Processus de LÃ©vy en Finance - Laboratoire de ProbabilitÃ©s et ...

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8 CONTENTS 2.1.3 Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 2.1.4 Numerical difficulties of least squares calibration . . . . . . . . . . . . . . 68 2.2 Selection of solutions using relative entropy . . . . . . . . . . . . . . . . . . . . . 69 2.3 The use of relative entropy for pricing and calibration: review of literature . . . . 72 2.3.1 Pricing with minimal entropy martingale measure . . . . . . . . . . . . . 72 2.3.2 Calibration via relative entropy minimization . . . . . . . . . . . . . . . . 75 2.3.3 The present study in the context of previous work on this subject . . . . 79 2.4 Relative entropy of Lévy processes . . . . . . . . . . . . . . . . . . . . . . . . . . 80 2.4.1 Properties of the relative entropy functional . . . . . . . . . . . . . . . . . 82 2.5 Regularizing the calibration problem . . . . . . . . . . . . . . . . . . . . . . . . . 85 2.5.1 Properties of regularized solutions . . . . . . . . . . . . . . . . . . . . . . 87 2.5.2 Convergence of regularized solutions . . . . . . . . . . . . . . . . . . . . . 92 3 Numerical implementation of the calibration algorithm 95 3.1 Discretizing the calibration problem . . . . . . . . . . . . . . . . . . . . . . . . . 96 3.2 Choice of the prior Lévy process . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 3.3 Choice of the regularization parameter . . . . . . . . . . . . . . . . . . . . . . . . 102 3.3.1 A posteriori parameter choice rules for attainable calibration problems . . 103 3.3.2 A posteriori parameter choice rule for non-attainable calibration problems 110 3.3.3 Computing the noise level . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 3.4 Choice of the weights of option prices . . . . . . . . . . . . . . . . . . . . . . . . 111 3.5 Numerical solution of calibration problem . . . . . . . . . . . . . . . . . . . . . . 113 3.5.1 Computing the calibration functional . . . . . . . . . . . . . . . . . . . . . 114 3.5.2 Computing the gradient of the calibration functional . . . . . . . . . . . . 117 3.5.3 Overview of the algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 3.6 Numerical and empirical tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 3.6.1 Tests on simulated data . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 3.6.2 Empirical properties of implied Lévy measures . . . . . . . . . . . . . . . 123 3.6.3 Testing time homogeneity . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
CONTENTS 9 II Multidimensional modelling with Lévy processes 131 4 Characterization of dependence of multidimensional Lévy processes 133 4.1 Introduction to dependence modelling . . . . . . . . . . . . . . . . . . . . . . . . 133 4.2 Dependence concepts for multidimensional Lévy processes . . . . . . . . . . . . . 138 4.3 Increasing functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 4.4 Lévy copulas for spectrally positive Lévy processes . . . . . . . . . . . . . . . . . 146 4.5 Lévy copulas for general Lévy processes . . . . . . . . . . . . . . . . . . . . . . . 152 4.6 Examples of Lévy copulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 5 Applications of Lévy copulas 165 5.1 Parametric families of Lévy copulas . . . . . . . . . . . . . . . . . . . . . . . . . . 167 5.2 Simulation of multidimensional dependent Lévy processes . . . . . . . . . . . . . 174 5.3 A two-dimensional variance gamma model for option pricing . . . . . . . . . . . . 182 Conclusions and perspectives 187 Bibliography 189 Subject index 199
Page 1 and 2: Thèse présentée pour obtenir le
Page 3: Abstract This thesis deals with the
Page 7: Contents Remarks on notation 11 Int
Page 12 and 13: 12 For {P n } n≥1 , P ∈ P(Ω) t
Page 14 and 15: 14 INTRODUCTION EN FRANCAIS corresp
Page 16 and 17: 16 INTRODUCTION EN FRANCAIS Lévy p
Page 18 and 19: 18 INTRODUCTION EN FRANCAIS Calibra
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Page 22 and 23: 22 INTRODUCTION EN FRANCAIS et Scai
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Page 28 and 29: 28 INTRODUCTION 7500 Taux de change
Page 30 and 31: 30 INTRODUCTION Lévy models by Fou
Page 33 and 34: Chapter 1 Lévy processes and exp-L
Page 35 and 36: 1.1. LEVY PROCESSES 35 Proposition
Page 37 and 38: 1.1. LEVY PROCESSES 37 The characte
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2.1. LEAST SQUARES CALIBRATION 59 m
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2.1. LEAST SQUARES CALIBRATION 61 i
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2.1. LEAST SQUARES CALIBRATION 63 w
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2.1. LEAST SQUARES CALIBRATION 65 A
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2.1. LEAST SQUARES CALIBRATION 67 i
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2.2. SELECTION USING RELATIVE ENTRO
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2.2. SELECTION USING RELATIVE ENTRO
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2.3. RELATIVE ENTROPY IN THE LITERA
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2.4. RELATIVE ENTROPY OF LEVY PROCE
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2.5. REGULARIZING THE CALIBRATION P
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Chapter 3 Numerical implementation
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3.1. DISCRETIZING THE CALIBRATION P
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3.2. CHOICE OF THE PRIOR 99 Since
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3.2. CHOICE OF THE PRIOR 101 twice
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3.3. CHOICE OF THE REGULARIZATION P
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3.4. CHOICE OF THE WEIGHTS 111 Then
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3.5. NUMERICAL SOLUTION 113 have th
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3.5. NUMERICAL SOLUTION 115 of X t
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3.5. NUMERICAL SOLUTION 117 since
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3.6. NUMERICAL AND EMPIRICAL TESTS
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Part II Multidimensional modelling
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134 CHAPTER 4. DEPENDENCE OF LEVY P
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166 CHAPTER 5. APPLICATIONS OF LEVY
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188 CONCLUSIONS AND PERSPECTIVES ge
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190 BIBLIOGRAPHY [9] O. E. Barndorf
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192 BIBLIOGRAPHY [33] F. Delbaen, P
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194 BIBLIOGRAPHY [57] P. Jorion, On
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196 BIBLIOGRAPHY [81] S. Raible, L
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198 BIBLIOGRAPHY
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200 INDEX tightness, 40 levycalibra
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