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

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118 CHAPTER 3. NUMERICAL IMPLEMENTATION and formulas (3.28–3.29) yield: where ψ A k ∂ ˜ζ T (u k ) ∂q m = T eiu krT +T ψ k { e x m+iu k x m − 1 − (1 + iu k )(e xm − 1) } iu k (1 + iu k ) = T (e xm − 1) eiu krT +T ψ k + T e xm eiukrT (e T ψ k − e T ψA k ) 1 + iu k iu k (1 + iu k ) − T e xm eiu krT (e T ψ k − e T ψA k ) iu k (1 + iu k ) e iu kx m + T e xm eiu krT +T ψk A (e iu kx m − 1) , iu k (1 + iu k ) = − A 2 u k(u k − i). Suppose that the grid in strike space is such that x 0 = m 0 d for some m 0 ∈ Z. Then for every j ∈ {0, . . . , M − 1}, M−1 ∑ DFT j [f k e iu M−1−kx m ] = e iu M−1x m Introducing additional notation: k=0 e −2πik(j−m−m 0)/M f k H k = eiu krT +ψ k T 1 + iu k G k = eiu krT +ψ A k T iu k (1 + iu k ) , we obtain the final formula for computing the gradient: ∂ˆ˜z T (x j ) ∂q m = ∆ 2π e−ix ju M−1 T (e xm − 1)DFT j [ w k e −ix 0∆k H M−1−k ] = e iu M−1x m DFT (j−m0 −m) mod M [f k ]. + T e xm (e iu M−1x m ˆ˜z T (x (j−m0 −m) mod M ) − ˆ˜z T (x j )) + ∆ ] 2π ei(xm−x j)u M−1 T e xm DFT (j−m0 −m) mod M [w k e −ix0∆k G M−1−k − ∆ [ ] 2π e−ix ju M−1 T e xm DFT j w k e −ix0∆k G M−1−k . (3.32) The time values ˆ˜z T have already been computed in all points of the grid when evaluating option prices, and the Fourier transforms of G k do not need to be reevaluated at each step of the algorithm because G k do not depend on q i . Therefore, compared to evaluating the functional alone, to compute the gradient of Ĵ α one only needs one additional fast Fourier transform per maturity date. The complexity of evaluating the gradient using the above formula is thus only about 1.5 times higher than that of evaluating the functional itself (because evaluating option prices requires two Fourier transforms per maturity date). If the gradient was to be evaluated numerically, the complexity would typically be M times higher. The analytic formula (3.32) therefore allows to reduce the overall running time of the calibration algorithm from several hours to less than a minute on a standard PC.
3.6. NUMERICAL AND EMPIRICAL TESTS 119 3.5.3 Overview of the algorithm Here is the final numerical algorithm, as implemented in the computer program levycalibration, used to run the tests of Section 3.6. • Fix the prior using one of the methods described in Section 3.2. In the tests below, a user-specified prior was taken. • Compute the weights of market option prices (Section 3.4) and estimate the noise level (Section 3.3.3). • Use one of the a posteriori methods of Section 3.3.1 to compute an optimal regularization parameter α ∗ achieving trade-off between precision and stability. The optimal α ∗ is computed by bisection, minimizing ˜J α several times for different values of α with low precision. In the tests below we have always been able to choose α ∗ using the discrepancy principle (3.13) with c 1 = 1.1 and c 2 = 1.3, so there was no need to resort to the alternative scheme (3.19). • Minimize ˜J α ∗ with high precision to find the regularized solution Q ∗ . 3.6 Numerical and empirical tests Our numerical tests, performed using the levycalibration program, fall into two categories. First, to assess the accuracy and numerical stability of our method, we tested it on option prices produced by a known exponential-Lévy model (Section 3.6.1). We then applied our algorithm to real options data, using the prices of European options on different European stock market indices, provided by Thomson Financial R and studied the properties of Lévy measures, implied by market data. 3.6.1 Tests on simulated data A compound Poisson example: Kou’s model In the first series of tests, option prices were generated using Kou’s jump diffusion model [62] with a diffusion part with volatility σ 0 = 10% and a Lévy density given by Equation (1.17).
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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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168 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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