  In the present paper we present a finite element approach for 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 Levy process of
compound Poisson type, while the volatility behavior 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 Levy models, the option pricing problem can be formulated in terms of an
integro-differential equation: for the Bates model the unknown F(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 Levy measure of
the jumps. In this paper we will present a variational formulation of the
problem suitable for a finite element approach. The numerical results obtained
for european options will be compared with those obtained with different
methods.
