =

·∫0

bT ∗j−1s ds+

·∫0

c1/2s dWT ∗j−1s +

·∫0

∫R

x(µL − νT∗j−1)(ds,dx). (4.45)

Let LM,T ∗j−1 denote the martingale part of LT∗j−1 , i.e. LM,T ∗j−1 is a semi-

martingale with predictable characteristics T(LM,T ∗j−1 |IPT ∗j−1) = (0, C, νT

∗j−1).

Now, the dynamics of L(·, T ∗j )−1 under IPT ∗j−1is

L(t, T ∗j )−1 = L(0, T ∗j )−1 exp

(−

t∫0

bL(s, T ∗j , T∗j−1)ds−

t∫0

λ(s, T ∗j )dLT ∗j−1s

)

= L(0, T ∗j )−1 exp

( t∫0

bL(s, T ∗j , T∗j−1)ds+

t∫0

λ(s, T ∗j )dLT ∗j−1s

)

=: L(t, T ∗j ), (4.46)

where LT∗j−1 := −LM,T ∗j−1 is the dual process of LM,T ∗j−1 and its triplet of

semimartingale characteristics T(LT∗j−1 |IPT ∗j−1

), using Proposition 1.3 with

f ≡ 1, is (0, C, νT∗j−1), where

1A(x) ∗ νT∗j−1 = 1A(−x) ∗ νT

∗j−1 . (4.47)

Page 118: Applications of semimartingales and L´evy processes in ...webdoc.sub.gwdg.de/ebook/dissts/Freiburg/Papapantoleon2006.pdf · gales as driving processes. Another one, was to consider

110 4. LEVY TERM STRUCTURE MODELS

Furthermore, we define the drift term

bL(s, T ∗j , T∗j−1) := −bL(s, T ∗j , T

∗j−1)− λ(s, T ∗j )b

T ∗j−1s . (4.48)

The following simple calculation shows that the drift term bL(s, T ∗j , T∗j−1)

corresponding to L(t, T ∗j ), is of the same form as in (4.18). Keeping in mind

the form of νT∗j−1 or, equivalently λT

∗j−1 , we have

bL(s, T ∗j , T∗j−1)

(4.18)=

(4.44)−1

2(λ(s, T ∗j )

)2cs

+∫R

(eλ(s,T ∗j )x − 1− xλ(s, T ∗j )eλ(s,T ∗j )x

)λT ∗j−1s (dx)

(4.44)= −1

2(λ(s, T ∗j )

)2cs

−∫R

(e−λ(s,T ∗j )x − 1 + xλ(s, T ∗j )

)λT ∗j−1s (dx)

(4.47)= −1

2(λ(s, T ∗j )

)2cs

−∫R

(eλ(s,T ∗j )x − 1− xλ(s, T ∗j )

)λT ∗j−1s (dx). (4.49)

This concludes the proof, since

C = B(0, T ∗j−1)KL(0, T ∗j ) IEeIPT∗j−1

[δ∗j(K−1 − L(T ∗j , T

∗j )−1

)+]= B(0, T ∗j−1) IEeIPT∗

j−1

[δ∗j(L(0, T ∗j )− L(T ∗j , T

∗j ))+]

,

where L(T ∗j , T∗j ) := KL(0, T ∗j )L(T ∗j , T

∗j ) and noting that the dynamics of

L(·, T ∗j ) is given by (4.46) and (4.49).

4.4.3. Duality in the Levy forward price model. In this section,we state a duality relationship between call and put options on the forwardprice. 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 floorletsin the forward price model.

We denote the time-T0 value of a call option on the forward price withstrike K, which is settled in arrears at time T ∗j−1, by

C(F 0T ∗j,K;C, νT

∗j−1

)= B(0, T ∗j−1) IEIPT∗

j−1

[(F (T ∗j , T

∗j , T

∗j−1)−K)+

]where F 0

T ∗j:= F (0, T ∗j , T

∗j−1). Note that the drift characteristic of the driving

process is determined by the other two characteristics (C, νT∗j−1) and the

volatility structure λ(·, T ∗j ), using equation (4.22). Similar notation will beused for a put option on the forward price.

Theorem 4.14. Assume that the forward process is modeled accordingto the Levy forward price model. Then, we can relate the values of call and

Page 119: Applications of semimartingales and L´evy processes in ...webdoc.sub.gwdg.de/ebook/dissts/Freiburg/Papapantoleon2006.pdf · gales as driving processes. Another one, was to consider

4.5. VALUATION OF COMPOSITIONS 111

put options on the forward price via the following duality:

C(F 0T ∗j,K;C, νT

∗j−1

)= P

(K, F 0

T ∗j;C, νT

∗j−1

)where f(s, x) = exp(λ(s, T ∗j )x) and 1A(x) ∗ νT

∗j−1 = 1A(−x)f ∗ νT

∗j−1, A ∈

B(R\0).

Proof. The proof is analogous to the proof of Theorem 4.13 and there-fore omitted for the sake of brevity.

4.5. Valuation of compositions

We begin by describing the structure of the composition and the payoffof an option on the composition. Consider a discrete tenor structure 0 =T0 < T1 < · · · < TN < TN+1 = T ∗, where the accrual factor for the timeperiod [Ti, Ti+1] is δi = Ti+1 − Ti, i ∈ 0, 1, . . . , N. The composition paysa floating rate, typically the LIBOR, compounded on several consecutivedates. The rates are fixed at the dates si ≤ Ti and the composition is

N∏i=1

(1 + δiL(si, Ti)

);

therefore, the composition equals an investment of one currency unit at theLIBOR rate for N consecutive periods. The value of the composition issubjected to a cap (or floor) denoted by K and is settled in arrears, at timeT ∗. Hence, a cap on the composition pays off at maturity(

N∏i=1

(1 + δiL(si, Ti)

)−K

)+

,

while the payoff of a floor on the composition is(K −

N∏i=1

(1 + δiL(si, Ti)

))+

.

