Finite Difference Methods

Abstract

Here we give a brief introduction to finite difference methods. We first explain the implicit method; then we move to the explicit method. The former is more robust, in that it converges to the solution of a partial differential equation as the discrete increments of the state variables approach zero. The latter, instead, does not necessarily converge to that solution, but is simpler; besides, it bears a strong resemblance to the trinomial lattice approach. Along the way we explain the main differences between the two. As an application, we are going to evaluate a simple real option of the European type. Both the explicit and the implicit method involve considering three nodes at one date and one node at another contiguous date. Given their relative advantages and weaknesses, there have been efforts to devise better schemes. One such scheme considers the three points at the two dates. This is the so-called Crank-Nicolson method, which is subject to lower truncation errors than the other two methods. Last, we illustrate its use in two examples.