A Fractional Version of the Stochastic Alpha–Beta–Rho (SABR) Model within a Fast-varying Stochastic Environment (FVSE)

This paper proposes a fractional extension of the SABR model that better captures the irregular behavior of volatility, particularly its long-memory and rough characteristics. Relying on empirical evidence showing that log-volatility follows a fractional Brownian motion dynamics with a low Hurst exponent, and embedding volatility within a fast-varying stochastic environment, the model is formulated through a fractional stochastic differential equation combining fractional dynamics with high-frequency stochastic drivers. This approach improves the modeling of implied volatility surfaces (smile and skew) while preserving analytical tractability for practical applications such as calibration and options pricing. The proposed fractional SABR model thus proves to be robust in capturing volatility smiles and realistic in accounting for long-memory volatility dynamics, offering a promising framework for market calibration, volatility forecasting, and further theoretical or numerical developments.