Smart Expansion and Fast Calibration for Jump Diffusion

26 Pages Posted: 1 Jan 2008 Last revised: 30 Dec 2009

Eric Benhamou

Université Paris Est - Université Paris Est-Creteil

Emmanuel Gobet

Ecole Polytechnique, Paris - Centre de Mathematiques Appliquees

Mohammed Miri

Thomson Reuters

Date Written: September 17, 2008

Abstract

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.

Keywords: asymptotic expansion, Malliavin calculus, volatility skew and smile, small diffusion process, small jump frequency/size

JEL Classification: G13

Suggested Citation

Benhamou, Eric and Gobet, Emmanuel and Miri, Mohammed, Smart Expansion and Fast Calibration for Jump Diffusion (September 17, 2008). Available at SSRN: https://ssrn.com/abstract=1079627 or http://dx.doi.org/10.2139/ssrn.1079627

Eric Benhamou

Université Paris Est - Université Paris Est-Creteil ( email )

61 avenue du Général de Gaulle
Créteil, 940000
France

Emmanuel Gobet

Ecole Polytechnique, Paris - Centre de Mathematiques Appliquees ( email )

Palaiseau Cedex, 91128
France

Mohammed Miri (Contact Author)

Thomson Reuters ( email )

6 Bd Haussman
France, FL 75009
France

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