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112 CHAPTER 3. NUMERICAL IMPLEMENTATION C Least squares calibration Noisy market data K Figure 3.3: Estimating the noise level in the data. can be assessed from the bid-ask spreads by choosing w i := w (CM bid(T i, K i ) − CM ask(T i, K i )) 2 , where w is the normalization constant, determined from ∑ i w i = 1. However, the bid and ask quotes are not always available from option price data bases. On the other hand, it is known that at least for the options that are not too far from the money, the bid-ask spread in implied volatility units is of order of 1%. This means that to have errors proportional to bid-ask spreads, one must minimize the differences of implied volatilities and not those of the option prices. However, this would prohibitively increase the computational burden since one would have to invert numerically the Black-Scholes formula once for each data point at each minimization step. each A feasible solution is to minimize the squared differences of option prices weighted by the Black Scholes “vegas” evaluated at the points corresponding to market option prices. Denoting the implied volatility computed in the model Q for strike K and maturity T by Σ Q (T, K) and the corresponding market implied volatility by Σ M (T, K), we
3.5. NUMERICAL SOLUTION 113 have the following approximation: N∑ (Σ Q (T i , K i ) − Σ M (T i , K i )) 2 i=1 ≈ = N∑ i=1 ( ) ∂Σ 2 ∂C (C M(T i , K i ))|C Q (T i , K i ) − C M (T i , K i )| N∑ (C Q (T i , K i ) − C M (T i , K i )) 2 Vega 2 , (3.24) (Σ M (T i , K i )) i=1 where Vega denotes the derivative of the Black-Scholes option price with respect to volatility: Vega(σ) = S √ ( ( ) 1 S T n σ √ T log Ke −rT + 1 ) 2 σ√ T , with n denoting the CDF of the standard normal distribution. Therefore, by using w i = w Vega 2 i one can achieve the correct weighting of option prices without increasing the computational burden 1 (because w i can be computed in advance.) 3.5 Numerical solution of calibration problem Once the (discrete) prior P = P (A, ν P , γ P ), the regularization parameter α and the weights w i have been fixed, it remains to find the numerical solution of the calibration problem (2.27). We construct an approximate solution of the calibration problem (2.27) by minimizing an approximation of the calibration functional J α (Q), denoted by Ĵ α (Q) := ‖C M − ĈQ ‖ 2 w + αI(Q|P ), (3.25) where ĈQ is the approximate option price defined by Equation (3.31) below. The minimization is done over all Lévy processes Q ∈ M ∩ L + B with Lévy measures of the form (3.2). The calibration functional therefore becomes a function of a finite number of arguments: Ĵ α (Q) = Ĵα(q 0 , . . . , q M−1 ), (3.26) To minimize (3.26) we use a variant of the popular Broyden-Fletcher-Goldfarb-Shanno (BFGS) variable metric algorithm. For a description of the algorithm see [80]. This algorithm requires 1 Since vegas can be very small for out of the money options, to avoid giving them too much weight one should impose a lower bound on the vegas used for weighting, i.e., use weights of the form w i = constant C ∼ 10 −2 ÷ 10 −1 . w (Vega i ∧C) 2 with some
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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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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
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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
Page 131: Part II Multidimensional modelling
Page 134 and 135: 134 CHAPTER 4. DEPENDENCE OF LEVY P
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162 CHAPTER 4. DEPENDENCE OF LEVY P
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164 CHAPTER 4. DEPENDENCE OF LEVY P
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166 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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