Notice that without the cap (resp. floor), the payoff of the composition wouldsimply be that of a floating rate note. Similarly, if we only consider a singlecompounding date, then we are dealing with a caplet (resp. floorlet), withstrike K := K−1

δ .In the following sections, we present methods for the valuation of a

cap on the composition in the Levy-driven forward rate and forward priceframeworks. The value of a floor on the composition can either be deducedvia analogous valuation methods (cf. also Chapter 3) or via the cap-floorparity for compositions, which reads

C(T ∗;K) = F(T ∗;K) +B(0, T1)−KB(0, T ∗).

Here C(T ∗;K) and F(T ∗;K) denote the time-T0 value of a cap, resp. floor,on the composition with cap, resp. floor, equal to K.

Page 120: Applications of semimartingales and L´evy processes in ...webdoc.sub.gwdg.de/ebook/dissts/Freiburg/Papapantoleon2006.pdf · gales as driving processes. Another one, was to consider

112 4. LEVY TERM STRUCTURE MODELS

4.5.1. Forward rate framework. In this section we derive an explicitformula for the valuation of a cap on the composition in the Levy forwardrate model, making use of the methods developed in Chapter 3. As a specialcase, we get valuation formulae for caplets in the Levy forward rate frame-work that generalize the results of Eberlein and Kluge (2006a), since we donot require the existence of a Lebesgue density (cf. Remark 3.4).

Firstly, we calculate the quantity that appears in the composition. Byan elementary calculation, we have that

B(si, Ti)B(si, Ti+1)

=B(0, Ti)B(0, Ti+1)

exp

si∫0

A(s, Ti, Ti+1)ds−si∫

0

Σ(s, Ti, Ti+1)dLs

.

Using the fact that 1 + δiL(si, Ti) = B(si,Ti)B(si,Ti+1) we immediately get

N∏i=1

(1+δiL(si, Ti)

)=

N∏i=1

B(si, Ti)B(si, Ti+1)

=B(0, T1)B(0, T ∗)

× exp

N∑i=1

si∫0

A(s, Ti, Ti+1)ds−N∑i=1

si∫0

Σ(s, Ti, Ti+1)dLs

.Next, we define the forward measure associated with the date T ∗ via theRadon–Nikodym derivative

dIPT ∗dIP

:=1

BMT ∗B(0, T ∗)

= exp

− T ∗∫0

A(s, T ∗)ds+

T ∗∫0

Σ(s, T ∗)dLs

.

The measures IP and IPT ∗ are equivalent, since the density is strictly positive;moreover, we immediately note that IE

[1

BMT∗B(0,T ∗)

]= 1. The density process

related to the change of measure is given by the restriction of the Radon–Nikodym derivative to the σ-field Ft, t ≤ T ∗, therefore

IE[dIPT ∗dIP

∣∣∣Ft] =B(t, T ∗)

BMt B(0, T ∗)

= exp

− t∫0

A(s, T ∗)ds+

t∫0

Σ(s, T ∗)dLs

.

This allows us to determine the tuple of functions that characterize theprocess L under this change of measure and we can conclude, using The-orems III.3.24 and II.4.15 in Jacod and Shiryaev (2003), that the drivingprocess L = (Lt)t∈[0,T ∗] remains a time-inhomogeneous Levy process underthe measure IPT ∗ .

According to the fundamental theorem of asset pricing the price of anoption on the composition is equal to its discounted expected payoff under

Page 121: Applications of semimartingales and L´evy processes in ...webdoc.sub.gwdg.de/ebook/dissts/Freiburg/Papapantoleon2006.pdf · gales as driving processes. Another one, was to consider

4.5. VALUATION OF COMPOSITIONS 113

the martingale measure. Combined with the forward measure defined above,this gives

C(T ∗;K) = IEIP

1BMT ∗

(N∏i=1

B(si, Ti)B(si, Ti+1)

−K

)+

= B(0, T ∗)IEIPT∗

( N∏i=1

B(si, Ti)B(si, Ti+1)

−K

)+

= B(0, T ∗)IEIPT∗

[(expH −K)+

],

where the random variable H is defined as

H := logB(0, T1)B(0, T ∗)

+N∑i=1

si∫0

A(s, Ti, Ti+1)ds−N∑i=1

si∫0

Σ(s, Ti, Ti+1)dLs.

Let us denote by MT ∗H the moment generating function of H under the

measure IPT ∗ . The next theorem provides an analytical expression for thevalue of a cap on the composition. Before that, we provide an expression forMT ∗H (z) for suitable complex arguments z.

Lemma 4.15. Let M and ε be suitably chosen such that Σ(s, T ) ≤M ′ forall s, T ∈ [0, T ∗] and Σ(s, Ti+1)1[si,si+1](s) ≤ M ′′ for all s, si, Ti+1 ∈ [0, T ∗],where 0 < M ′′ < M ′ < M and M

M ′′ > N + 1. Then, for each R ∈ I2 =[1 − M−M ′′(N+1)

M ′+M ′′(N+1) , 1 + M−M ′′(N+1)M ′+M ′′(N+1) ], we have that MT ∗

H (R) < ∞ and forevery z ∈ C with <z = R

MT ∗H (z) = Zz exp

T ∗∫0

(z

N∑i=1

A(s, Ti, Ti+1)1[0,si](s)− θs

(Σ(s, T ∗)

)

+θs(Σ(s, T ∗)− z

N∑i=1

Σ(s, Ti, Ti+1)1[0,si](s)))

ds

where Z := B(0,T1)

B(0,T ∗) .

