ADI finite difference schemes for option pricing in the Heston model\n with correlation

Abstract

This paper deals with the numerical solution of the Heston partial\ndifferential equation that plays an important role in financial option pricing,\nHeston (1993, Rev. Finan. Stud. 6). A feature of this time-dependent,\ntwo-dimensional convection-diffusion-reaction equation is the presence of a\nmixed spatial-derivative term, which stems from the correlation between the two\nunderlying stochastic processes for the asset price and its variance.\n Semi-discretization of the Heston PDE, using finite difference schemes on a\nnon-uniform grid, gives rise to large systems of stiff ordinary differential\nequations. For the effective numerical solution of these systems, standard\nimplicit time-stepping methods are often not suitable anymore, and tailored\ntime-discretization methods are required. In the present paper, we investigate\nfour splitting schemes of the Alternating Direction Implicit (ADI) type: the\nDouglas scheme, the Craig & Sneyd scheme, the Modified Craig & Sneyd scheme,\nand the Hundsdorfer & Verwer scheme - each of which contains a free parameter.\n ADI schemes were not originally developed to deal with mixed\nspatial-derivative terms. Accordingly, we first discuss the adaptation of the\nabove four ADI schemes to the Heston equation. Subsequently, we present various\nnumerical examples with realistic data sets from the literature, where we\nconsider European call options as well as down-and-out barrier options.\nCombined with ample theoretical stability results for ADI schemes that have\nrecently been obtained in In 't Hout & Welfert (2007, Appl. Numer. Math.), we\narrive at three ADI schemes that all prove to be very effective in the\nnumerical solution of the Heston PDE with a mixed derivative term.\n