  We present high-order compact schemes for a linear second-order parabolic
partial differential equation (PDE) with mixed second-order derivative terms in
two spatial dimensions. The schemes are applied to option pricing PDE for a
family of stochastic volatility models. We use a non-uniform grid with more
grid-points around the strike price. The schemes are fourth-order accurate in
space and second-order accurate in time for vanishing correlation. In our
numerical convergence study we achieve fourth-order accuracy also for non-zero
correlation. A combination of Crank-Nicolson and BDF-4 discretisation is
applied in time. Numerical examples confirm that a standard, second-order
finite difference scheme is significantly outperformed.

Computational Finance;Numerical Analysis