  This paper deals with the numerical solution of the Heston partial
differential equation that plays an important role in financial option pricing,
Heston (1993, Rev. Finan. Stud. 6). 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 a
non-uniform grid, 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 equation. 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, Appl. Numer. Math.), 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.
