European Barrier Options: Analytical and Numerical Valuation

Nils Rosendahl

Friday, 29 January 2010, 13:30
Matfys library

European barrier options offer investors opportunities to invest at a lower cost compared to the corresponding ordinary European options. European barrier options can either be of out type, meaning that the option becomes worthless if the barrier is crossed, or of in type meaning that a barrier has to be crossed, otherwise the payoff is zero. This paper derives the well known closed form solution for European barrier options, where the underlying is assumed to follow a geometric Brownian motion. Two numerical methods are applied to approximate the price of a barrier option , the Monte Carlo method and the finite difference method. The Monte Carlo method is first implemented for a barrier option written on an asset that is assumed to follow the so called Cox Ingersoll Ross dynamics as well as a Poisson driven jump process. The finite difference method is used to valuate a barrier option on an underlying asset that follows a geometric Brownian motion. Two implementations of the explicit finite difference method are made and compared, where the second is an improvement of the first in the sense that it on forehand can determine the stability of the implementation.