Fuzzy Pricing and Empirical Analysis of Binary Options Under Long Memory

To address the limitations of traditional binary option pricing models in capturing the long-range dependence characteristics of financial time series and in neglecting the cognitive fuzziness of parameters, this paper proposes an innovative pricing model that integrates mixed fractional Brownian motion with triangular fuzzy numbers. Mixed fractional Brownian motion is employed to characterize the long-range dependence of stock prices, while triangular fuzzy numbers are used to describe the fuzziness inherent in stock prices and volatility. Based on this framework, a fuzzy pricing model is constructed, and explicit expressions for the price cutsets are derived. Theoretical analysis and numerical experiments show that the influence of the Hurst exponent on option prices is modulated by the exercise time. Empirical results reveal that the underlying asset price exhibits significant long-range dependence, supporting the use of mixed fractional Brownian motion for modeling. Compared with the Black–Scholes model, the fuzzy price interval from the proposed framework covers 98.00% of actual price observations, while the Black–Scholes-based interval covers 84.00%. The interval is also narrower, indicating higher efficiency in uncertainty representation.