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

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94 CHAPTER 2. THE CALIBRATION PROBLEM Choose a subsequence of {Q δ k}, converging weakly to Q ∗ ∈ M ∩ L + B . To simplify notation, this subsequence is denoted again by {Q δ k} k≥1 . Substituting r = δ into Equation (2.34) and making k tend to infinity shows that Together with Lemma 2.2 this implies that lim sup ‖C Qδk − C M ‖ 2 ≤ ‖C Q+ − C M ‖ 2 . k ‖C Q∗ − C M ‖ 2 ≤ ‖C Q+ − C M ‖ 2 , hence Q ∗ is a least squares solution. By weak lower semicontinuity of I (cf. Lemma 2.11) and using (2.36), I(Q ∗ |P ) ≤ lim inf k I(Q δ k |P ) ≤ lim sup I(Q δ k |P ) ≤ I(Q + |P ), k which means that Q ∗ is a MELSS. The last assertion of the theorem follows from the fact that in this case every subsequence of {Q δ k} has a further subsequence converging toward Q + .
Chapter 3 Numerical implementation of the calibration algorithm Before solving the calibration problem (2.27) numerically, we reformulate it as follows: • The calibration problem (2.27) is expressed in terms of the characteristic triplets (A, ν Q , γ Q ) and (A, ν P , γ P ) of the prior Lévy process and the solution. This can be done using Equations (1.1), (1.25) and (1.26) (option pricing by Fourier transform) and Equation (2.19) (expression of the relative entropy in terms of characteristic triplets). • The Lévy measure ν Q is discretized (approximated by a finite-dimensional object). This discussed in detail in Section 3.1. The prior Lévy process P is a crucial ingredient that must be specified by the user. Section 3.2 suggests different ways to do this and studies the effect of a misspecification of the prior on the solutions of the calibration problem. Section 3.3 discusses the methods to choose the regularization parameter α based on the data and Section 3.4 treats the choice of weights w i of different options. Section 3.5 details the numerical algorithm that we use to solve the calibration problem once the prior, the regularization parameter and the weights have been fixed. The calibration algorithm, including the automatic choice of the regularization parameter, has been implemented in a computer program levycalibration and various tests have been carried out, both on simulated option prices (computed in a known exponential Lévy model) and using real market data. Section 3.6 discusses the results of these tests. 95
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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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Page 131: Part II Multidimensional modelling
Page 134 and 135: 134 CHAPTER 4. DEPENDENCE OF LEVY P
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144 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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