Fair Volatility: A Multifractional Model for Realized Volatility

Volatility is an important measure of market uncertainty and risk. For decades, realized volatility has been computed from the squared returns. Recent research, however, has highlighted several deficiencies in traditional volatility measures.

Reference [1] continues this line of inquiry, identifying three key inefficiencies in conventional volatility estimation,

• Volatility is path-independent and blind to temporal dependence and non-stationarity,

• Its relevance collapses in derivative-intensive strategies, where volatility often represents opportunity rather than risk,

• It lacks an absolute benchmark, providing no guidance on what level of volatility is economically fair in efficient markets.

To address these issues, the author introduces the Hurst–Hölder exponent within the Multifractional Process with Random Exponent (MPRE) framework, incorporating it into the stochastic equation describing asset dynamics. This relationship leads to a formal definition of fair volatility—the level of volatility implied under market efficiency, where prices follow semi-martingale dynamics.

The authors pointed out,

This work establishes the Hurst-Hölder exponent as a superior, informationally equivalent substitute for volatility in financial risk measurement, provided price dynamics are locally fractional. Its adoption offers three principal advantages:

• Path-Dependent Risk. It directly quantifies path roughness, capturing deviations from semi-martingale behavior that volatility alone cannot. It moves beyond measuring mere variability to diagnosing the type and intensity of randomness.

• Absolute Benchmarking. Its value is intrinsically meaningful. Unlike volatility, which requires relative comparison, the exponent provides an absolute scale anchored by the martingale benchmark of H(t) =1/2.

• Theoretical Synthesis. It provides a contribution to resolve the apparent dichotomy between market efficiency and behavioral finance. These are not opposing models but alternating market phases, dynamically captured by the exponent’s fluctuation around its efficient equilibrium.

The convertibility of Hurst-Hölder exponent into realized volatility is established by Proposition 1. This enables the determination of a confidence interval around the volatility level that prevails under conditions of informational market efficiency, that is, when prices exhibit sub-martingale behavior. This benchmark is the level that we termed fair volatility.

Theoretically, the results are noteworthy, as the paper contributes to the literature addressing inefficiencies in traditional volatility measures and extends the Black–Scholes–Merton framework by accounting for the behavior (mean-reverting or trending) of the underlying asset.

However, practical applications of this approach remain to be seen. Given that the authors have already defined fair volatility, it would be valuable to see trading strategies developed around this concept and their performance evaluated in real markets.

Let us know what you think in the comments below or in the discussion forum.

References

[1] Sergio Bianchi, Daniele Angelini, Fair Volatility: A Framework for Reconceptualizing Financial Risk, 2025, arXiv:2509.18837