A Generalized SMART Fuzzy Disjunction of Volatility Indicators Applied to Option Pricing in a Binomial Model

In this paper we extend our previous contributions on the elicitation of the fuzzy volatility membership function in option pricing models. More specifically we generalize the SMART disjunction for a multi-model volatility behavior (Uniform, LogNormal, Gamma, ...) and within a double-source (direct vs. indirect) information set. The whole procedure is then applied to the Cox-Ross-Rubinstein framework for option pricing on the S&P500 Index where the historical volatility, computed from the Index returns' time series, and the VIX Index observed data are respectively considered as the direct and indirect sources of knowledge. A suitable distance among the resulting fuzzy option prices and the market bid-ask spread make us appreciate the proposed procedure against the classical fuzzy mean.