An extremely efficient numerical method for pricing options in the Black-Scholes model with jumps

We propose a new numerical method for pricing options in the Black-Scholes model with jumps. Specifically, we consider the partial integro-differential problem that yields the option price, and we solve it by means of a finite difference scheme that combines a fixed-point iteration technique and a repeated space-time Richardson extrapolation procedure. Such an approach turns out to be not only extremely accurate and fast but also very simple to implement, since the use of fast convolution techniques for handling the jump integral is not required. Numerical experiments are presented in which vanilla, barrier, and American options are considered.