Interval LU-fuzzy arithmetic in the Black and Scholes option pricing

In financial markets people have to cope with a lot of uncertainty while making decisions. Many models have been introduced in the last years to handle vagueness but it is very difficult to capture together all the fundamental characteristics of real markets. Fuzzy modeling for finance seems to have some challenging features describing the financial markets behavior; in this paper we show that the vagueness induced by the fuzzy mathematics can be relevant in modelling objects in finance, especially when a flexible parametrization is adopted to represent the fuzzy numbers. Fuzzy calculus for financial applications requires a big amount of computations and the LU-fuzzy representation produces good results due to the fact that it is computationally fast and it reproduces the essential quality of the shape of fuzzy numbers involved in computations. The paper considers the Black and Scholes option pricing formula, as long as many other have done in the last few years. We suggest the use of the LU-fuzzy parametric representation for fuzzy numbers, introduced in Guerra and Stefanini and improved in Stefanini, Sorini and Guerra, in the framework of the Black and Scholes model for option pricing, everywhere recognized as a benchmark; the details of the computations by the interval fuzzy arithmetic approach and an illustrative example are also incuded.