Proof. Fix an R ∈ I2. Then, for z ∈ C with <z = R, and denoting byΣ(s, Ti+1) =

∑Ni=0 Σ(s, Ti+1)1[si,si+1](s), we get that∣∣∣<(− z

N∑i=1

Σ(s, Ti, Ti+1)1[0,si](s))

+ Σ(s, T ∗)∣∣∣

=∣∣∣<(z N∑

i=0

Σ(s, Ti+1)1[si,si+1](s)− zΣ(s, T ∗))

+ Σ(s, T ∗)∣∣∣

=∣∣∣<((1− z)

(Σ(s, T ∗)− Σ(s, Ti+1)

))+ Σ(s, Ti+1)

∣∣∣≤ |1−R||Σ(s, T ∗)− Σ(s, Ti+1)|+ |Σ(s, Ti+1)|

≤ M −M ′′(N + 1)M ′ +M ′′(N + 1)

(M ′ +M ′′(N + 1)) +M ′′(N + 1) = M. (4.50)

Page 122: Applications of semimartingales and L´evy processes in ...webdoc.sub.gwdg.de/ebook/dissts/Freiburg/Papapantoleon2006.pdf · gales as driving processes. Another one, was to consider

114 4. LEVY TERM STRUCTURE MODELS

Now, define the constants

Z0 := exp

z( logB(0, T1)B(0, T ∗)

+

T ∗∫0

N∑i=1

A(s, Ti, Ti+1)1[0,si](s)ds)

and Z1 := Z0 × exp(−∫ T ∗0 A(s, T ∗)ds

). Hence, the moment generating

function of H is

MT ∗H (z) = IEIPT∗

[exp(zH)

]= IEIPT∗

exp

z( logB(0, T1)B(0, T ∗)

+N∑i=1

si∫0

A(s, Ti, Ti+1)ds

−N∑i=1

si∫0

Σ(s, Ti, Ti+1)dLs

)= exp

− T ∗∫0

A(s, T ∗)ds

×Z0

× IEIP

exp

−z N∑i=1

si∫0

Σ(s, Ti, Ti+1)dLs +

T ∗∫0

Σ(s, T ∗)dLs

= Z1IEIP

exp

T ∗∫0

(−z

N∑i=1

Σ(s, Ti, Ti+1)1[0,si](s) + Σ(s, T ∗)

)dLs

= Z1 exp

T ∗∫0

(θs

(− z

N∑i=1

Σ(s, Ti, Ti+1)1[0,si](s) + Σ(s, T ∗)))

ds,

where for the last equality we have applied Lemma 4.1, which is justified by(4.50). In addition, we get that MT ∗

H (R) < ∞ for R ∈ I2 and therefore themoment generating function of H can be extended to the complex plane forz ∈ C with <z ∈ I2.

Theorem 4.16. Assume that bond prices are modeled according to theLevy forward rate model. The price of a cap on the composition is

C(T ∗;K) =B(0, T ∗)

∫R

MT ∗H (R− iu)

K1+iu−R

(iu−R)(1 + iu−R)du,

where MT ∗H is given by Lemma 4.15 and R ∈ I1 ∩I2 = (1, 1 + M−M ′′(N+1)

M ′+M ′′(N+1) ].

Proof. Since the prerequisites of Theorem 3.2 are satisfied for R ∈I1 ∩ I2, we immediately have that

C(T ∗;K) = B(0, T ∗)IEIPT∗

[(eH −K

)+]=B(0, T ∗)

∫R

MT ∗H (R− iu)

K1+iu−R

(iu−R)(1 + iu−R)du,

Page 123: Applications of semimartingales and L´evy processes in ...webdoc.sub.gwdg.de/ebook/dissts/Freiburg/Papapantoleon2006.pdf · gales as driving processes. Another one, was to consider

4.5. VALUATION OF COMPOSITIONS 115

where we have also used Example 3.15, hence I1 = (1,∞).

4.5.2. Forward price framework. The aim of this section is to derivean explicit formula for the valuation of a cap on the composition in the Levyforward price model. Once again, the valuation formulae will be based onthe methods developed in Chapter 3.

We begin by noticing that the quantity that appears in the compositioncan be expressed in terms of forward prices, since

1 + δiL(·, Ti) = F (·, Ti, Ti+1),

and the forward prices are the modeling object in this framework. We knowthat each forward price process evolves as a martingale under its correspond-ing forward measure; moreover, we know that all forward price processes aredriven by the same time-inhomogeneous Levy process (see also Remark 4.6).Therefore, we will carry out the following program to arrive at the valuationformulae:

(1) lift all forward price processes from their forward measure to theterminal 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 con-nection between the Brownian motions and the compensators under thedifferent measures, cf. equations (4.23) and (4.24), we get that

F (t, T ∗j , T∗j−1) = F (0, T ∗j , T

∗j−1) exp

t∫0

b(s, T ∗j , T∗j−1)ds+

t∫0

λ(s, T ∗j )dLT ∗j−1s

= F (0, T ∗j , T

∗j−1) exp

t∫0

b(s, T ∗j , T∗)ds+

t∫0

λ(s, T ∗j )dLT∗

s

.

(4.51)

Here LT∗is the driving time-inhomogeneous Levy process with IPT ∗-canonical

decomposition

LT∗

t =

t∫0

√csdW T ∗

s +

t∫0

∫R

x(µL − νT∗)(ds,dx), (4.52)

and the drift term of the forward process F (·, T ∗j , T ∗j−1) under the terminalmeasure IPT ∗ , is

b(s, T ∗j , T∗) = −cs

(12(λ(s, T ∗j ))2 + λ(s, T ∗j )

j−1∑k=1

λ(s, T ∗k )

)

−∫R

((exλ(s,T ∗j ) − 1

)ex

Pj−1k=1 λ(s,T ∗k ) − xλ(s, T ∗j )

)λT

∗s (dx).

(4.53)

Page 124: Applications of semimartingales and L´evy processes in ...webdoc.sub.gwdg.de/ebook/dissts/Freiburg/Papapantoleon2006.pdf · gales as driving processes. Another one, was to consider

116 4. LEVY TERM STRUCTURE MODELS

It is immediately obvious from (4.51), (4.52) and (4.53) that F (·, T ∗j , T ∗j−1)is not a IPT ∗-martingale, unless j = 1 (where we use the convention that∑0

j=1 = 0).Now, the composition takes the following form

N∏i=1

(1 + δiL(si, Ti)

)=

N∏j=1

F (s∗j , T∗j , T

∗j−1)

=B(0, T ∗N )B(0, T ∗)

(4.54)

× exp

N∑j=1

s∗j∫0

b(s, T ∗j , T∗)ds+

N∑j=1

s∗j∫0

λ(s, T ∗j )dLT∗

s

.

where s∗j = sN+1−j , j ∈ 1, · · · , N. Define the random variable

H := logB(0, T ∗N )B(0, T ∗)

+N∑j=1

s∗j∫0

b(s, T ∗j , T∗)ds+

N∑j=1

s∗j∫0

λ(s, T ∗j )dLT∗

s (4.55)

and now we can express the option on the composition as an option depend-ing on this random variable. The next theorem provides a formula for thevaluation of the composition.

