The convergence of a penalty method for solving the discrete regularized American option valuation problem is studied.

Sufficient conditions are derived which both guarantee convergence of the nonlinear penalty iteration and ensure that the iterates converge monotonically to the solution. In addition, we show that the penalty iteration has finite termination, which means that the American Constraint can be enforced with very high precsion with a small number of iterations.

The efficiency and quality of solutions obtained using the implicit penalty method are compared with those produced with the commonly used technique of handling the American constraint explicitly. Convergence rates are studied as the timestep and mesh size tend to zero.