9. The Partial Differential Equation Approach Under Geometric Brownian Motion

The Partial Differential Equation (PDE) Approach is one of the techniques in solving the pricing equations for financial instruments. The solution technique of the PDE approach is the Fourier transform, which reduces the problem of solving the PDE to one of solving an ordinary differential equation (ODE). The Fourier transform provides quite a general framework for solving the PDEs of financial instruments when the underlying asset follows a jump-diffusion process and also when we deal with American options. This chapter illustrates that in the case of geometric Brownian motion, the ODE determining the transform can be solved explicitly. It shows how the PDE approach is related to pricing derivatives in terms of integration and expectations under the risk-neutral measure.