ADAPTIVE FRACTAL DYNAMICS: A TIME-VARYING HURST APPROACH TO VOLATILITY MODELING IN EQUITY MARKETS

Abe Webb^*

• Westchester Research, Los Angeles, United States

We propose a dynamic fractional volatility model that incorporates a time-varying Hurst exponent estimated via Daubechies-4 wavelet analysis on 252-day rolling windows to capture evolving market memory effects in equity markets. This approach overcomes the limitations of traditional GARCH-type (Engle, 1982) and static fractional volatility models (Baillie et al., 1996), which assume a constant memory parameter and struggle during regime shifts and market stress. The model is applied to daily closing prices of the S&P 500 Index over 1,258 trading days from January 1, 2015 to December 31, 2019, yielding statistically significant improvements in forecasting performance. Empirical results indicate a 12.3% reduction in RMSE and a 9.8% improvement in MAPE, with an out-of-sample R-squared exceeding 0.72 compared to benchmark models. Maximum likelihood estimation with Fisher scoring is used for daily parameter updates, ensuring the model remains responsive to rapidly changing market conditions. Additionally, the model achieves an average absolute option pricing error of 1.8%, markedly lower than that of traditional specifications. These enhancements are further corroborated by pairwise Diebold-Mariano tests, which confirm the statistical significance of the forecast improvements. Overall, this framework offers a rigorous and computationally efficient method for real-time volatility forecasting that delivers substantial benefits for risk management, derivative pricing, and automated trading strategies, thereby extending the theoretical foundations established by Cont (2005), Mandelbrot andHudson (2004), andPercival andWalden (2000).