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European Barrier Options: Analytical and Numerical Valuation

### Nils Rosendahl

**Friday, 29 January 2010, 13:30**

Matfys library

**Abstract**:

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.