A discrete-time mathematical model for American put option with uncertainty is presented, and the randomness and fuzziness are evaluated by both probabilistic expectation and fuzzy expectation defined by a possibility measure from the viewpoint of fuzzy expectation, taking account of decision-maker's subjective judgment. An optimality equation for the optimal stopping problem in a fuzzy stochastic process is derived and an optimal exercise time is given for the American put option. It is shown that the optimal fuzzy price is a solution of the optimality equation under a reasonable assumption. The writer's (seller's) permissible range of optimal expected price in the American put option is presented and the meaning and properties of the optimal expected prices are discussed in a numerical example.