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

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126 CHAPTER 3. NUMERICAL IMPLEMENTATION 5 4.5 4 3.5 11 May 2001, 8 days 11 June 2001, 4 days 11 July 2001, 9 days Prior 4 3.5 3 11 May 2001, 36 days 11 June 2001, 39 days 11 July 2001, 37 days Prior 3 2.5 2.5 2 2 1.5 1.5 1 1 0.5 0.5 0 −0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 Figure 3.11: Results of calibration at different dates for shortest (left) and second shortest (right) maturity. DAX index options. Recall that the prior density is Gaussian, which shows up as a symmetric parabola on log scales. The area under the curves shown here is to be interpreted as the (risk neutral) jump intensity. While the shape of the curve does vary slightly across calendar time, the intensity stays surprisingly stable: its numerical value is empirically found to be λ ≃ 1, which means around one jump a year. Of course note that this is the risk neutral intensity: jump intensities are not invariant under equivalent change of measures. Moreover this illustrates that a small intensity of jumps λ can be sufficient for explaining the shape of the implied volatility skew for small maturities. Therefore the motivation of infinite activity processes from the point of view of option pricing does not seem clear to us. On the other hand, from the modelling perspective infinite intensity processes do present a particular interest since they do not have a diffusion component and therefore do not suffer from the problem of redundancy of small jumps and diffusion, described above, leading to more parsimonious and identifiable models. 3.6.3 Testing time homogeneity While the literature on jump processes in finance has focused on time homogeneous (Lévy) models, practitioners have tended to use time dependent jump or volatility parameters. Despite the fact that, as our non-parametric analysis and several empirical studies by other authors [21, 67] have shown, Lévy processes reproduce the implied volatility smile for a single maturity
3.6. NUMERICAL AND EMPIRICAL TESTS 127 0.28 Implied volatility 0.26 0.24 0.22 0.2 0.18 0.16 0 Maturity 0.5 1 7000 6500 6000 Strike 5500 5000 0.3 Implied volatility 0.28 0.26 0.24 0.22 0.2 0.18 0.16 0 Maturity 0.5 1 7000 6500 6000 Strike 5500 Implied volatility 0.28 0.26 0.24 0.22 0.2 0.18 0.16 0 5000 0.5 Maturity 1 7000 6500 6000 Strike 5500 5000 Figure 3.12: Top: Market implied volatility surface. Bottom left: implied volatility surface in an exponential Lévy model, calibrated to market prices of the first maturity. Bottom right: implied volatility surface in an exponential Lévy model, calibrated to market prices of the last maturity.
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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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Page 85 and 86: 2.5. REGULARIZING THE CALIBRATION P
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Page 99 and 100: 3.2. CHOICE OF THE PRIOR 99 Since
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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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Page 131: Part II Multidimensional modelling
Page 134 and 135: 134 CHAPTER 4. DEPENDENCE OF LEVY P
Page 166 and 167: 166 CHAPTER 5. APPLICATIONS OF LEVY
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176 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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