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124 CHAPTER 3. NUMERICAL IMPLEMENTATION 0.4 0.38 Variance gamma Calibrated with σ=0.25 Calibrated with σ=0.2 15 True VG Levy measure Calibrated with σ=0.25 Calibrated with σ=0.2 0.36 10 0.34 0.32 5 0.3 0.28 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 0 −0.5 0 0.5 Figure 3.8: Calibration of options simulated from a variance gamma model using a finite intensity prior with different values of diffusion volatility σ 0 . Left: calibration quality (implied volatilities as a function of strike prices). Right: Calibrated Lévy measures. Increasing the diffusion coefficient decreases the intensity of small jumps in the calibrated measure. graphs show that using an exponential Lévy model one can calibrate with high precision the prices of a set of options with common maturity. This conclusion contradicts the findings of Medvedev and Scaillet [70] who observe that “jumps in returns do not help explaining the slope of implied volatilities at the money”. It is important to note, however, that Medvedev and Scaillet used short term options on S&P 500 index whereas our analysis is based on DAX index options. Figure 3.9, left graph, illustrates the typical shape of risk neutral Lévy densities obtained from our data set. It is readily seen that the Lévy measures obtained are far from being symmetric: the distribution of jump sizes is highly skewed toward negative values. Figure 3.11 shows the same result across calendar time, showing that this asymmetry persists across time. This effect also depends on the maturity of options in question: for longer maturities (see Figure 3.11, right graph) the support of the Lévy measure extends further to the left. Most of the densities we obtained are bimodal with one mode corresponding to small jumps that are hard to separate from a diffusion component and another clearly distinguishable mode corresponding to a large negative jump that can reflect the market participants’ “fear of crash” [11]. The logarithmic scale in the right graph of Figure 3.9 allows the tails to be seen more clearly.
3.6. NUMERICAL AND EMPIRICAL TESTS 125 4 3.5 3 2.5 2 1.5 1 0.5 Maturity 8 days Maturity 36 days Maturity 71 days Prior 10 1 Maturity 8 days Maturity 36 days 10 0 Maturity 71 days Prior 10 −1 10 −2 10 −3 10 −4 10 −5 0 −1 −0.5 0 0.5 10 −6 −1 −0.5 0 0.5 Figure 3.9: Lévy measures calibrated to DAX options on the same calendar date for three different maturities, linear and logarithmic scale. 1.1 1 0.9 0.8 Maturity 8 days, market Maturity 8 days, model Maturity 36 days, market Maturity 36 days, model Maturity 71 days, market Maturity 71 days, model 0.7 0.6 0.5 0.4 0.3 0.2 0.1 3500 4000 4500 5000 5500 6000 6500 7000 7500 Figure 3.10: Calibration quality for different maturities: market implied volatilities for DAX options against model implied volatilities. Each maturity has been calibrated separately.
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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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Page 97 and 98: 3.1. DISCRETIZING THE CALIBRATION P
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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174 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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