Theorem 4.17. Let the forward prices be modeled according to the Levyforward process framework. Then, the price of a cap on the composition is

C(T ∗;K) =B(0, T ∗)

∫R

MH(R− iu)K1+iu−R

(iu−R)(1 + iu−R)du (4.56)

where the moment generating function of H is given by Lemma 4.18 andR ∈ (1, MM ′ ].

Proof. The option on the composition is priced under the terminalforward martingale measure IPT ∗ . Using (4.54) and (4.55), we can expressthe cap on the composition as a call option depending on the random variableH. Then we get

C(T ∗;K) = B(0, T ∗)IEIPT∗

N∏j=1

F (s∗j , T∗j , T

∗j−1)−K

+= B(0, T ∗)IEIPT∗

[(eH −K

)+]=B(0, T ∗)

∫R

MH(R− iu)K1+iu−R

(iu−R)(1 + iu−R)du

where we have applied Theorem 3.2 and used Example 3.15.

Lemma 4.18. Let M and ε be suitably chosen such that |∑N

k=1 λ(s, Tk)| ≤M ′ for some M ′ < M and for all s ∈ [0, T ∗]. Then, for each R ∈ [0, MM ′ ] we

Page 125: Applications of semimartingales and L´evy processes in ...webdoc.sub.gwdg.de/ebook/dissts/Freiburg/Papapantoleon2006.pdf · gales as driving processes. Another one, was to consider

4.5. VALUATION OF COMPOSITIONS 117

have that MH(R) < ∞, and for every z ∈ C with <z ∈ [0, MM ′ ] the momentgenerating function of H is

MH(z) = Zz exp

T ∗∫0

(z

N∑j=1

b(s, T ∗j , T∗) + θT

∗s

(z

N∑j=1

λ(s, T ∗j )))

ds

,

where Z = B(0,T ∗N )

B(0,T ∗) and θT∗

s is the cumulant associated with the triplet(0, cs, λT

∗s ).

Proof. Fix an R ∈ [0, MM ′ ] and then for z ∈ C with <z = R we get∣∣∣∣∣<(z

N∑k=1

λ(s, Tk)

)∣∣∣∣∣ = R

∣∣∣∣∣N∑k=1

λ(s, Tk)

∣∣∣∣∣ ≤ M

M ′M′ = M. (4.57)

Now, define the constant

Z2 :=(B(0, T ∗N )B(0, T ∗1 )

)zexp

z N∑j=1

s∗j∫0

b(s, T ∗j , T∗)ds

.

Then

MH(z) = IEIPT∗

[exp(zH)

]= Z2IEIPT∗

exp

z N∑j=1

s∗j∫0

λ(s, T ∗j )dLT∗

s

= Z2IEIPT∗

exp

T ∗∫0

z

N∑j=1

λ(s, T ∗j )1[0,s∗j ](s)dLT ∗s

= Z2 exp

T ∗∫0

θT∗

s

(z

N∑j=1

λ(s, T ∗j ))ds

where for the last equality we have applied Lemma 4.1, which is justified by(4.57). Note also that λ(s, T ∗j ) = 0 for s > s∗j , which is the fixing date forthe rate; accordingly, b(s, T ∗j , T

∗) = 0 for s > s∗j , cf. (4.53).In addition, we get that MH(R) < ∞ for R ∈ [0, MM ′ ] and therefore the

moment generating function of H can be extended to the complex plane forz ∈ C with <z ∈ [0, MM ′ ].

Page 126: Applications of semimartingales and L´evy processes in ...webdoc.sub.gwdg.de/ebook/dissts/Freiburg/Papapantoleon2006.pdf · gales as driving processes. Another one, was to consider
Page 127: Applications of semimartingales and L´evy processes in ...webdoc.sub.gwdg.de/ebook/dissts/Freiburg/Papapantoleon2006.pdf · gales as driving processes. Another one, was to consider

APPENDIX A

Transformations

We use the well-known relationships between the LIBOR, the forwardprice and the bond price, to transform a caplet into a call option on theforward price or a put option on a bond. Similarly, a floorlet is transformedinto a put option on the forward price or a call option on a bond.

Let T0 < T1 < T2 < · · · < TN < TN+1 = T ∗ denote a discrete tenorstructure where δi = Ti+1 − Ti, i ∈ 0, 1, . . . , N. The time-Ti+1 payoff of acaplet settled in arrears at time Ti+1, is

Nδi(L(Ti, Ti)−K)+

where K is the strike rate and N is the notional amount.Now, using the relationship between the LIBOR and the forward price,

i.e. F (Ti, Ti, Ti+1) = 1 + δiL(Ti, Ti), we can rewrite the payoff of a caplet asa call option on the forward price. We have

Nδi(L(Ti, Ti)−K)+ = Nδi

(F (Ti, Ti, Ti+1)− 1

δi−K

)+

= N(F (Ti, Ti, Ti+1)−K)+, (A.1)

where K = 1 + δiK.Moreover, the payoff N(F (Ti, Ti, Ti+1)−K)+ settled at time Ti+1 is equal

to the payoff NB(Ti, Ti+1)(F (Ti, Ti, Ti+1) − K)+, settled at time Ti. Us-ing the relationship between forward and bond prices, i.e. F (Ti, Ti, Ti+1) =B(Ti, Ti)/B(Ti, Ti+1), we have

NB(Ti, Ti+1)(F (Ti, Ti, Ti+1)−K)+ = NB(Ti, Ti+1)(

B(Ti, Ti)B(Ti, Ti+1)

−K)+

= N(1−KB(Ti, Ti+1))+

= N(K−B(Ti, Ti+1))+, (A.2)

where K = K−1 and N = NK.

