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44 CHAPTER 1. LEVY PROCESSES AND EXP-LEVY MODELS where {N t } t≥0 is the Poisson process counting the jumps of X and Y i are jump sizes (i.i.d. variables). In the Merton model [71], which is the first model of this type, suggested in the literature, jumps in the log-price X are assumed to have a Gaussian distribution: Y i ∼ N(µ, δ 2 ). In the risk-neutral version the characteristic exponent of the log stock price takes the following form: ( ψ(u) = − Au2 2 + u 2 A /2+iµu λ{e−δ2 − 1} − iu 2 + /2+µ λ(eδ2 − 1)) . (1.16) In the Kou model [62], jump sizes are distributed according to an asymmetric Laplace law with a density of the form ν 0 (dx) = [pλ + e −λ +x 1 x>0 + (1 − p)λ − e −λ −|x| 1 x 0, λ − > 0 governing the decay of the tails for the distribution of positive and negative jump sizes and p ∈ [0, 1] representing the probability of an upward jump. The probability distribution of returns in this model has semi-heavy (exponential) tails. In the above two models, the dynamical structure of the process is easy to understand and describe, since the distribution of jump sizes is known. They are easy to simulate and efficient Monte Carlo methods for pricing path-dependent options have been developed. Models of this type also perform quite well for the purposes of implied volatility smile interpolation. However, they do not lead to closed-form densities: statistical estimation and computation of moments or quantiles may be quite difficult. The second category consists of models with infinite number of jumps in every interval, which we will call infinite activity or infinite intensity models. In these models, one does not need to introduce a Brownian component since the dynamics of jumps is already rich enough to generate nontrivial small time behavior [21] and it has been argued [21, 43, 66] that such models give a more realistic description of the price process at various time scales. In addition, many models from this class can be constructed via Brownian subordination, which gives them additional tractability compared to jump-diffusion models. Table 1.1 compares the advantages and drawbacks of these two categories. It should be kept in mind that since the price process is observed on a discrete grid, it is difficult if not impossible to see empirically to which category the price process belongs. The choice is more a question of modelling convenience than an empirical one.
1.3. EXP-LEVY MODELS IN FINANCE 45 Table 1.1: Compound Poisson or infinite intensity: a comparison of two modelling approaches Jump-diffusion models Must contain a Brownian component. Jumps are rare events. Distribution of jump sizes is known. Perform well for implied volatility smile interpolation. Densities not known in closed form. Easy to simulate. Infinite intensity models Brownian component is not needed. The process moves only by jumps and deterministic linear drift. “Distribution of jump sizes” does not exist: jumps arrive infinitely often. Give a realistic description of the historical price process. Closed form densities available in some cases. In some cases can be represented via Brownian subordination, which gives additional tractability.
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
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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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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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