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128 CHAPTER 3. NUMERICAL IMPLEMENTATION quite well, when it comes to calibrating several maturities at the same time, the calibration by Lévy processes becomes much less precise. This is clearly seen from the three graphs of Figure 3.12. The top graph shows the market implied volatilities for four maturities and different strikes. The bottom left graphs depicts implied volatilities, computed in an exponential Lévy model calibrated using our nonparametric algorithm to the first maturity present in the market data. One can see that while the calibration quality is acceptable for the first maturity, it quickly deteriorates as the time to maturity increases: the smile in an exponential Lévy model flattens too fast. The same effect can be observed in the bottom right graph: here, the model was calibrated to the last maturity, present in the data. As a result, the calibration quality is poor for the first maturity: the smile in an exponential Lévy model is more pronounced and its shape does not resemble that of the market. It is difficult to calibrate an exponential Lévy model to options of several maturities because due to independence and stationarity of their increments, Lévy processes have a very rigid term structure of cumulants. In particular, the skewness of a Lévy process is proportional to the inverse square root of time and the excess kurtosis is inversely proportional to time [68]. A number of empirical studies have compared the term structure of skewness and kurtosis implied in market option prices to the skewness and kurtosis of Lévy processes. Bates [13], after an empirical study of implicit kurtosis in $/DM exchange rate options concludes that “while implicit excess kurtosis does tend to increase as option maturity shrinks, . . . , the magnitude of maturity effects is not as large as predicted [by a Lévy model]”. For stock index options, Madan and Konikov [68] report even more surprising results: both implied skewness and kurtosis actually decrease as the length of the holding period becomes smaller. It should be mentioned, however, that implied skewness/kurtosis cannot be computed from a finite number of option prices with high precision. Our non-parametric approach allows to investigate time homogeneity by calibrating the Lévy measure separately to various option maturities. Figure 3.9 shows Lévy measures obtained by running the algorithm for options of different maturity. The hypothesis of time homogeneity would imply that all the curves are the same, which is apparently not the case here. However, computing the areas under the curves yields similar jump intensities across maturities: this result can be interpreted by saying that the risk neutral jump intensity is relatively stable through time while the shape of the (normalized) jump size density can actually change. The
3.6. NUMERICAL AND EMPIRICAL TESTS 129 50 30−day ATM options 450−day ATM options 45 40 35 30 25 20 15 2/01/1996 2/01/1997 2/01/1998 30/12/1998 Figure 3.13: Implied volatility of at the money European options on CAC40 index. type of time dependence is therefore more complicated than simply a time dependent intensity. A second major difficulty arising while trying to calibrate an exponential Lévy model is the time evolution of the smile. Exponential Lévy models belong to the class of so called “sticky moneyness” models, meaning that in an exponential Lévy model, the implied volatility of an option with given moneyness (strike price to spot ratio) does not depend on time. This can be seen from the following simple argument. In an exponential Lévy model Q, the implied volatility σ of a call option with moneyness m, expiring in τ years, satisfies: e −rτ E Q [(S t e rτ+Xτ − mS t ) + |F t ] = e −rτ σ2 rτ+σWτ − E[(S t e 2 τ − mS t ) + |F t ] Due to the independent increments property, S t cancels out and we obtain an equation for the implied volatility σ which does not contain t or S t . Therefore, in an exp-Lévy model this implied volatility does not depend on date t or stock price S t . This means that once the smile has been calibrated for a given date t, its shape is fixed for all future dates. Whether or not this is true in real markets can be tested in a model-free way by looking at the implied volatility of at the money options with the same maturity for different dates. Figure 3.13 depicts the behavior of implied volatility of two at the money options on the CAC40 index, expiring in 30 and 450 days. Since the maturities of available options are different for different dates, to obtain the implied volatility of an option with fixed maturity T for each date, we have taken two maturities, present in the data, closest to T from above and below: T 1 ≤ T and T 2 > T .
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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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2.3. RELATIVE ENTROPY IN THE LITERA
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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
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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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178 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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