A new evaluation of mean value for fuzzy numbers and its application to American put option under uncertainty

Abstract

This paper discusses two topics on fuzzy random variables in decision making. One is a new evaluation method of fuzzy random variables, and the other is to present a mathematical model in financial engineering by fuzzy random variables. The evaluation method is introduced as mean values defined by fuzzy measures, and it is also applicable to fuzzy numbers and fuzzy stochastic process defined by fuzzy random variables. The other is to apply the method to an American put option with uncertainty formulated as an optimal stopping problem for fuzzy random variables, and the randomness and fuzziness are estimated by the probabilistic expectation and the mean values. The optimal expected price of the American put option is given by the mean values with decision maker's subjective parameters. Numerical examples are given to illustrate our idea.