### Abstract:

This dissertation considers a problem of pricing barrier options when the dynamics of the asset prices (X(t)) are driven by the mean reverting process, the market/asset price X(t) is obtain from mean reversion model and a Black-Scholes PDE model for pricing barrier options under mean reversion model is obtained upon using Itoˆ formula. Through the Homotopy Analysis Method (HAM) the price of the chosen barrier option (up−and−out European call) that satisfies the Black-Scholes PDE model was deter- mined. Thus, through HAM we can determine approximated prices of barrier options when the dynamics of the prices are driven by the mean reverting process (Liao, 2004).
Lastly the analysis is conducted to observe the behaviour of the option price when value of one parameter increases while the value of the other two parameters remain constant. The analysis shows that the option price tends to increase with the increase of the value of the parameter for the case of volatility and degree of mean reversion while for interest rate the option price decreases when interest rate increases. In all cases it is observed that early exercise is better than late exercise to owner of the option since the option price tends to decrease as time increases also to minimize risk owner of the option should exercise the option when the volatility of the market become large.
It is recommended that in future, areas of interest for research related to this study are; first, finding the option price by using direct integration after obtaining a reflection principle which is useful in determining the joint distribution of the Itoˆ integral and secondly, finding the price which is a closed form solution by using Laplace transform.