  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.
