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42 CHAPTER 1. LEVY PROCESSES AND EXP-LEVY MODELS where for some θ ∈ R. A Q = A, ν Q = e −x2 +θx ν, (1.11) ∫ γ Q = γ + Aθ + x(e −x2 +θx − 1)ν(dx) (1.12) |x|≤1 In other words, this proposition shows that exponential Lévy models are arbitrage-free in the following (not mutually exclusive) cases: • X has infinite variation ∫ 1 −1 |x|ν(dx) = ∞ and/or A > 0. • X has both positive and negative jumps. • X has positive jumps and negative drift or negative jumps and positive drift. In the sequel, we will denote the set of Lévy processes satisfying at least one of these conditions by L NA and the corresponding exponential Lévy models will be called arbitrage-free exponential Lévy models. The first part of this proposition was proved in [55], see also [25, Theorem 4.6]. Since the second part will also be used below, we give a proof here. Proof of Proposition 1.8. For any θ ∈ R we denote the probability measure under which X has the characteristic triplet (1.11)–(1.12) by Q θ . From Proposition 1.5 it follows easily that for every θ ∈ R, both Q θ ≪ P and P ≪ Q θ . Therefore, we only need to prove that for some θ, the triplet (A, ν Qθ , γ Qθ ) satisfies the martingale condition (1.2), or equivalently, that the equation has a solution, where f(θ) := A 2 + Aθ + ∫ |x|≤1 f(θ) + γ = 0 (1.13) x(e −x2 +θx − 1)ν(dx) + ∫ ∞ −∞ (e x − 1 − x1 |x|≤1 )e −x2 +θx ν(dx) The dominated convergence theorem implies that f is continuous and differentiable and that f ′ (θ) = A + ∫ ∞ −∞ x(ex − 1)e −x2 +θx ν(dx) ≥ 0, which means that f(θ) is an increasing function. Moreover, if A > 0 or if ν((0, ∞)) > 0 and ν((−∞, 0)) > 0 then f ′ is everywhere bounded from below by a positive number. Therefore in these cases lim θ→+∞ f(θ) = +∞, lim θ→−∞ f(θ) = −∞ and Equation (1.13) admits a solution.
1.3. EXP-LEVY MODELS IN FINANCE 43 It remains to treat the case when A = 0 and ν is supported by one of the half-axes. Suppose that ν((−∞, 0)) = 0 and ν((0, ∞)) > 0. In this case lim θ→+∞ f(θ) = +∞ and lim f(θ) = − xν(dx). θ→−∞ ∫0
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 26 and 27: 26 INTRODUCTION EN FRANCAIS dans un
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
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Page 41: 1.2. EXPONENTIAL LEVY MODELS 41 (b)
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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
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Page 55 and 56: 1.4. PRICING EUROPEAN OPTIONS 55 Si
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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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