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164 CHAPTER 4. DEPENDENCE OF LEVY PROCESSES and therefore (4.32) holds for ˜F . However, ˜F is also the copula of X. Indeed, let I ⊂ {1, . . . , d} nonempty and (x i ) i∈I ∈ (R \ {0}) |I| . Two situations are possible: • For every i ∈ I, Ũi(x i ) = U(x i ). Then it is easy to see that Equation (4.24) holds with F replaced by ˜F . • For some k ∈ I, U k (x k ) = 0. Then ˜F I ((U i (x i )) i∈I ) = 0, but on the other hand |U I ((x i ) i∈I )| ≤ |U k (x k )| = 0, and (4.24) also holds. The “if” part. Let X have α-stable margins and a homogeneous Lévy copula. Then the marginal tail integrals of X satisfy (4.34), and since by Lemma 4.7, the Lévy measure of every set of the form [a, b) can be expressed as a linear combination of tail integrals, the Lévy measure of X has the property (4.33) and we conclude by Theorem 14.3 in [87] that X is α-stable.
Chapter 5 Applications of Lévy copulas To apply Lévy copulas to multidimensional financial problems with dependence between assets, three kinds of tools are required: 1. Parametric families of Lévy copulas. Parsimonious models are needed because one typically does not have enough information about the dependence to estimate many parameters or proceed with a non-parametric approach. 2. Algorithms allowing to compute various quantities within a Lévy copula model (e.g., option prices, risk measures etc.) 3. Estimation methods for Lévy copula models. In this chapter we give an answer to the first two questions. Section 5.1 describes several methods to construct parametric families of Lévy copulas and gives examples of such families. In Section 5.2 we show how Lévy copulas can be used to simulate multidimensional Lévy processes with dependence between components. This simulation algorithm enables us to compute the quantities of interest using the Monte Carlo method. Estimation methods for Lévy copulas are the topic of our current research and we do not discuss them here. A fundamental advantage of the Lévy copula approach compared to ordinary copulas is the possibility to work with several time scales at the same time. When the time scale is fixed, the dependence structure of returns can be described with an ordinary copula. However, on a different time scale the copula will not be the same (cf. Example 4.1). On the other hand, a 165
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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
Page 113 and 114: 3.5. NUMERICAL SOLUTION 113 have th
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Page 119 and 120: 3.6. NUMERICAL AND EMPIRICAL TESTS
Page 131: Part II Multidimensional modelling
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Page 188 and 189: 188 CONCLUSIONS AND PERSPECTIVES ge
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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