Negative coefficients in two-factor option pricing models

The importance of positive coefficients in numerical schemes is frequently emphasized in the finance literature. This topic is explored in detail in this paper, in the particular context of two factor models. First, several two factor lattice type methods are derived using a finite difference/finite element methodology. Some of these methods have negative coefficients, but are nevertheless stable and consistent. Second, we outline the conditions under which finite volume/element methods applied to two factor option pricing partial differential equations give rise to discretizations with positive coefficients. Numerical experiments indicate that constructing a mesh which satisfies positive coefficient conditions may not only be unnecessary, but in some cases even 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 conditions as the mesh spacing approaches zero. This finding is of significance since, for arbitrary diffusion tensors, it may not be possible to construct a positive coefficient discretization for a given set of nodes. Acknowledgment: This work was supported by the Natural Sciences and Engineering Research Council of Canada, the Social Sciences and Humanities Research Council of Canada, and the Royal Bank of Canada. ∗A previous version of this paper was entitled “Diffusion Operators and Meshes in Option Pricing”. †rzvan@bear.com ‡paforsyt@elora.math.uwaterloo.ca §kvetzal@watarts.uwaterloo.ca