This thesis is concerned with the numerical solution of the American option valuation problem formulated as a free boundary/initial value model. While other studies have focused on modified pricing model, formulating the problem as a non-linear model, using the front-fix method to fix the moving boundary, or trying to find semi-/analytical solutions to the problem, we introduce and analyze a front-tracking (FT) finite difference method (FDM) based on original Black-Scholes Model. The basis of the B-S Model, FDM, FT and options theory will be introduced. The numerical experiments performed indicate that the front tracking method considered is an efficient alternative for approximating simultaneously the option value and optimal exercise boundary functions associated with the valuation problem. We also extend the study to pricing options with stochastic volatility using Heston Model, as well as valuation of multi-asset options.