119

Page 128: Applications of semimartingales and L´evy processes in ...webdoc.sub.gwdg.de/ebook/dissts/Freiburg/Papapantoleon2006.pdf · gales as driving processes. Another one, was to consider
Page 129: Applications of semimartingales and L´evy processes in ...webdoc.sub.gwdg.de/ebook/dissts/Freiburg/Papapantoleon2006.pdf · gales as driving processes. Another one, was to consider

APPENDIX B

An application of Ito’s formula

We apply Ito’s formula for semimartingales, cf. Theorem I.4.57 in Jacodand Shiryaev (2003) or Lemma A.5 in Goll and Kallsen (2000), to calculatethe canonical decomposition of an exponential semimartingale.

1. Consider a stochastic basis (Ω,F , (Ft)0≤t≤T , P ) and a semimartin-gale H = (Ht)0≤t≤T with triplet of predictable characteristics T(H|P ) =(B,C, ν). Assume that H is an exponentially special semimartingale andeH ∈ Mloc(P ). Therefore, applying Theorem 2.19 in Kallsen and Shiryaev(2002a), we have that

eH = E(Hc +

ex − 1

1 + W∗ (µH − ν)

), (B.1)

where Wt :=∫

(ex − 1)ν(t × dx).Applying Ito’s formula to the function f(x) = ex, since f ∈ C2, and

using the canonical decomposition of H, cf. (1.2), we get

eH = 1 +

·∫0

eHs−dHs +12

·∫0

eHs−d〈Hc〉s

+

·∫0

∫R

(eHs−+x − eHs− − eHs−x

)µH(ds,dx)

=

·∫0

eHs−dBs +12

·∫0

eHs−d〈Hc〉s

+

·∫0

∫R

(eHs−+x − eHs− − eHs−h(x)

)µH(ds,dx)

+

·∫0

eHs−dHcs +

·∫0

∫R

eHs−h(x)(µH − ν)(ds,dx). (B.2)

Therefore, we can conclude that

(eH)c

=

·∫0

eHs−dHcs . (B.3)

121

Page 130: Applications of semimartingales and L´evy processes in ...webdoc.sub.gwdg.de/ebook/dissts/Freiburg/Papapantoleon2006.pdf · gales as driving processes. Another one, was to consider

122 B. AN APPLICATION OF ITO’S FORMULA

2. Consider the setting of Chapter 2, section 2.5. We have that Z =E(Xα), where

Xα =

·∫0

ασsdWs +

·∫0

∫R

(eαx − 1)(µL − ν)(ds,dx), (B.4)

where L = (Lt)0≤t≤T is a time-inhomogeneous Levy process. Therefore, weimmediately get that

Zc =

·∫0

Zs−dXα,cs =

·∫0

Zs−ασsdWs. (B.5)

3. Finally, consider the setting of Chapter 4, section 4.4, where we havethat Z = E(X), where

X =

·∫0

Σ(s, T, U)√csdWs +

·∫0

∫R

(exΣ(s,T,U) − 1)(µL − ν)(ds,dx), (B.6)

where L = (Lt)0≤t≤T is again a time-inhomogeneous Levy process. Similarly,we can immediately conclude that

Zc =

·∫0

Zs−dXcs =

·∫0

Zs−Σ(s, T, U)√csdWs. (B.7)

Page 131: Applications of semimartingales and L´evy processes in ...webdoc.sub.gwdg.de/ebook/dissts/Freiburg/Papapantoleon2006.pdf · gales as driving processes. Another one, was to consider

Bibliography

Abramowitz, M. and I. Stegun (Eds.) (1968). Handbook of MathematicalFunctions (5th ed.). Dover.

Andreasen, J. (1998). The pricing of discretely sampled Asian andlookback options: a change of numeraire approach. J. Comput. Fi-nance 2 (1), 5–30.

Applebaum, D. (2004). Levy Processes and Stochastic Calculus. Cam-bridge University Press.

Barndorff-Nielsen, O. E. (1998). Processes of normal inverse Gaussiantype. Finance Stoch. 2, 41–68.

Barndorff-Nielsen, O. E. and N. Shephard (2001). Non-GaussianOrnstein–Uhlenbeck-based models and some of their uses in financialeconomics. J. Roy. Statist. Soc. Ser. B 63, 167–241.

Bates, D. S. (1997). The skewness premium: option pricing under asym-metric processes. In P. Ritchken, P. P. Boyle, and G. Pennacchi (Eds.),Advances in Futures and Options Research, Volume 9, pp. 51–82. El-sevier.

Bellini, F. and M. Frittelli (2002). On the existence of minimax martingalemeasures. Math. Finance 12, 1–21.

Belomestny, D. and M. Reiß (2006). Spectral calibration of exponentialLevy models. Finance Stoch. 10, 449–474.

Belomestny, D. and J. Schoenmakers (2006). A jump-diffusion LIBORmodel and its robust calibration. WIAS Preprint No. 1113.

Benhamou, E. (2002). Fast Fourier transform for discrete Asian options.J. Comput. Finance 6 (1), 49–68.

Bertoin, J. (1996). Levy processes. Cambridge University Press.Bjork, T. (2004). Arbitrage Theory in Continuous Time (2nd ed.). Oxford

University Press.Bjork, T., G. Di Masi, Y. Kabanov, and W. Runggaldier (1997). Towards

a general theory of bond markets. Finance Stoch. 1, 141–174.Black, F. (1976). The pricing of commodity contracts. J. Financ. Econ. 3,

167–179.Black, F. and M. Scholes (1973). The pricing of options and corporate

liabilities. J. Polit. Econ. 81, 637–654.Borovkov, K. and A. Novikov (2002). On a new approach to calculating

expectations for option pricing. J. Appl. Probab. 39, 889–895.Boyarchenko, S. I. and S. Z. Levendorskiı (2002). Barrier options and

touch-and-out options under regular Levy processes of exponentialtype. Ann. Appl. Probab. 12, 1261–1298.

Brace, A., D. Gatarek, and M. Musiela (1997). The market model ofinterest rate dynamics. Math. Finance 7, 127–155.

