Smart expansion and fast calibration for jump diffusion

Eric Benhamou (1) , Emmanuel Gobet (2) , Mohammed Miri (1, 2)

1 Pricing Partners
2 MATHFI - Mathématiques financières

Using Malliavin calculus techniques, we derive an analytical formula for the price of European options, for any model including local volatility and Poisson jump process. We show that the accuracy of the formula depends on the smoothness of the payoff function. Our approach relies on an asymptotic expansion related to small diffusion and small jump frequency/size. Our formula has excellent accuracy (the error on implied Black-Scholes volatilities for call option is smaller than 2 bp for various strikes and maturities). Additionally, model calibration becomes very rapid.

Mots clés: asymptotic expansion; Malliavin calculus; volatility skew and smile; small diffusion process; small jump frequency/size

Domaines: Probabilités [math.PR]; Economies et finances

Eric Benhamou, Emmanuel Gobet, Mohammed Miri. Smart expansion and fast calibration for jump diffusion. Finance and Stochastics, 2009, 13 (4), pp.563-589. ⟨10.1007/s00780-009-0102-3⟩. ⟨hal-00200395v2⟩