Fair Volatility: A Framework for Reconceptualizing Financial Risk

Paper: arXiv 2509.18837

Authors: Sergio Bianchi, Daniele Angelini

Abstract

Volatility is the canonical measure of financial risk, a role largely inherited from Modern Portfolio Theory. Yet, its universality rests on restrictive efficiency assumptions that render volatility, at best, an incomplete proxy for true risk. This paper identifies three fundamental inconsistencies: (i) volatility is path-independent and blind to temporal dependence and non-stationarity; (ii) its relevance collapses in derivative-intensive strategies, where volatility often represents opportunity rather than risk; and (iii) it lacks an absolute benchmark, providing no guidance on what level of volatility is economically fair'' in efficient markets. To address these limitations, we propose a new paradigm that reconceptualizes risk in terms of predictability rather than variability. Building on a general class of stochastic processes, we derive an analytical link between volatility and the Hurst-Holder exponent within the Multifractional Process with Random Exponent (MPRE) framework. This relationship yields a formal definition of fair volatility’’, namely the volatility implied under market efficiency, where prices follow semi-martingale dynamics. Extensive empirical analysis on global equity indices supports this framework, showing that deviations from fair volatility provide a tractable measure of market inefficiency, distinguishing between momentum-driven and mean-reverting regimes. Our results advance both the theoretical foundations and empirical assessment of financial risk, offering a definition of volatility that is efficiency-consistent and economically interpretable.

Complexity vs Empirical Score

• Math Complexity: 7.5/10
• Empirical Rigor: 7.0/10
• Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper provides a strong theoretical framework connecting Hurst-Hölder regularity to a novel concept of ‘fair volatility’ using Multifractional Processes with Random Exponents (MPRE). It backs up these advanced mathematical derivations with robust empirical analysis across 14 international equity indices to distinguish between market efficiency regimes.

Research Flowchart

  flowchart TD
    A["Research Goal: Resolve limitations of volatility as risk<br/>by defining efficiency-consistent 'Fair Volatility'"] --> B["Identify Volatility Limits:<br/>Path-independence, derivative misfits & lack of benchmarks"]
    B --> C["Theoretical Framework:<br/>Model stochastic processes via MPRE and Hurst-Hölder exponent"]
    D["Inputs: Global Equity Indices Data"] --> E["Analytical & Empirical Analysis:<br/>Derive Fair Volatility link and test market efficiency"]
    C --> E
    E --> F["Outcomes: Analytical definition of Fair Volatility<br/>& classification of momentum vs. mean-reverting regimes"]

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2 of 307 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.