123

Page 132: Applications of semimartingales and L´evy processes in ...webdoc.sub.gwdg.de/ebook/dissts/Freiburg/Papapantoleon2006.pdf · gales as driving processes. Another one, was to consider

124 Bibliography

Brigo, D. and F. Mercurio (2006). Interest Rate Models – Theory andPractice (2nd ed.). Springer.

Carr, P. (1994). European put call symmetry. Preprint, Cornell University.Carr, P. and M. Chesney (1996). American put call symmetry. Preprint,

H.E.C.Carr, P., K. Ellis, and V. Gupta (1998). Static hedging of exotic options.

J. Finance 53, 1165–1190.Carr, P., H. Geman, D. B. Madan, and M. Yor (2002). The fine structure

of asset returns: an empirical investigation. J. Business 75, 305–332.Carr, P., H. Geman, D. B. Madan, and M. Yor (2003). Stochastic volatility

for Levy processes. Math. Finance 13, 345–382.Carr, P. and D. B. Madan (1999). Option valuation using the fast Fourier

transform. J. Comput. Finance 2 (4), 61–73.Cerny, A. (2007). Optimal continuous-time hedging with leptokurtic re-

turns. Math. Finance 17, 175–203.Chesney, M. and R. Gibson (1995). State space symmetry and two factor

option pricing models. In P. P. Boyle, F. A. Longstaff, and P. Ritchken(Eds.), Advances in Futures and Options Research, Volume 8, pp. 85–112.

Cont, R. and P. Tankov (2003). Financial Modelling with Jump Processes.Chapman and Hall/CRC Press.

Cont, R. and P. Tankov (2004). Nonparametric calibration of jump-diffusion option pricing models. J. Comput. Finance 7 (3), 1–49.

Cont, R. and P. Tankov (2006). Retrieving Levy processes from optionprices: regularization of an ill-posed inverse problem. SIAM J. ControlOptim. 45, 1–25.

Corcuera, J. M., D. Nualart, and W. Schoutens (2005). Completion of aLevy market by power-jump assets. Finance Stoch. 9, 109–127.

Delbaen, F. and W. Schachermayer (1994). A general version of the fun-damental theorem of asset pricing. Math. Ann. 300, 463–520.

Delbaen, F. and W. Schachermayer (1998). The fundamental theoremof asset pricing for unbounded stochastic processes. Math. Ann. 312,215–250.

Detemple, J. (2001). American options: symmetry properties. In J. Cvi-tanic, E. Jouini, and M. Musiela (Eds.), Option Pricing, Interest Ratesand Risk Management, pp. 67–104. Cambridge University Press.

Doetsch, G. (1950). Handbuch der Laplace-Transformation. Birkhauser.Dupire, B. (1994). Pricing with a smile. Risk 7, 18–20.Eberlein, E. (2001). Application of generalized hyperbolic Levy mo-

tions to finance. In O. E. Barndorff-Nielsen, T. Mikosch, and S. I.Resnick (Eds.), Levy Processes: Theory and Applications, pp. 319–336.Birkhauser.

Eberlein, E. and J. Jacod (1997). On the range of options prices. FinanceStoch. 1, 131–140.

Eberlein, E., J. Jacod, and S. Raible (2005). Levy term structure models:no-arbitrage and completeness. Finance Stoch. 9, 67–88.

Eberlein, E. and U. Keller (1995). Hyperbolic distributions in finance.Bernoulli 1, 281–299.

Page 133: Applications of semimartingales and L´evy processes in ...webdoc.sub.gwdg.de/ebook/dissts/Freiburg/Papapantoleon2006.pdf · gales as driving processes. Another one, was to consider

Bibliography 125

Eberlein, E. and W. Kluge (2006a). Exact pricing formulae for caps andswaptions in a Levy term structure model. J. Comput. Finance 9 (2),99–125.

Eberlein, E. and W. Kluge (2006b). Valuation of floating range notes inLevy term structure models. Math. Finance 16, 237–254.

Eberlein, E. and W. Kluge (2007). Calibration of Levy term structuremodels. In R. Elliott, M. Fu, R. Jarrow, and J. Y. Yen (Eds.),Festschrift Dilip Madan. (forthcoming).

Eberlein, E., W. Kluge, and A. Papapantoleon (2006). Symmetries in Levyterm structure models. Int. J. Theor. Appl. Finance 9, 967–986.

Eberlein, E. and F. Ozkan (2005). The Levy LIBOR model. FinanceStoch. 9, 327–348.

Eberlein, E. and A. Papapantoleon (2005a). Equivalence of floating andfixed strike Asian and lookback options. Stochastic Process. Appl. 115,31–40.

Eberlein, E. and A. Papapantoleon (2005b). Symmetries and pricing ofexotic options in Levy models. In A. Kyprianou, W. Schoutens, andP. Wilmott (Eds.), Exotic Option Pricing and Advanced Levy Models,pp. 99–128. Wiley.

Eberlein, E., A. Papapantoleon, and A. N. Shiryaev (2006). On the dualityprinciple in option pricing: semimartingale setting. FDM Preprint 92.

Eberlein, E. and K. Prause (2002). The generalized hyperbolic model:financial derivatives and risk measures. In H. Geman, D. Madan,S. Pliska, and T. Vorst (Eds.), Mathematical Finance-Bachelier Con-gress 2000, pp. 245–267. Springer Verlag.

Eberlein, E. and S. Raible (1999). Term structure models driven by generalLevy processes. Math. Finance 9, 31–53.

Eberlein, E. and E. A. v. Hammerstein (2004). Generalized hyperbolicand inverse Gaussian distributions: limiting cases and approximationof processes. In R. Dalang, M. Dozzi, and F. Russo (Eds.), Seminaron Stochastic Analysis, Random Fields and Applications IV, Progressin Probability 58, pp. 221–264. Birkhauser.

Fajardo, J. and E. Mordecki (2006a). Skewness premium with Levy pro-cesses. Working paper, IBMEC.

Fajardo, J. and E. Mordecki (2006b). Symmetry and duality in Levy mar-kets. Quant. Finance 6, 219–227.

Feller, W. (1971). An Introduction to Probability Theory and Its Applica-tions (2nd ed.), Volume II. Wiley.

