20.1 Introduction and Objectives
In this chapter we give an overview of linear one-factor partial differential equations that are used to price derivatives. Mathematically, these are time-dependent convection–diffusion–reaction equations. We study their qualitative properties including transforming them to more manageable forms. In most cases these PDEs do not have an analytical solution and for this reason we use numerical methods. In this book we have chosen the popular Finite Difference Method (FDM) because it is a mature branch of numerical analysis, it is easy to implement and it has good run-time performance and accuracy characteristics. We are interested in the PDE that describes the Black–Scholes equation and its generalisations and many of our results are applicable to a range of one-factor and two-factor PDEs.
The goal of this chapter is to introduce enough mathematics to model partial differential equations, approximate them by FDM algorithms and then map these algorithms to C++. In particular, we focus on option pricing using the Black–Scholes PDE. We assume that the reader has some knowledge of PDEs and FDM. For background, see Thomas (1995) for a discussion of FDM for time-dependent parabolic PDEs and their numerical approximation by FDM and Duffy (2006) for applications to computational finance, including a numerical analysis of finite difference schemes.
A good way to prepare for the material ...