The numerical solution of two-factor option pricing models

Abstract

This work develops a nonconservative finite volume approach for solving two-dimensional partial differential equation option pricing models. The finite volume method is more flexible than finite difference schemas which are often described in the finance literature and frequently used in practice. Moreover, the finite volume method is more flexible than finite difference schemas which are often described in the finance literature and frequently used in practice. Moreover, the finite volume method naturally handles cases where the underlying partial differential equation becomes convection dominated or degenerate. This work will demonstrate how a variety of two-dimensional valuation problems can all be solved using the same approach. The generality of the approach is in part due to the fact that changes caused by different model specifications are localized.

For convection dominated pricing problems, a compact positive coefficient scheme is developed. The positive coefficient scheme allows accurate solutions of degenerate problems to be obtained with essentially the same computational cost as nondegenerate problems.

The conditions under which finite volume/element methods, when applied to the two-factor option pricing equation, give rise to discretizations with positive coefficients are also outlined in this work. The importance of positive coefficients in numerical schemes is often stressed in the finance literature. Numerical experiments indicate that constructing a mesh which satisfies the positive coefficient conditions may not be necessary, and in some cases appears to even be detrimental. As well, it is shown that schemes with negative coefficients due to the discretization of the diffusion term satisfy approximate local maximum and minimum principles as the mesh spacing approaches zero. This finding is of significance since, for arbitrary diffusion tensors, it may not always be possible to construct a positive coefficient discretization for a given set of nodes.

In addition, it is shown that several lattice methods are equivalent to known finite difference/element schemes.