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

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Chapter 4 Characterization of dependence of multidimensional Lévy processes 4.1 Introduction to dependence modelling Many financial applications require a multidimensional model with jumps, taking into account the dependence between components. However, such models are more difficult to construct than one-dimensional ones and the applications continue to be dominated by (geometric) Brownian motion. A simple method to introduce jumps into a multidimensional model is to take a multivariate Brownian motion and time change it with a one-dimensional increasing Lévy process. This approach, advocated in [36, 79], allows to construct multidimensional versions of many popular one-dimensional models, including variance gamma, normal inverse Gaussian and generalized hyperbolic process. The principal advantage of this method is its simplicity and analytic tractability; in particular, processes of this type are easy to simulate. However, the range of dependence patterns that one can obtain using this approach is quite limited (for instance, independence is not included), and all components must follow the same parametric model (e.g., either all of the components are variance gamma or all of the components are normal inverse Gaussian etc.) To be more specific, suppose that two stock price processes {St 1 } t≥0 and {St 2 } t≥0 are mod- 133
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Thèse présentée pour obtenir le
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Abstract This thesis deals with the
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Contents Remarks on notation 11 Int
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CONTENTS 9 II Multidimensional mode
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12 For {P n } n≥1 , P ∈ P(Ω) t
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14 INTRODUCTION EN FRANCAIS corresp
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16 INTRODUCTION EN FRANCAIS Lévy p
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18 INTRODUCTION EN FRANCAIS Calibra
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20 INTRODUCTION EN FRANCAIS • Si
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22 INTRODUCTION EN FRANCAIS et Scai
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24 INTRODUCTION EN FRANCAIS Ce rés
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26 INTRODUCTION EN FRANCAIS dans un
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28 INTRODUCTION 7500 Taux de change
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30 INTRODUCTION Lévy models by Fou
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Chapter 1 Lévy processes and exp-L
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1.1. LEVY PROCESSES 35 Proposition
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1.1. LEVY PROCESSES 37 The characte
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1.1. LEVY PROCESSES 39 On the other
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1.2. EXPONENTIAL LEVY MODELS 41 (b)
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1.3. EXP-LEVY MODELS IN FINANCE 43
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1.3. EXP-LEVY MODELS IN FINANCE 45
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1.4. PRICING EUROPEAN OPTIONS 47 wi
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1.4. PRICING EUROPEAN OPTIONS 49 Pr
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1.4. PRICING EUROPEAN OPTIONS 51 Nu
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1.4. PRICING EUROPEAN OPTIONS 53 By
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1.4. PRICING EUROPEAN OPTIONS 55 Si
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Chapter 2 The calibration problem a
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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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Page 81 and 82: 2.4. RELATIVE ENTROPY OF LEVY PROCE
Page 83 and 84: 2.4. RELATIVE ENTROPY OF LEVY PROCE
Page 85 and 86: 2.5. REGULARIZING THE CALIBRATION P
Page 95 and 96: Chapter 3 Numerical implementation
Page 97 and 98: 3.1. DISCRETIZING THE CALIBRATION P
Page 99 and 100: 3.2. CHOICE OF THE PRIOR 99 Since
Page 101 and 102: 3.2. CHOICE OF THE PRIOR 101 twice
Page 103 and 104: 3.3. CHOICE OF THE REGULARIZATION P
Page 111 and 112: 3.4. CHOICE OF THE WEIGHTS 111 Then
Page 113 and 114: 3.5. NUMERICAL SOLUTION 113 have th
Page 115 and 116: 3.5. NUMERICAL SOLUTION 115 of X t
Page 117 and 118: 3.5. NUMERICAL SOLUTION 117 since
Page 119 and 120: 3.6. NUMERICAL AND EMPIRICAL TESTS
Page 131: Part II Multidimensional modelling
Page 135 and 136: 4.1. INTRODUCTION 135 as jump times
Page 137 and 138: 4.1. INTRODUCTION 137 distribution
Page 139 and 140: 4.2. DEPENDENCE CONCEPTS FOR LEVY P
Page 141 and 142: 4.3. INCREASING FUNCTIONS 141 Clear
Page 143 and 144: 4.3. INCREASING FUNCTIONS 143 for x
Page 145 and 146: 4.3. INCREASING FUNCTIONS 145 are o
Page 147 and 148: 4.4. LEVY COPULAS: SPECTRALLY POSIT
Page 153 and 154: 4.5. LEVY COPULAS: GENERAL CASE 153
Page 159 and 160: 4.6. EXAMPLES OF LEVY COPULAS 159 X
Page 161 and 162: 4.6. EXAMPLES OF LEVY COPULAS 161 w
Page 163 and 164: 4.6. EXAMPLES OF LEVY COPULAS 163 C
Page 165 and 166: Chapter 5 Applications of Lévy cop
Page 167 and 168: 5.1. PARAMETRIC FAMILIES 167 The ap
Page 169 and 170: 5.1. PARAMETRIC FAMILIES 169 for u
Page 171 and 172: 5.1. PARAMETRIC FAMILIES 171 2.5 2.
Page 173 and 174: 5.1. PARAMETRIC FAMILIES 173 Lévy
Page 175 and 176: 5.2. SIMULATION OF DEPENDENT LEVY P
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5.3. TWO-DIMENSIONAL VARIANCE GAMMA
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5.3. TWO-DIMENSIONAL VARIANCE GAMMA
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Conclusions and perspectives In the
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Bibliography [1] L. Ambrosio, N. Fu
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BIBLIOGRAPHY 191 [21] P. Carr, H. G
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BIBLIOGRAPHY 193 [45] P. Glasserman
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BIBLIOGRAPHY 195 [69] C. Mancini, D
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BIBLIOGRAPHY 197 [93] M. Teboulle a
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Index a posteriori parameter choice
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