This paper presents a comparative effectiveness of stochastic approximation method and pseudo inversion method for American option valuation under the Black-Scholes model. The stochastic approximation method and pseudo inversion method base its analysis on a drifted financial derivative system. With finer discretization, space nodes and time nodes, we demonstrate that the drifted financial derivative system can be efficiently and easily solved with high accuracy, by using a stochastic approximation method and pseudo inversion method. The stochastic approximation method proves to be faster in pricing an American options than the pseudo inversion method which needs the system to be stabilized for its accuracy. An illustrative example is given in concrete setting.