In this paper we introduce two methods for the efficient and accurate numerical solution of Black–Scholes models of American options: a penalty method and a front-fixing scheme. In the penalty approach the free and moving boundary is removed by adding a small, continuous penalty term to the Black–Scholes equation. The problem can then be solved on a fixed domain, thus removing the difficulties associated with a moving boundary. To gain insight into the accuracy of the method, we apply it to similar situations where the approximate solutions can be compared with analytical solutions. For explicit, semi-implicit and fully implicit numerical schemes we prove that the numerical option values generated by the penalty method mimic the basic properties of the analytical solution to the American option problem. In the front-fixing method we apply a change of variables to transform the American put problem into a nonlinear parabolic differential equation posed on a fixed domain. We propose both an implicit and an explicit scheme for solving this latter equation. Finally, the performance of the schemes is illustrated using a series of numerical experiments.