Smart Expansion and Fast Calibration for Jump Diffusion
26 Pages Posted: 1 Jan 2008 Last revised: 30 Dec 2009
Date Written: September 17, 2008
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
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