Option pricing models for fuzzy decision making in financial engineering

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.