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166 CHAPTER 5. APPLICATIONS OF LEVY COPULAS Lévy copula of a Lévy process is not linked to any given time scale; it describes the dependence structure of the entire process. One example of a financial modelling problem where Lévy copulas naturally appear is the pricing of multi-asset options on underlyings, described by exponential Lévy models. In this setting the parameters of the marginal Lévy processes can be calibrated from the prices of European options, quoted at the market, and the dependence structure will typically be estimated from the historical time series of returns. Since the sampling rate of returns is different from the maturity of traded options as well as from the maturity of the basket option that one wants to price, ordinary copulas are not well suited for this problem. Pricing of basket options in a Lévy copula model is discussed in more detail in Section 5.3. Lévy copulas can also be useful outside the realm of financial modelling. Other contexts where modelling dependence in jumps is required are portfolios of insurance claims and models of operational risk. Consider an insurance company with two subsidiaries, in France and in Germany. The aggregate loss process of the French subsidiary is modelled by the subordinator {X t } t≥0 and the loss process of the German one is {Y t } t≥0 . The nature of processes X and Y may be different because the subsidiaries may not be working in the same sector and many risks that cause losses are local. However, common risks like floods and pan-European windstorms will lead to a certain degree of dependence between the claims. In this setting it is convenient to model the dependence between X and Y using a Lévy copula on [0, ∞] 2 . In this modelling approach, the two-dimensional Lévy measure of (X, Y ) is known and the overall loss distribution and ruin probability can be computed. Another example where jump processes naturally appear is given by models of operational risk. The 2001 Basel agreement defines the operational risk as “the risk of direct and indirect loss resulting from inadequate or failed internal processes, people and systems or from external events” and allows banks to use internal loss data to compute regulatory capital requirements. Taking into account the dependence between different business lines in this computation, due to a diversification effect, may lead to substantial reduction of regulatory capital [14]. Aggregate loss processes from different business lines can be dynamically modelled by subordinators and the dependence between them can be accounted for using a Lévy copula on [0, ∞] 2 .
5.1. PARAMETRIC FAMILIES 167 The applications of Lévy copulas are not limited to constructing exponential Lévy models. Lévy copula models can be time changed to obtain multidimensional analogs of stochastic volatility models discussed in [22]. More generally, since a large class of Markov processes or even semimartingales behaves locally as a Lévy process in the sense that its dynamics can be described by a drift rate, a covariance matrix, and a Lévy measure, which may all change randomly through time (cf. e.g. [54], II.2.9, II.4.19), Lévy copulas could be used to describe dependence between processes of these types. 5.1 Parametric families of Lévy copulas Our first result is a method to construct Lévy copulas on [0, ∞] d from ordinary copulas. Theorem 5.1 (Construction of Lévy copulas from ordinary copulas). Let C be a copula on [0, 1] d and φ : [0, 1] → [0, ∞] be a strictly increasing continuous function with φ(1) = ∞, φ(0) = 0, having nonnegative derivatives of orders up to d on (0, 1). Then F (u 1 , . . . , u d ) := φ(C(φ −1 (u 1 ), . . . , φ −1 (u d ))) is a Lévy copula on [0, ∞] d . Proof. First, note that φ −1 is well defined and satisfies φ −1 (0) = 0 and φ −1 (∞) = 1. Therefore, properties 1 and 2 of Definition 4.14 are clear, and in view of Equation (4.5), property 4 also holds. It remains to show that F is a d-increasing function, and since φ −1 is strictly increasing, it suffices to prove that the function φ(C(u 1 , . . . , u d )) is d-increasing on [0, 1] d . To this end, let us show by induction on d that if H : [0, 1] d → [0, 1] is d-increasing and grounded and ψ : [0, 1] → [0, ∞] is a continuous increasing function with ψ(0) = 0 and positive derivatives of orders up to d on (0, 1) then ψ(H) is also d-increasing and grounded. For d = 1, the result is clear. Suppose d ≥ 2. For k = 1, . . . , d let a k , b k ∈ [0, 1] with a k ≤ b k . The function ˜H(u 2 , . . . , u d ) := ψ(H(b 1 , u 2 , . . . , u d )) − ψ(H(a 1 , u 2 , . . . , u d )) satisfies V ψ(H) (|a 1 , b 1 | × · · · × |a d , b d |) = V ˜H(|a 2 , b 2 | × · · · × |a d , b d |), hence, it remains to prove that ˜H is d − 1-increasing. It can be represented as follows (“∗”
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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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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
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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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