Pricing European options based on the fuzzy pattern of Black–Scholes formula

The application of fuzzy sets theory to the Black–Scholes formula is proposed in this paper. Owing to the fluctuation of financial market from time to time, some input parameters in the Black–Scholes formula cannot always be expected in the precise sense. Therefore, it is natural to consider the fuzzy interest rate, fuzzy volatility and fuzzy stock price. The fuzzy pattern of Black–Scholes formula and put–call parity relationship are then proposed in this paper. Under these assumptions, the European option price will turn into a fuzzy number, so that a financial analyst who can pick any European option price with an acceptable belief degree for the later use. In order to obtain the belief degree, an optimization problem has to be solved.