  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.
