Using fuzzy sets theory and Black–Scholes formula to generate pricing boundaries of European options

The application by using extension principle in fuzzy sets theory to the Black–Scholes formula is proposed in this paper. Owing to the fluctuation of financial market from time to time, the risk-free interest rate, volatility and stock price may occur imprecisely in the real world. Therefore, it is natural to consider the fuzzy interest rate, fuzzy volatility and fuzzy stock price in the financial market. Under these assumptions, the European call and put option price will turn into the fuzzy numbers, and the extension principle will be invoked to generate the pricing boundaries of the European call and put options. This will make the financial analyst who can pick any reasonable European option price with an acceptable belief degree for his/her later financial analysis. In order to obtain the belief degree, an optimization problem has to be solved.