The main result of this paper is a probabilistic proof of the penalty method for approximating the price of an American put in the Black-Scholes market. The method gives a parametrized family of partial differential equations, and by varying the parameter the corresponding solutions converge to the price of an American put. For each PDE the parameter may be interpreted as a rationality parameter of the holder of the option. The method may be extended to other valuation situations given as an optimal stopping problem with no explicit solution. The method may also be used for valuations where actors do not behave completely rationally but instead have randomness affecting their choices. The rationality parameter is a measure for this randomness.
