This study compares dynamic delta hedging under standard Brownian motion (Bm) and fractional Brownian motion (fBm) during the 2008 financial crisis and the COVID-19 recession.

Unlike standard Bm, fBm incorporates long-range dependence through the Hurst exponent, allowing it to better capture persistent market dynamics during turbulent periods.

Using S&P 500 and NYSE data, the results show that fBm generally improves hedging efficiency, reduces downside risk, and eliminates large negative P&L outliers when the Hurst exponent is near or above 0.5.

However, when the exponent falls below 0.5, the framework can increase volatility and tail risk, emphasizing the importance of calibration.

The study also highlights that better hedging does not necessarily require dramatically better forecasts.

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