American option pricing is a popular topic and there are many efficient methods to price the option. In this thesis, an American option is priced by Least-Square Monte Carlo Method and the pseudo-random numbers are modified with the quasirandom numbers to replace the pseudo-random numbers, this purpose try to make the calculation more accurate. On the other hand, Black-Scholes model has been widely used in option pricing, but the fat-tailed phenomenon often exists in the financial markets. Option pricing with Black-Scholes model is based on a normal distribution, if a distribution is actually a fat-tailed one, the model will under-price options that are far out of the money, which affects the validity of this formula. Therefore, Merton’s Jump-Diffusion model is well considered, which contains a jump component in the stock price process, that is to say more suitable to model the financial markets.