Lévy Processes in Finance: Theory, Numerics, and Empirical Facts
Dissertation zur Erlangung des Doktorgrades
der Mathematischen Fakultät
der Albert-Ludwigs-Universität Freiburg i. Br.

vorgelegt von
Sebastian Raible
Januar 2000

Preface

Lévy processes are an excellent tool for modelling price processes in mathematical finance. On the
one hand, they are very flexible, since for any time increment
t any infinitely divisible distribution
can be chosen as the increment distribution over periods of time
t. On the other hand, they have a
simple structure in comparison with general semimartingales. Thus stochastic models based on Lévy
processes often allow for analytically or numerically tractable formulas. This is a key factor for practical
applications.

This thesis is divided into two parts. The first, consisting of Chapters 1, 2, and 3, is devoted to the study
of stock price models involving exponential Lévy processes. In the second part, we study term structure
models driven by Lévy processes. This part is a continuation of the research that started with the author's
diploma thesis Raible (1996) and the article Eberlein and Raible (1999).

Contents

Preface iii
1 Exponential Lévy Processes in Stock Price Modeling 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Exponential Lévy Processes as Stock Price Models . . . . . . . . . . . . . . . . . . . . 2
1.3 EsscherTransforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.4 OptionPricing byEsscherTransforms . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.5 ADifferential Equation for theOptionPricingFunction . . . . . . . . . . . . . . . . . . 12
1.6 ACharacterization of theEsscherTransform. . . . . . . . . . . . . . . . . . . . . . . . 14
2 On the Lévy Measure
of Generalized Hyperbolic Distributions 21
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.2 Calculating theLévyMeasure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.3 EsscherTransforms and theLévyMeasure . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.4 FourierTransformof theModifiedLévyMeasure . . . . . . . . . . . . . . . . . . . . . 28
2.4.1 The Lévy Measure of a Generalized Hyperbolic Distribution . . . . . . . . . . . 30
2.4.2 AsymptoticExpansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.4.3 Calculating theFourier Inverse . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.4.4 SumRepresentations forSomeBessel Functions . . . . . . . . . . . . . . . . . 37
2.4.5 ExplicitExpressions for theFourierBacktransform . . . . . . . . . . . . . . . . 38
2.4.6 Behavior of the Density around the Origin . . . . . . . . . . . . . . . . . . . . . 38
2.4.7 NIG Distributions as a Special Case . . . . . . . . . . . . . . . . . . . . . . . . 40
2.5 Absolute Continuity and Singularity for Generalized Hyperbolic Lévy Processes . . . . 41
2.5.1 ChangingMeasures byChangingTriplets . . . . . . . . . . . . . . . . . . . . . 41
2.5.2 Allowed andDisallowedChanges ofParameters . . . . . . . . . . . . . . . . . 42
2.6 The GH Parameters  and  as Path Properties . . . . . . . . . . . . . . . . . . . . . . . 47
2.6.1 Determination of  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
2.6.2 Determination of  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
2.6.3 Implications andVisualization . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
2.7 Implications forOption Pricing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3 Computation of European Option Prices
Using Fast Fourier Transforms 61
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.2 Definitions andBasicAssumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
3.3 ConvolutionRepresentation forOptionPricing Formulas . . . . . . . . . . . . . . . . . 63
3.4 Standard andExoticOptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
3.4.1 PowerCallOptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
3.4.2 PowerPutOptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
3.4.3 AsymptoticBehavior of theBilateralLaplaceTransforms . . . . . . . . . . . . 67
3.4.4 Self-Quanto Calls and Puts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
3.4.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
3.5 Approximation of theFourier Integrals bySums . . . . . . . . . . . . . . . . . . . . . . 69
3.5.1 FastFourierTransform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
3.6 Outline of the Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
3.7 Applicability to Different Stock Price Models . . . . . . . . . . . . . . . . . . . . . . . 72
3.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
4 The Lévy Term Structure Model 77
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
4.2 Overviewof theLévyTermStructureModel . . . . . . . . . . . . . . . . . . . . . . . . 79
4.3 The Markov Property of the Short Rate: Generalized Hyperbolic Driving Lévy Processes 81
4.4 AffineTermStructures intheLévyTermStructureModel . . . . . . . . . . . . . . . . . 85
4.5 Differential Equations for theOptionPrice . . . . . . . . . . . . . . . . . . . . . . . . . 87
5 Bond Price Models: Empirical Facts 93
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
5.2 LogReturns in theGaussianHJMModel . . . . . . . . . . . . . . . . . . . . . . . . . 93
5.3 TheDataset and itsPreparation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
5.3.1 Calculating Zero Coupon Bond Prices and Log Returns From the Yields Data . . 95
5.3.2 AFirstAnalysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
5.4 Assessing the Goodness of Fit of the Gaussian HJM Model . . . . . . . . . . . . . . . . 99
5.4.1 VisualAssessment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
5.4.2 Quantitative Assessment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
5.5 Normal Inverse Gaussian as Alternative Log Return Distribution . . . . . . . . . . . . . 103
5.5.1 VisualAssessment ofFit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
5.5.2 Quantitative Assessment of Fit . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
5.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
6 Lévy Term Structure Models: Uniqueness of the Martingale Measure 109
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
6.2 The Björk/Di Masi/Kabanov/Runggaldier Framework . . . . . . . . . . . . . . . . . . . 110
6.3 TheLévyTermStructureModel as aSpecialCase . . . . . . . . . . . . . . . . . . . . . 111
6.3.1 GeneralAssumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
6.3.2 Classification in the Björk/Di Masi/Kabanov/Runggaldier Framework . . . . . . 111
6.4 SomeFacts fromStochasticAnalysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
6.5 Uniqueness of theMartingaleMeasure . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
6.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
7 Lévy Term-Structure Models: Generalization to Multivariate Driving Lévy Processes and
Stochastic Volatility Structures 125
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
7.2 Constructing Martingales of Exponential Form . . . . . . . . . . . . . . . . . . . . . . 125
7.3 ForwardRates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
7.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
A Generalized Hyperbolic and CGMY Distributions and Lévy Processes 137
A.1 Generalized Hyperbolic Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
A.2 Important Subclasses ofGH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
A.2.1 Hyperbolic Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
A.2.2 Normal Inverse Gaussian (NIG) Distributions . . . . . . . . . . . . . . . . . . . 139
A.3 The Carr-Geman-Madan-Yor (CGMY) Class of Distributions . . . . . . . . . . . . . . . 139
A.3.1 Variance Gamma Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
A.3.2 CGMY Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
A.3.3 Reparameterization of the Variance Gamma Distribution . . . . . . . . . . . . . 143
A.4 Generation of (Pseudo-)Random Variables . . . . . . . . . . . . . . . . . . . . . . . . . 145
A.5 Comparison of NIG and Hyperbolic Distributions . . . . . . . . . . . . . . . . . . . . . 147
A.5.1 Implications for Maximum Likelihood Estimation . . . . . . . . . . . . . . . . 148
A.6 GeneralizedHyperbolic LévyMotion . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
B Complements to Chapter 3 151
B.1 Convolutions andLaplace transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
B.2 Modeling theLogReturn on aSpotContract Instead of aForwardContract . . . . . . . 152
Index 160
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