Finite Difference Methods

We review the form of the Black-Scholes-Merton PDE, the general form of its boundary conditions, and a transformation that converts the PDE into the canonical heat equation form. We compare the stability and local error estimates for three schemes: an explicit scheme, an implicit scheme, and the Crank-Nicolson scheme. The last of these is both unconditionally stable and rapidly converging. We then demonstrate how these methods, having been developed in quite a general setting, may be applied to the valuation of European options and barrier options and implement a variation that allows us to numerically approximate the optimal exercise boundary and hence to value American and Bermudan options.