Valuing Derivatives Using Finite Difference Methods

Abstract

The valuation of options and other contingent claims typically requires the use of computational methods. A variety of computational methods are available. Of these, finite difference methods are among the most well known, most reliable, and most flexible. Finite difference techniques approximate the solution to the partial differential equation that Black, Scholes, and Merton demonstrated that any contingent claim must satisfy when the assumptions they made about stock price dynamics hold.1 The user of the method can control the accuracy of the approximation, with more accurate approximations requiring greater use of computational resources. Finite difference methods are an essential tool in any derivatives modeler’s kit because they can be adapted to a wide variety of problems, including the valuation of options that permit early exercise and many exotic derivatives. This chapter discusses the use of finite difference methods to value contingent claims. I first discuss several basic valuation methods—explicit integration, Monte Carlo integration, and finite difference methods—to put the latter in a more general context. I then set out the common finite difference methods—explicit, implicit, and Crank-Nicolson—used to solve for one-dimensional problems. I then discuss the use of finite difference methods in multidimensional problems, such as options on multiple underlying assets. Finally, I examine the pros and cons of finite difference methods relative to alternative numerical techniques.