Processus de LÃ©vy en Finance - Laboratoire de ProbabilitÃ©s et ...

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76 CHAPTER 2. THE CALIBRATION PROBLEM a Lévy process under a consistent minimal entropy martingale measure and no analytic results similar to Theorem 2.7 are available for the constrained case. For example, let {X t } t≥0 be a real-valued Lévy process on (A, ν, γ) such that for every t, X t = N ′ t − N ′′ t , (2.17) where N ′ and N ′′ are independent Poisson processes with intensity 1 under P . It follows from Proposition 1.5 that the set of Lévy processes equivalent to P and satisfying the martingale condition (1.2) contains all processes under which X has the form (2.17) where N ′ and N ′′ are still independent and have respective intensities λ ′ = λ and λ ′′ = eλ for some λ > 0. The set of all equivalent martingale measures under which X remains a Lévy process is thus parametrized by one parameter λ. If one allows X to be an additive process (process with independent increments, continuous in probability) under the new measure, the class of equivalent martingale measures is much larger: it follows from Theorem IV.4.32 in [54] that N ′ and N ′′ can now have variable (but deterministic) intensities λ ′ (t) = λ(t) and λ ′′ (t) = eλ(t) for some function λ : [0, T ∞ ] → (0, ∞). It is clear that many market data sets (involving several maturities) can be reproduced by an equivalent martingale measure, under which X is an additive process but not by a martingale measure under which X is a Lévy process, which implies that under the consistent minimal entropy martingale measure X will not be a Lévy process. Stutzer [91] suggests a three-step algorithm for numerical evaluation of derivative prices under the CMEMM in a model with a single time horizon T . This method allows to compute prices of European options with maturity T , consistent with prices of market-quoted European options with the same maturity. First, possible asset price values at time T , {P h } N h=1 and the corresponding probabilities ˆπ(h) are estimated nonparametrically using a histogram estimator from the historical return values. Second, one needs to find the probabilities π ∗ (h), that satisfy the martingale constraint and the pricing constraints and have the smallest relative entropy with respect to ˆπ. In this simplified one period model the martingale condition reduces to a single constraint that the discounted expectation of stock price under π ∗ be equal to its present value. The European options expiring at T can then be priced using these martingale probabilities π ∗ . In this paper, Stutzer suggests an information theoretic rationale for using the relative entropy (also called Kullback-Leibler information criterion) for pricing and calibration. From a Bayesian point of view, the historical prices constitute a prior information about the future
2.3. RELATIVE ENTROPY IN THE LITERATURE 77 risk-neutral distribution of asset prices. This prior must be updated to take into account the martingale constraint and the observed option prices, and it is natural to demand that the updated distribution incorporate no additional information other than the martingale constraint and the pricing constraints. One must therefore minimize some quantitative measure of relative information of the two distributions, and it has been shown (see [52]) that every measure of information satisfying a set of natural axioms must be proportional to Kullback-Leibler relative entropy. Because it only allows to reconstruct the asset price distribution for one specified time horizon, and only takes into account the prices of options that expire at this horizon, Stutzer’s method does not provide any information about the risk-neutral process and thus cannot be used to price any derivative that depends on the stock price at times other than T . Avellaneda et al. [3] pursue the same logic further and propose a method allowing to construct a discrete approximation of the law of the process underlying the observed option prices. They consider N < ∞ fixed trajectories {X 1 , . . . , X N } that the price process can take, simulated beforehand from a prior model. The new state space Ω ′ thus contains a finite number of elements: Ω ′ = {X 1 , . . . , X N }. The new prior P ′ is the uniform law on Ω ′ : P ′ (X i ) = 1 N and the paper suggests to calibrate the weights (probabilities) of these trajectories q i := Q ′ (X i ) to reproduce market-quoted option prices correctly. Minimizing relative entropy I(Q ′ |P ′ ) is then equivalent to maximizing the entropy of Q ′ and the calibration problem becomes: maximize − N∑ q i log q i under constraints E Q′ [H j ] = C j , j = 1, . . . , M, i=1 where H j are terminal payoffs and C j the observed market prices of M traded options. Denoting by g ij the payoff of j-th option on the i-th trajectory, and introducing Lagrange multipliers λ 1 , . . . , λ M , Avellaneda et al. reformulate the calibration problem as a minimax problem: ⎧ ⎨ N ( min max λ q ⎩ − ∑ M∑ N ) ⎫ ∑ ⎬ q i log q i + λ j q i g ij − C j ⎭ . i=1 The inner maximum can be computed analytically and, since it is taken over linear functions of λ 1 , . . . , λ M , yields a convex function of Lagrange multipliers. The outer minimum can then j=1 be evaluated numerically using a gradient descent method. This technique (weighted Monte Carlo) is attractive for numerical computations but has a number of drawbacks from the theoretical viewpoint. i=1 First, the result of calibration is a
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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
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
Page 39 and 40: 1.1. LEVY PROCESSES 39 On the other
Page 41 and 42: 1.2. EXPONENTIAL LEVY MODELS 41 (b)
Page 43 and 44: 1.3. EXP-LEVY MODELS IN FINANCE 43
Page 45 and 46: 1.3. EXP-LEVY MODELS IN FINANCE 45
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
Page 53 and 54: 1.4. PRICING EUROPEAN OPTIONS 53 By
Page 55 and 56: 1.4. PRICING EUROPEAN OPTIONS 55 Si
Page 57 and 58: Chapter 2 The calibration problem a
Page 59 and 60: 2.1. LEAST SQUARES CALIBRATION 59 m
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Page 65 and 66: 2.1. LEAST SQUARES CALIBRATION 65 A
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Page 73 and 74: 2.3. RELATIVE ENTROPY IN THE LITERA
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Page 85 and 86: 2.5. REGULARIZING THE CALIBRATION P
Page 95 and 96: Chapter 3 Numerical implementation
Page 97 and 98: 3.1. DISCRETIZING THE CALIBRATION P
Page 99 and 100: 3.2. CHOICE OF THE PRIOR 99 Since
Page 101 and 102: 3.2. CHOICE OF THE PRIOR 101 twice
Page 103 and 104: 3.3. CHOICE OF THE REGULARIZATION P
Page 111 and 112: 3.4. CHOICE OF THE WEIGHTS 111 Then
Page 113 and 114: 3.5. NUMERICAL SOLUTION 113 have th
Page 115 and 116: 3.5. NUMERICAL SOLUTION 115 of X t
Page 117 and 118: 3.5. NUMERICAL SOLUTION 117 since
Page 119 and 120: 3.6. NUMERICAL AND EMPIRICAL TESTS
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3.6. NUMERICAL AND EMPIRICAL TESTS
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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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