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186 CHAPTER 5. APPLICATIONS OF LEVY COPULAS
Conclusions and perspectives In the first part of this thesis we have solved, using entropic regularization, the ill-posed problem of calibrating an exponential Lévy model to options data and proposed a stable numerical method for computing this solution. Applying our method to prices of index options allowed us to estimate the risk-neutral Lévy measures, implied by market prices. This object is the analog, for exponential Lévy models, of implied volatility, used in the Black-Scholes framework. Our empirical results allow to make a number of important conclusions. First, using an exponential Lévy model one can calibrate with high precision the prices of a set of options with common maturity. Moreover, high quality of calibration is achieved already by using finite-intensity Lévy processes. Therefore, from the point of view of option pricing the imperative for using more complex infinite-intensity models is not clear. The third conclusion is that even in the nonparametric setting it is impossible, using an exponential Lévy model, to calibrate accurately the prices of stock index options of several maturities at the same time: options of different maturities produce different implied Lévy measures. This confirms the observation already made by several authors [13, 68] that the framework of exponential Lévy models is not sufficiently flexible to reproduce the term structure of implied volatilities correctly. In view of the above conclusions, we plan to continue the line of research initiated by this thesis, by extending its results to models of stock price behavior that do allow to describe the entire term structure of implied volatilities, e.g. models based on additive processes (processes with independent but not stationary increments) and hybrid models including both jumps and stochastic volatility. The second important direction of future research is to investigate the impact of our calibration methodology on the methods of hedging in presence of jumps in stock prices. In the second part of this thesis we introduced the notion of Lévy copula, providing a 187
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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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2.4. RELATIVE ENTROPY OF LEVY PROCE
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2.4. RELATIVE ENTROPY OF LEVY PROCE
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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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Page 188 and 189: 188 CONCLUSIONS AND PERSPECTIVES ge
Page 190 and 191: 190 BIBLIOGRAPHY [9] O. E. Barndorf
Page 192 and 193: 192 BIBLIOGRAPHY [33] F. Delbaen, P
Page 194 and 195: 194 BIBLIOGRAPHY [57] P. Jorion, On
Page 196 and 197: 196 BIBLIOGRAPHY [81] S. Raible, L
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Page 200: 200 INDEX tightness, 40 levycalibra
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