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26 INTRODUCTION EN FRANCAIS dans un tel modèle en utilisant la méthode de Monte Carlo. Pour estimer l’importance de la dépendance dans les queues de distribution pour la valorisation des options, je prends deux jeux de paramètres η et θ, qui correspondent tous les deux à la corrélation de rendements de 50% mais ont les structures de dépendance différentes. Les prix des options sur panier pour les deux jeux de paramètres diffèrent de 10% à la monnaie, ce qui montre que la dépendance dans les queues de distribution est importante à prendre en compte pour la valorisation d’options sur panier, et que les copules de Lévy sont un outil adapté pour décrire cette dépendance.
Introduction Lévy processes are defined as stochastic processes with stationary and independent increments: if {X t } t≥0 is a Lévy process, then X t −X s with t > s is independent of the history of the process up to time s, and its law only depends on t − s but not on t or s separately. This property of increments suggests an analogy with linear functions: one can say that Lévy processes are, in some sense, “linear processes”. Despite this apparent simplicity, Lévy processes have many interesting properties and constitute an exciting field of study: this is shown by the recent publication of several monographs (see for example [17, 87]) and by the success of the series of international conferences on Lévy processes and applications (see [6]). From the point of view of financial modelling, Lévy processes provide a class of models with jumps that is both sufficiently rich to reproduce empirical data and simple enough to do many computations analytically. The interest of jump models in finance is mainly due to three reasons. First, in a model with continuous paths like a diffusion model, the price process behaves locally like a Brownian motion and the probability that the stock moves by a large amount over a short period of time is very small, unless one fixes an unrealistically high value of the volatility of volatility. Therefore, in such models the prices of short term out of the money options should be much lower than what one observes in real markets. On the other hand, if stock prices are allowed to jump, even when the time to maturity is very short, there is a non-negligible probability that after a sudden change in the stock price the option will move in the money. Second, from the hedging point of view, continuous models of stock price behavior generally lead to a complete market or to a market that can be made complete by adding one or two 27
Page 1 and 2: Thèse présentée pour obtenir le
Page 3: Abstract This thesis deals with the
Page 7 and 8: Contents Remarks on notation 11 Int
Page 9: CONTENTS 9 II Multidimensional mode
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
Page 20 and 21: 20 INTRODUCTION EN FRANCAIS • Si
Page 22 and 23: 22 INTRODUCTION EN FRANCAIS et Scai
Page 24 and 25: 24 INTRODUCTION EN FRANCAIS Ce rés
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
Page 39 and 40: 1.1. LEVY PROCESSES 39 On the other
Page 41 and 42: 1.2. EXPONENTIAL LEVY MODELS 41 (b)
Page 43 and 44: 1.3. EXP-LEVY MODELS IN FINANCE 43
Page 45 and 46: 1.3. EXP-LEVY MODELS IN FINANCE 45
Page 47 and 48: 1.4. PRICING EUROPEAN OPTIONS 47 wi
Page 49 and 50: 1.4. PRICING EUROPEAN OPTIONS 49 Pr
Page 51 and 52: 1.4. PRICING EUROPEAN OPTIONS 51 Nu
Page 53 and 54: 1.4. PRICING EUROPEAN OPTIONS 53 By
Page 55 and 56: 1.4. PRICING EUROPEAN OPTIONS 55 Si
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Page 59 and 60: 2.1. LEAST SQUARES CALIBRATION 59 m
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Page 69 and 70: 2.2. SELECTION USING RELATIVE ENTRO
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Page 73 and 74: 2.3. RELATIVE ENTROPY IN THE LITERA
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2.3. RELATIVE ENTROPY IN THE LITERA
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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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2.4. 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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