Frittelli, M. (2000). The minimal entropy martingale measure and thevaluation problem in incomplete markets. Math. Finance 10, 39–52.

Fujiwara, T. and Y. Miyahara (2003). The minimal entropy martingalemeasures for geometric Levy processes. Finance Stoch. 7, 509–531.

Geman, H., N. El Karoui, and J.-C. Rochet (1995). Changes ofnumeraire, changes of probability measures and option pricing. J.Appl. Probab. 32, 443–458.

Goll, T. and J. Kallsen (2000). Optimal portfolios for logarithmic utility.Stochastic Process. Appl. 89, 31–48.

Page 134: Applications of semimartingales and L´evy processes in ...webdoc.sub.gwdg.de/ebook/dissts/Freiburg/Papapantoleon2006.pdf · gales as driving processes. Another one, was to consider

126 Bibliography

Goll, T. and L. Ruschendorf (2001). Minimax and minimal distance mar-tingale measures and their relationship to portfolio optimization. Fi-nance Stoch. 5, 557–581.

Greenwood, P. and J. Pitman (1980a). Fluctuation identities for Levyprocesses and splitting at the maximum. Adv. Appl. Probab. 12, 893–902.

Greenwood, P. and J. Pitman (1980b). Fluctuation identities for randomwalk by path decomposition at the maximum. Adv. Appl. Probab. 12,291–293.

Gushchin, A. A. and E. Mordecki (2002). Bounds on option prices forsemimartingale market models. Proc. Steklov Inst. Math. 237, 73–113.

Gut, A. (1995). An Intermediate Course in Probability. Springer.Hakala, J. and U. Wystup (2002). Heston’s stochastic volatility model ap-

plied to foreign exchange options. In J. Hakala and U. Wystup (Eds.),Foreign Exchange Risk, pp. 267–282. Risk Publications.

Harrison, J. M. and S. R. Pliska (1981). Martingales and stochastic inte-grals in the theory of continous trading. Stochastic Process. Appl. 11,215–260.

Harrison, M. J. and D. Kreps (1979). Martingales and arbitrage in multi-period securities markets. J. Econ. Theory 20, 381–408.

Haug, E. G. (2002). A look in the antimatter mirror. Wilmott Magazine,September, 38–42.

Heath, D., R. Jarrow, and A. Morton (1992). Bond pricing and the termstructure of interest rates: a new methodology for contingent claimsvaluation. Econometrica 60, 77–105.

Henderson, V. and R. Wojakowski (2002). On the equivalence of floating-and fixed-strike Asian options. J. Appl. Probab. 39, 391–394.

Henrard, M. (2005). Swaptions: 1 price, 10 deltas, and ... 6 1/2 gammas.Working paper.

Hubalek, F., J. Kallsen, and L. Krawczyk (2006). Variance-optimal hedg-ing for processes with stationary independent increments. Ann. Appl.Probab. 16, 853–885.

Hunt, P. J. and J. E. Kennedy (2004). Financial Derivatives in Theoryand Practice (2nd ed.). Wiley.

Jacod, J. (1979). Calcul Stochastique et Problemes de Martingales. LectureNotes Math. 714. Springer.

Jacod, J. (1980). Integrales stochastiques par rapport a une semi-martingale vectorielle et changements de filtration. In Seminaire deProbabilites XIV, 1978/79, Lecture Notes Math. 784, pp. 161–172.Springer.

Jacod, J. and A. N. Shiryaev (2003). Limit Theorems for Stochastic Pro-cesses (2nd ed.). Springer.

Jamshidian, F. (1997). LIBOR and swap market models and measures.Finance Stoch. 1, 293–330.

Kallsen, J. (2000). Optimal portfolios for exponential Levy processes.Math. Meth. Oper. Res. 51, 357–374.

Kallsen, J. (2006). A didactic note on affine stochastic volatility models.In Y. Kabanov, R. Lipster, and J. Stoyanov (Eds.), From Stochastic

Page 135: Applications of semimartingales and L´evy processes in ...webdoc.sub.gwdg.de/ebook/dissts/Freiburg/Papapantoleon2006.pdf · gales as driving processes. Another one, was to consider

Bibliography 127

Calculus to Mathematical Finance: The Shiryaev Festschrift, pp. 343–368. Springer.

Kallsen, J. and A. N. Shiryaev (2002a). The cumulant process and Ess-cher’s change of measure. Finance Stoch. 6, 397–428.

Kallsen, J. and A. N. Shiryaev (2002b). Time change representation ofstochastic integrals. Theory Probab. Appl. 46, 522–528.

Kallsen, J. and P. Tankov (2006). Characterization of dependence ofmultidimensional Levy processes using Levy copulas. J. MultivariateAnal. 97, 1551–1572.

Keller, U. (1997). Realistic modelling of financial derivatives. Ph. D. the-sis, University of Freiburg.

Keller-Ressel, M. (2006). Non-parametric calibration of the Barndorff-Nielsen–Shephard model. Working paper, TU Vienna.

Kluge, W. (2005). Time-inhomogeneous Levy processes in interest rateand credit risk models. Ph. D. thesis, University of Freiburg.

Kluge, W. and A. Papapantoleon (2006). Valuation of compositions inLevy term structure models. Working paper, University of Freiburg.

Kou, S. G. and H. Wang (2003). First passage times of a jump diffusionprocess. Adv. Appl. Probab. 35, 504–531.

Kou, S. G. and H. Wang (2004). Option pricing under a double exponentialjump diffusion model. Manag. Sci. 50, 1178–1192.

Kreps, D. (1981). Arbitrage and equilibrium in economies with infinitelymany commodities. J. Math. Econ. 8, 15–35.

Kuchler, U. and S. Tappe (2006). Bilateral Gamma distributions and pro-cesses in financial mathematics. Working paper, University of Munich.

Kyprianou, A. E. (2006). Introductory Lectures on Fluctuations of LevyProcesses with Applications. Springer.

Kyprianou, A. E., W. Schoutens, and P. Wilmott (Eds.) (2005). ExoticOption Pricing and Advanced Levy Models. Wiley.

Kyprianou, A. E. and B. A. Surya (2005). On the Novikov–Shiryaev op-timal stopping problem in continuous time. Elect. Comm. Probab. 10,146–154.

