This paper deals with American call and put options. Determining the fair price and the free boundary of an American option is a very difficult problem since they depends on each other. This paper presents numerical algorithms of finite element method based on the three-level scheme to compute both the price and the free boundary. One algorithm is designed for American call options and the other one for American put options. These algorithms are formulated on the system of the Jamshidian equation for the option price and the free boundary. Here, the Jamshidian equation is of a kind of the nonhomogeneous Black-Scholes equations. We prove the existence and uniqueness of the numerical solution by the Lax-Milgram lemma and carried out extensive numerical experiments to compare with various methods.