A new application of fuzzy set theory to the Black–Scholes option pricing model

Abstract

The Black-Scholes Option pricing model (OPM) developed in 1973 has always been taken as the cornerstone of option pricing model. The generic applications of such a model are always restricted by its nature of not being suitable for fuzzy environment since the decision-making problems occurring in the area of option pricing are always with a feature of uncertainty. When an investor faces an option-pricing problem, the outcomes of the primary variables depend on the investor's estimation. It means that a person's deduction and thinking process uses a non-binary logic with fuzziness. Unfortunately, the traditional probabilistic B-S model does not consider fuzziness to deal with the aforementioned problems. The purpose of this study is to adopt the fuzzy decision theory and Bayes' rule as a base for measuring fuzziness in the practice of option analysis. This study also employs 'Fuzzy Decision Space' consisting of four dimensions, i.e. fuzzy state; fuzzy sample information, fuzzy action and evaluation function to describe the decision of investors, which is used to derive a fuzzy B-S OPM under fuzzy environment. Finally, this study finds that the over-estimation exists in the value of risk interest rate, the expected value of variation stock price, and in the value of the call price of in the money and at the money, but under-estimation exists in the value of the call price of out of the money without a consideration of the fuzziness. ey without a consideration of the fuzziness.