European call options under uncertainty

A model of European options with uncertainty of both randomness and fuzziness in output is presented, by introducing fuzzy logic to the stochastic financial model. The randomness and fuzziness in the systems are evaluated by both probabilistic expectation and fuzzy expectation, taking account of seller's/buyer's subjective judgment. Prices of European call/put options with uncertainty are given and their valuation and properties are discussed under a reasonable assumption. This paper demonstrates Black-Scholes formula to give rational expected price of the European options and buyer's/seller's permissible range of expected prices. The meaning and properties of rational expected prices are discussed in a numerical example. The hedging strategies are also considered for marketability of the European options for portfolio selection.