Option pricing in a stochastic and fuzzy environment

A mathematical model for American put option and European options 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 judgement. Two models in American options with uncertainty, the case with an expiration date and the perpetual option case, are dealt with under a reasonable assumption. Prices of European call/put options with uncertainty are given and their valuation and properties are discussed under a reasonable assumption. The buyer's/writer's (seller's) rational range of optimal expected price in the options are presented and the meaning and properties of the optimal expected prices are discussed. The hedging strategies are also considered for marketability of the European options for portfolio selection.