Financial Derivative and Energy Market Valuation: Theory and Implementation in MATLAB

Chapter 6: Finite Difference Methods

6.1 Introduction

The finite difference method approximates the evolution of a derivative price on a grid that typically represents time and the underlying asset price but can include other factors such as the risk-free rate or volatility. The evolution is dictated by the particular partial differential equation (PDE) of the asset model and the derivative. The standard approach is to start with the known values at expiration, and to repeatedly solve the set of derivative values at the previous time step. The partial derivatives in the PDE are approximated with explicit values available at the forward time step or with implicit values not yet known at the current time step. The explicit approach directly parallels a trinomial tree, and the numerical implementation is simple. The main constraint for the explicit finite difference scheme is that asset price spacing relative to the time step interval must be less than the Courant stability limit. The asset price spacing relative to the time step interval ratio can be selected to mimic a normal distribution in both the explicit PDE and the trinomial tree. The implicit PDE method removes this stability limit, but a pseudoinversion technique is necessary to solve for the interdependent values within the same time layer on the grid.

In general, the finite difference method is effective for solving derivatives that can be described moving backwards in time such as American options. Additionally, Greeks ...