This paper deals with the numerical solution of the Heston partial differential equation (PDE) that plays an important role in financial option pricing theory, Heston (1993). A feature of this time-dependent, two-dimensional convection-diffusion-reaction equation is the presence of a mixed spatial-derivative term, which stems from the correlation between the two underlying stochastic processes for the asset price and its variance.
Semi-discretization of the Heston PDE, using finite difference schemes on non-uniform grids, gives rise to large systems of stiff ordinary differential equations. For the effective numerical solution of these systems, standard implicit time-stepping methods are often not suitable anymore, and tailored time-discretization methods are required. In the present paper, we investigate four splitting schemes of the Alternating Direction Implicit (ADI) type: the Douglas scheme, the Craig-Sneyd scheme, the Modified Craig-Sneyd scheme, and the Hundsdorfer–Verwer scheme, each of which contains a free parameter.
ADI schemes were not originally developed to deal with mixed spatial-derivative terms. Accordingly, we first discuss the adaptation of the above four ADI schemes to the Heston PDE. Subsequently, we present various numerical examples with realistic data sets from the literature, where we consider European call options as well as down-and-out barrier options. Combined with ample theoretical stability results for ADI schemes that have recently been obtained in In ’t Hout & Welfert (2007, 2009) we arrive at three ADI schemes that all prove to be very effective in the numerical solution of the Heston PDE with a mixed derivative term. It is expected that these schemes will be useful also for general two-dimensional convection-diffusion-reaction equations with mixed derivative terms.