Quantitative Finance

7Approximating PDEs: Finite Difference Methods

7.1 Introduction

In this chapter, we discuss a different approach for approximating the solutions of partial differential equations (PDEs), namely, finite difference methods and functional approximation methods. The finite difference methods approximate the differential operator by replacing the derivatives in the equation using differential quotients. The domain is partitioned in space and time, and approximations of the solution are computed at the space and time points. For the functional approximation method, we will study the numerical problems arising in the computation of the implied volatilities and the implied volatility surface.

In the previous chapter, the central idea in the construction of trees is to divide the time to maturity into intervals of length . The Finite Difference methods take this idea one step further dividing the space into intervals as well. Specifically it divides the space domain into equally spaced nodes at distance apart, and the time domain into equally spaced nodes a distance apart in the return ...