A Finite Element Discretization Method for Option Pricing with the Bates Model

In the present paper we present a finite element approach to option pricing in the framework of a well-known stochastic volatility model with jumps, the Bates model. In this model the asset log-returns are assumed to follow a jump-diffusion model where the jump component consists of a Lévy process of compound Poisson type, while the volatility dynamics is described by a stochastic differential equation of CIR type, with a mean-reverting drift term and a diffusion component correlated with that of the log-returns. Like in all the Lévy models, the option pricing problem can be formulated in terms of an integro-differential equation: for the Bates model the unknown C(S, V, t) (the option price) of the pricing equation depends on three independent variables and the differential operator part turns out to be of parabolic kind, while the nonlocal integral operator is calculated with respect to the Lévy measure of the jumps. In this paper we will present a variational formulation of the problem suitable for a finite element approach. After evaluating European options as a benchmark, the method will be applied to Barrier (Down and Out) options.