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184 CHAPTER 5. APPLICATIONS OF LEVY COPULAS 0.04 0.035 Tail dependence No tail dependence 0.04 0.035 Tail dependence No tail dependence 0.03 0.03 0.025 0.025 0.02 0.02 0.015 0.015 0.01 0.01 0.005 0.005 0 0.95 1 1.05 0 0.95 1 1.05 Figure 5.6: Prices of options on weighted average (left) and of best-of options (right) for two different dependence patterns. Pattern 1 Strong tail dependence: θ = 10 and η = 0.75. The scatter plot of returns is shown in Figure 5.5, left graph. Although the signs of returns may be different, the probability that the returns will be large in absolute value simultaneously in both components is very high. Pattern 2 Weak tail dependence: θ = 0.61 and η = 0.99. The scatter plot of returns in shown in Figure 5.5, right graph. With this dependence structure the returns typically have the same sign but their absolute values are not correlated. In each of the two cases, a sample of 1000 realizations of the couple (XT 1 , X2 T ) with T = 0.02 (one-week options) was simulated using the procedure described in Example 5.3. The cutoff parameter τ (see Equation (5.14)) was taken equal to 1000, which lead to limiting the average number of jumps for each trajectory to about 40. For this value of τ, Ui −1 (τ) is of order of 10 −19 for both assets. Since for the variance gamma model the convergence of U −1 to zero as τ → ∞ is exponential, the error resulting from the truncation of small jumps is of the same order, hence, negligible. Figure 5.6 shows the prices of basket options, computed for different strikes with dependence patterns given above. The initial asset prices were S0 1 = S2 0 = 1, and the interest rate was taken to be r = 0.03. For the option on weighted average, the weights w i were both equal to 0.5 and for the best-of option the coefficient was N = 1. The prices of European options, used for
5.3. TWO-DIMENSIONAL VARIANCE GAMMA MODEL 185 variance reduction, were computed using the Fourier transform algorithm described in Chapter 1. The standard deviation of Monte Carlo estimates of option prices was below 2 · 10 −4 at the money in all cases. The difference between option prices computed with and without tail dependece is clearly important for both types of options: as seen from Figure 5.6, neglecting tail dependence may easily lead to a 10% error on the option price at the money. On the other hand, this example shows that using Lévy copulas allows to take into account the tail dependence and discriminate between two situations that would be undistinguishable in a log-normal framework.
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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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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
Page 198 and 199: 198 BIBLIOGRAPHY
Page 200: 200 INDEX tightness, 40 levycalibra
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