In this thesis we provide a detailed analysis of the power penalty method for American options using the theory of viscosity solutions. The method is a recent improvement of the classical penalty method, and in the thesis is motivated from quasi-variational inequality formulation. The first main result implies that in the class of viscosity solutions, there exists exactly one solution to the penalized equation which satisfies a natural growth condition. We used a comparison arguments and Perron's method to derive such a theorem. Secondly, using so called half-relaxed (weak) limit method, we proved that the power penalty approach converges uniformly to American option value (viscosity solution of the variational inequality). The convergence takes place in the space of locally bounded functions, and the error bounds of the convergence are derived. The order of the convergence rate allows us to achieve the required accuracy of the approximate solution with comparable large penalty term. We emphasize that the proof of the error estimates is based on the comparison arguments and therefore improves the present findings in the literature. We proposed "an easy to implement" numerical scheme for computing the value of American option and we compared it with the two other schemes which have appeared in the literature before. The results confirm the theoretical findings and the effectiveness and usefulness of the method.