ABSTRACT
In this chapter we will discuss the use of the finite difference techniques discussed in Chapter 3 for pricing derivatives under models for which a partial differential equation (PDE) describing derivative prices can be formulated. We start with geometric Brownian motion. As discussed in Section 1.2.1, under geometric Brownian motion the price of an asset follows the following stochastic differential equation (SDE):
dSt = (r − q)Stdt+ σStdWt (4.1) And we know the value of options on that asset satisfy the Black-Scholes partial differential equation
∂v
∂t +
σ2S2
∂2v
∂S2 + (r − q)S ∂v
∂S = rv(S, t) (4.2)
with terminal and boundary conditions which are dependent on the type of the option we wish to price.
