Market Application of the Fuzzy-Stochastic Approach in the Heston Option Pricing Model

The present study analyzes the extra insights that option pricing models may achieve when uncertainty about parameters is modeled through fuzzy numbers. Specifically, we consider the Heston stochastic volatility model, which assumes that stock price changes and their instantaneous variance evolve as a bivariate, possibly correlated, diffusive process. The original Heston model provides a quasi-closed formula for the pricing of several derivative products such as European options. By applying the fuzzy extension principle, we generalize the model to the case of fuzzy parameters; given their shape we are able to derive the membership of the fuzzy price of a European option. Finally, to understand the extent to which our approach might be useful in practice, we give a numerical illustration of our procedure on the S&P 500 and VIX indexes. As a by-product of our research, a simple estimation method is introduced to obtain (crisp) parameters in the Heston model under the risk-neutral measure and applied in the sequel of the paper to obtain alternative shapes for the fuzzy parameters of the model.