11-JAN-2024
These slides develop the connections between Bessel processes, fractional Brownian motion (fBm), and long-memory asset price models. The Hurst exponent $H \in (0,1)$ parameterizes the fractional GBM, with $H < 1/2$ (rough regime) linking to Bessel processes of dimension $\delta = 2(1-H)$ and explaining near-zero volatility clustering. European option pricing under fractional dynamics is derived via Wick–Itô calculus, yielding a closed-form fractional Black–Scholes formula. Sample paths, hyperbolic autocorrelation decay, and call price surfaces are illustrated numerically for contrasting values of $H$.
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