  We derive a new high-order compact finite difference scheme for option
pricing in stochastic volatility jump models, e.g. in Bates model. In such
models the option price is determined as the solution of a partial
integro-differential equation. The scheme is fourth order accurate in space and
second order accurate in time. Numerical experiments for the European option
pricing problem are presented. We validate the stability of the scheme
numerically and compare its performance to standard finite difference and
finite element methods. The new scheme outperforms a standard discretisation
based on a second-order central finite difference approximation in all our
experiments. At the same time, it is very efficient, requiring only one initial
$LU$-factorisation of a sparse matrix to perform the option price valuation.
Compared to finite element approaches, it is very parsimonious in terms of
memory requirements and computational effort, since it achieves high-order
convergence without requiring additional unknowns, unlike finite element
methods with higher polynomial order basis functions. The new high-order
compact scheme can also be useful to upgrade existing implementations based on
standard finite differences in a straightforward manner to obtain a highly
efficient option pricing code.
