This paper proposes the Hurst–Hölder exponent as a superior alternative to volatility for measuring financial risk, under locally fractional price dynamics.

Unlike volatility, which captures only variability, the exponent measures path roughness and deviations from semi-martingale behavior, providing insight into the nature and intensity of randomness.

The study further proves that the Hurst–Hölder exponent can be converted into realized volatility, enabling the construction of confidence intervals around a benchmark “fair volatility” consistent with informational efficiency.

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