Numerical Solutions of Financial Partial Differential Equations

Abstract

This paper provides a general survey of important PDE numerical solutions and studies in detail of certain numerical methods specific to finance with programming samples. These important numerical solutions for financial PDEs include finite difference, finite volume, and finite element. Finite difference is simple to discretize and easy to implement; however, explicit method is not guaranteed stable. The finite volume has an advantage over finite difference in that it does not require a structured mesh. If a structured mesh is used, as in most cases of pricing financial derivatives, the finite volume and finite difference method yield the same discretization equations. Finite difference method can be considered a special case of the finite element method. In general, the finite element method has better quality in obtaining an approximated solution when compared to the finite difference method. Since most PDEs in financial derivatives pricing have simple boundary conditions, the implicit method of finite difference is preferred to finite element method in applications of financial engineering.