Lipton, A. (2002). Assets with jumps. Risk 15 (9), 149–153.Lukacs, E. (1970). Characteristic Functions (2nd ed.). Griffin.Madan, D. B. and E. Seneta (1990). The variance gamma (VG) model for

share market returns. J. Business 63, 511–524.Margrabe, W. (1978). The value of an option to exchange one asset for

another. J. Finance 33, 177–186.Merton, R. C. (1973). Theory of rational option pricing. Bell J. Econ.

Manag. Sci. 4, 141–183.Miltersen, K. R., K. Sandmann, and D. Sondermann (1997). Closed form

solutions for term structure derivatives with log-normal interest rates.J. Finance 52, 409–430.

Muller, A. and D. Stoyan (2002). Comparison Methods for Stochastic Mod-els and Risks. Wiley.

Musiela, M. and M. Rutkowski (1997). Continuous-time term structuremodels: forward measure approach. Finance Stoch. 1, 261–291.

Page 136: Applications of semimartingales and L´evy processes in ...webdoc.sub.gwdg.de/ebook/dissts/Freiburg/Papapantoleon2006.pdf · gales as driving processes. Another one, was to consider

128 Bibliography

Musiela, M. and M. Rutkowski (2005). Martingale Methods in FinancialModelling (2nd ed.). Springer.

Nguyen-Ngoc, L. (2003). Exotic options in general exponential Levy mod-els. Prepublication no 850, Universite Paris VI.

Nguyen-Ngoc, L. and M. Yor (2005). Lookback and barrier options undergeneral Levy processes. In Y. Ait-Sahalia and L.-P. Hansen (Eds.),Handbook of Financial Econometrics. North-Holland. (forthcoming).

Nicolato, E. and E. Venardos (2003). Option pricing in stochastic volatilitymodels of the Ornstein–Uhlenbeck type. Math. Finance 13, 445–466.

Ozkan, F. and T. Schmidt (2005). Credit risk with infinite dimensionalLevy processes. Statist. Decisions 23, 281–299.

Peskir, G. and A. N. Shiryaev (2002). A note on the call-put parity anda call-put duality. Theory Probab. Appl. 46, 167–170.

Peskir, G. and A. N. Shiryaev (2006). Optimal Stopping and Free-Boundary Problems. Birkhauser.

Protter, P. (2004). Stochastic Integration and Differential Equations (3rded.). Springer.

Psychoyios, D., G. Skiadopoulos, and P. Alexakis (2003). A review ofstochastic volatility processes: properties and implications. J. Risk Fi-nance 4, 43–60.

Rachev, S. T. (Ed.) (2003). Handbook of Heavy Tailed Distributions inFinance. Elsevier.

Raible, S. (2000). Levy processes in finance: theory, numerics, and empir-ical facts. Ph. D. thesis, University of Freiburg.

Revuz, D. and M. Yor (1999). Continuous Martingales and BrownianMotion (3rd ed.). Springer.

Rudin, W. (1987). Real and Complex Analysis (3rd ed.). McGraw-Hill.Samorodnitsky, G. and M. Taqqu (1994). Stable non-Gaussian Random

Processes. Chapman and Hall.Samuelson, P. A. (1965). Rational theory of warrant pricing. Indust.

Manag. Rev. 6, 13–31.Sandmann, K., D. Sondermann, and K. R. Miltersen (1995). Closed form

term structure derivatives in a Heath–Jarrow–Morton model with log-normal annually compounded interest rates. In Proceedings of the Sev-enth Annual European Futures Research Symposium Bonn, pp. 145–165. Chicago Board of Trade.

Sato, K.-I. (1999). Levy Processes and Infinitely Divisible Distributions.Cambridge University Press.

Schilling, R. (2005). Measures, Integrals and Martingales. Cambridge Uni-versity Press.

Schlogl, E. (2002). A multicurrency extension of the lognormal interestrate market models. Finance Stoch. 6, 173–196.

Schoutens, W. (2002). The Meixner process: theory and applications infinance. In O. E. Barndorff-Nielsen (Ed.), Mini-proceedings of the 2ndMaPhySto Conference on Levy Processes, pp. 237–241.

Schoutens, W. (2003). Levy Processes in Finance: Pricing FinancialDerivatives. Wiley.

Page 137: Applications of semimartingales and L´evy processes in ...webdoc.sub.gwdg.de/ebook/dissts/Freiburg/Papapantoleon2006.pdf · gales as driving processes. Another one, was to consider

Bibliography 129

Schoutens, W. and J. L. Teugels (1998). Levy processes, polynomials andmartingales. Comm. Statist. Stochastic Models 14, 335–349.

Schroder, M. (1999). Changes of numeraire for pricing futures, forwardsand options. Rev. Financ. Stud. 12, 1143–1163.

Shepp, L. A. and A. N. Shiryaev (1994). A new look at pricing of the“Russian option”. Theory Probab. Appl. 39, 103–119.

Shiryaev, A. N. (1999). Essentials of Stochastic Finance: Facts, Models,Theory. World Scientific.

Shiryaev, A. N., Y. M. Kabanov, D. O. Kramkov, and A. Mel’nikov (1994).Toward the theory of pricing of options of both European and Amer-ican types. II. Continuous time. Theory Probab. Appl. 39, 61–102.

Skiadopoulos, G. (2001). Volatility smile consistent option models: a sur-vey. Int. J. Theor. Appl. Finance 4, 403–437.

Spitzer, F. (1964). Principles of Random Walk. Van Nostrand.Tankov, P. (2003). Dependence structure of spectrally positive multidi-

mensional Levy processes. Unpublished manuscript.Vanmaele, M., G. Deelstra, J. Liinev, J. Dhaene, and M. J. Goovaerts

(2006). Bounds for the price of discrete arithmetic Asian options. J.Comput. Appl. Math. 185, 51–90.

Vecer, J. (2002). Unified Asian pricing. Risk 15 (6), 113–116.Vecer, J. and M. Xu (2004). Pricing Asian options in a semimartingale

model. Quant. Finance 4 (2), 170–175.