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In this paper, efficient numerical methods are developed for the pricing of American options governed by the Black–Scholes equation. The front-fixing technique is first employed to transform the free boundary of optimal exercise prices to some a priori known temporal line for a one-dimensional parabolic problem via the change of variables. The perfectly matched layer (PML) technique is then applied to the pricing problem for the effective truncation of the semi-infinite domain. Finite element methods using the respective continuous and discontinuous Galerkin discretization are proposed for the resulting truncated PML problems related to the options and Greeks. The free boundary is determined by Newton’s method coupled with the discrete truncated PML problem. Stability and nonnegativeness are established for the approximate solution to the truncated PML problem. Under some weak assumptions on the PML medium parameters, it is also proved that the solution of the truncated PML problem converges to that of the unbounded Black–Scholes equation in the computational domain and decays exponentially in the perfectly matched layer. Numerical experiments are conducted to test the performance of the proposed methods and to compare them with some existing methods. [ABSTRACT FROM PUBLISHER]