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188 CONCLUSIONS AND PERSPECTIVES general framework in which the dependence structures of multidimensional Lévy processes can be described. Lévy copulas completely characterize the possible dependence patterns of Lévy processes in the sense that for every Lévy process, there exists a Lévy copula that describes its dependence structure and for every Lévy copula and every n one-dimensional Lévy processes, there exists an n-dimensional Lévy process with this Lévy copula and with margins given by these one-dimensional processes. Multidimensional Lévy process models for applications can thus be constructed by taking any n one-dimensional processes and a Lévy copula from a (possibly parametric) family. The simulation methods, developed in the last chapter of this thesis, allow to compute various quantities of interest in a Lévy copula model using the Monte Carlo method. The scope of potential applications of Lévy copula models in finance and other domains is large. Financial applications include basket option pricing, and portfolio management. Lévy copula models are also useful in insurance and in risk management, to model the dependence between loss processes of different business lines, and more generally, in all multivariate problems where dependence between jumps needs to be taken into account. From the point of view of applications, the next step is to develop the methods of estimating Lévy copula models from the data, using, for example, simulation-based techniques of statistical inference [51]. A more theoretical research direction that we currently pursue in collaboration with Jan Kallsen is to investigate the relation between the Lévy copula of a Lévy process and its probabilistic copula at a given time.
Bibliography [1] L. Ambrosio, N. Fusco, and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Oxford University Press, 2000. [2] L. Andersen and J. Andreasen, Jump-diffusion models: Volatility smile fitting and numerical methods for pricing, Rev. Derivatives Research, 4 (2000), pp. 231–262. [3] M. Avellaneda, R. Buff, C. Friedman, N. Grandchamp, L. Kruk, and J. Newman, Weighted Monte Carlo: a new technique for calibrating asset-pricing models, Int. J. Theor. Appl. Finance, 4 (2001), pp. 91–119. [4] M. Avellaneda, C. Friedman, R. Holmes, and D. Samperi, Calibrating volatility surfaces via relative entropy minimization, Applied Mathematical Finance, 4 (1997), pp. 37–64. [5] O. Barndorff-Nielsen, Processes of normal inverse Gaussian type, Finance Stoch., 2 (1998), pp. 41–68. [6] O. Barndorff-Nielsen, T. Mikosch, and S. Resnick, eds., Lévy Processes — Theory and Applications, Birkhäuser, Boston, 2001. [7] O. E. Barndorff-Nielsen, Normal inverse Gaussian distributions and stochastic volatility modelling, Scand. J. Statist., 24 (1997), pp. 1–13. [8] O. E. Barndorff-Nielsen and A. Lindner, Some aspects of Lévy copulas. Download from www-m4.ma.tum.de/m4/pers/lindner, 2004. 189
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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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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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136 CHAPTER 4. DEPENDENCE OF LEVY P
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Page 190 and 191: 190 BIBLIOGRAPHY [9] O. E. Barndorf
Page 192 and 193: 192 BIBLIOGRAPHY [33] F. Delbaen, P
Page 194 and 195: 194 BIBLIOGRAPHY [57] P. Jorion, On
Page 196 and 197: 196 BIBLIOGRAPHY [81] S. Raible, L
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Page 200: 200 INDEX tightness, 40 levycalibra
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