C# for Financial Markets

10

Advanced Lattices and Finite Difference Methods

10.1 INTRODUCTION AND OBJECTIVES

In this chapter we introduce a number of numerical methods to price derivatives. First, we discuss the trinomial method and we implement it in C#. It is more robust and accurate than the binomial method introduced in Chapter 9. Second, we introduce explicit finite difference methods and their relationship to the trinomial method, including the Alternating Direction Explicit (ADE) method which is an unconditionally stable and explicit finite difference method for n-factor derivative models. For a detailed discussion of ADE and some of its applications, see Pealat and Duffy 2011.

This chapter can be read from a number of viewpoints:

• We implement the trinomial method in C#. We can compare the solution with other documented solutions (for example, pseudocode, VBA and Matlab). The rationale is that we discuss a problem that readers already understand.
• We apply a number of design patterns to promote the flexibility of the C# implementations.
• A discussion of the ADE finite difference method and a comparison with the trinomial method. In particular, we see the advantages of using ADE in terms of performance, accuracy and ease of implementation. We discuss the mathematical foundations of ADE in Appendix 3.

This chapter is an introduction to explicit finite difference schemes for the one-factor Black-Scholes equation. The approach taken is to develop modular software to price one-factor European and American ...