European Options in BS Markets - Computational Methods for Quantitative Finance - page 47

Chapter 4
European Options in BS Markets

In the last chapters, we explained various methods to solve partial differential equa-
tions. These methods are now applied to obtain the price of a European option. We
assume that the stock price follows a geometric Brownian motion and show that the
option price satisfies a parabolic PDE. The unbounded log-price domain is local-
ized to a bounded domain and the error incurred by the truncation is estimated. It
is shown that the variational formulation has a unique solution and the discretiza-
tion schemes for finite element and finite differences are derived. Furthermore, we
describe extensions of the Black-Scholes model, like the constant elasticity of vari-
ance (CEV) and the local volatility model.

4.1 Black-Scholes Equation

Let X be the solution of the SDE (1.2), where we assume that the coefficients b, σ
are independent of time t and satisfy the assumptions of Theorem 1.2.6. Further
assume r(x) to be a bounded and continuous function modeling the riskless inter-
est rate. We want to compute the value of the option with payoff g which is the
conditional expectation

e − t r(X s ) d s g(X T )

x .

V(t,x) = E | X t = (4.1)

We show that V(t,x) is a solution of a deterministic partial differential equation.
Therefore, we first relate to the process X a differential operator A , the so-called

infinitesimal generator of the process X .

A

Proposition 4.1.1 Let denote the differential operator which is , for functions

C 2 (

f ∈ R ) with bounded derivatives , given by

1
2 σ 2 (x)∂ xx f(x)

( A f )(x) = + b(x)∂ x f(x). (4.2)