Hybrid Finite–Difference/Pseudospectral Methods for the Heston and Heston–Hull–White Partial Differential Equations

We propose a hybrid spatial finite-difference/pseudospectral discretization for European option-pricing problems under the Heston and Heston–Hull–White models. In the direction of the underlying asset, where the payoff profile is nonsmooth, we use a standard central second-order finite-difference scheme, whereas we use a Chebyshev collocation method in the other spatial dimensions. In the time domain, we employ alternating direction implicit schemes to efficiently decompose the system matrix into simpler one-dimensional problems. This approach allows us to compute numerical solutions, which are second-order accurate in time and exhibit spectral accuracy in the spatial domains except for the asset direction. The numerical experiments reveal that the proposed scheme outperforms the standard second-order finite-difference scheme in terms of accuracy versus runtime and shows an unconditionally stable behavior.