A discrete-time model of American put option in an uncertain environment

Abstract

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 permissible range of the writer’s (seller’s) optimal expected price in the American put option is presented, and the meaning and properties of the optimal expected prices are discussed in numerical examples. In a numerical example, the discrete-time approximation model is discussed for the continuous-time model.