  We propose a new high-order alternating direction implicit (ADI) finite
difference scheme for the solution of initial-boundary value problems of
convection-diffusion type with mixed derivatives and non-constant coefficients,
as they arise from stochastic volatility models in option pricing. Our approach
combines different high-order spatial discretisations with Hundsdorfer and
Verwer's ADI time-stepping method, to obtain an efficient method which is
fourth-order accurate in space and second-order accurate in time. Numerical
experiments for the European put option pricing problem using Heston's
stochastic volatility model confirm the high-order convergence.
