The evaluation of American options in a stochastic volatility model with jumps: An efficient finite element approach.

We consider the problem of pricing American options in the framework of a well-known stochastic volatility model with jumps, the Bates model. According to this model the asset price is described by a jump-diffusion stochastic differential equation in which the jump term consists of a Lévy process of compound Poisson type, while the volatility is modeled as a CIR-type process correlated with the asset price. Pricing American options under the Bates model requires to solve a partial integro-differential equation with final condition and boundary conditions prescribed on a free boundary. In this paper a numerical method to solve such a problem is proposed. In particular, first of all, using a Richardson extrapolation technique, the problem is reduced to a problem with fixed boundary. Then the problem obtained is solved using an ad-hoc finite element method which efficiently combines an implicit/explicit time stepping, an operator splitting technique, and a non-uniform mesh of right-angled triangles. Numerical experiments are presented showing that the option pricing algorithm developed in this paper is extremely accurate and fast. In particular it is significantly more efficient than other numerical methods that have recently been proposed for pricing American options under the Bates model.