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arxiv: 2509.18837 · v3 · submitted 2025-09-23 · 💱 q-fin.MF

Fair Volatility: A Framework for Reconceptualizing Financial Risk

Sergio Bianchi , Daniele Angelini This is my paper

Pith reviewed 2026-05-18 15:09 UTC · model grok-4.3

classification 💱 q-fin.MF
keywords volatilitymarket efficiencyHurst-Holder exponentsemi-martingale dynamicsmultifractional processfinancial riskmarket inefficiency
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The pith

Volatility is reconceptualized as the level implied by semi-martingale dynamics under market efficiency.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Traditional volatility fails as a complete risk measure because it is path-independent, offers no absolute benchmark for what level is economically fair, and often misrepresents opportunity rather than risk in derivative strategies. The paper establishes an analytical link between volatility and the Hurst-Holder exponent inside the multifractional process with random exponent framework. This link produces a definition of fair volatility as the specific volatility value that would arise if prices followed semi-martingale dynamics, the mathematical condition tied to market efficiency. Empirical checks across global equity indices then show that departures from this fair level track inefficiency and separate momentum-driven regimes from mean-reverting ones.

Core claim

Within the multifractional process with random exponent framework, volatility is analytically tied to the Hurst-Holder exponent. The resulting relationship supplies a formal definition of fair volatility as the volatility level that is consistent with semi-martingale price dynamics under market efficiency.

What carries the argument

The multifractional process with random exponent (MPRE), which supplies the analytical bridge between observed volatility and the local Hurst-Holder exponent to isolate the fair-volatility component under semi-martingale dynamics.

If this is right

  • Deviations from fair volatility quantify departures from market efficiency.
  • Markets can be classified into momentum-driven versus mean-reverting regimes according to the sign and size of these deviations.
  • Risk assessment gains an absolute, efficiency-consistent benchmark rather than a relative one.
  • Derivative strategies can be re-evaluated by treating volatility as a predictability signal instead of raw variability.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same fair-volatility benchmark could be applied to other asset classes such as bonds or currencies to detect analogous inefficiency patterns.
  • Portfolio construction rules might treat sustained deviations from fair volatility as an explicit signal alongside traditional factors.
  • Aggregate fair-volatility deviations across many assets could serve as a system-level indicator for monitoring overall market stress.

Load-bearing premise

Market efficiency corresponds exactly to prices following semi-martingale dynamics.

What would settle it

A direct empirical check would test whether, during periods widely viewed as efficient, realized volatility equals the value obtained by substituting the observed Hurst-Holder exponent into the semi-martingale relation derived from the MPRE model.

Figures

Figures reproduced from arXiv: 2509.18837 by Daniele Angelini, Sergio Bianchi.

Figure 1
Figure 1. Figure 1: Path-dependent risk characteristics: (Top panel) IID Gaussian noise; (Upper-mid panel) Inde [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Resetting autocorrelation: (Top left panel) fractional Gaussian noise with parameter [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 9
Figure 9. Figure 9: KOSPI Index [PITH_FULL_IMAGE:figures/full_fig_p015_9.png] view at source ↗
Figure 8
Figure 8. Figure 8: KLCI Index [PITH_FULL_IMAGE:figures/full_fig_p015_8.png] view at source ↗
Figure 15
Figure 15. Figure 15: FSSTI Index [PITH_FULL_IMAGE:figures/full_fig_p016_15.png] view at source ↗
Figure 14
Figure 14. Figure 14: SHCOMP Index [PITH_FULL_IMAGE:figures/full_fig_p016_14.png] view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper claims that traditional volatility is an incomplete risk measure due to path-independence, limited relevance in derivative strategies, and absence of an absolute benchmark for 'fair' levels. It derives an analytical relationship between volatility and the random Hurst-Holder exponent inside the Multifractional Process with Random Exponent (MPRE) framework, then defines 'fair volatility' as the value consistent with semi-martingale dynamics under market efficiency. Empirical tests on global equity indices are presented to show that deviations from this benchmark quantify inefficiency and separate momentum from mean-reverting regimes.

Significance. If the MPRE link is shown to be independent of the random-exponent law and if the semi-martingale-efficiency identification is justified from no-arbitrage or information principles, the framework would supply a theoretically grounded, efficiency-consistent volatility benchmark with direct implications for inefficiency measurement. The empirical distinction between regimes could be useful for practical risk assessment, though its value hinges on the robustness of the central derivation.

major comments (2)
  1. [Abstract] Abstract (paragraph on the MPRE link and fair volatility definition): the identification of semi-martingale dynamics with market efficiency is invoked to define fair volatility but is not derived from no-arbitrage conditions or from an information-based efficiency criterion; this mapping is load-bearing for the entire construction.
  2. [Abstract] Abstract (MPRE framework paragraph): the claimed analytical link between volatility and the Hurst-Holder exponent is asserted to yield the fair-volatility definition, yet it is not shown to be independent of the specific probability law chosen for the random exponent or of the local regularity conditions that actually render the MPRE a semi-martingale.
minor comments (2)
  1. The abstract refers to 'extensive empirical analysis' without specifying the equity indices, sample periods, or statistical procedures used to distinguish regimes; these details are needed to evaluate the strength of the empirical support.
  2. Notation for the random Hurst-Holder exponent and the precise definition of the MPRE should be introduced with explicit equations early in the manuscript to allow readers to follow the derivation.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and insightful comments on our manuscript. The points raised highlight important aspects of the theoretical foundations that we address below. We believe the clarifications will improve the presentation without altering the core contributions.

read point-by-point responses

  1. Referee: [Abstract] Abstract (paragraph on the MPRE link and fair volatility definition): the identification of semi-martingale dynamics with market efficiency is invoked to define fair volatility but is not derived from no-arbitrage conditions or from an information-based efficiency criterion; this mapping is load-bearing for the entire construction.

    Authors: We agree that an explicit link strengthens the argument. The semi-martingale characterization follows from the Fundamental Theorem of Asset Pricing: under the no-arbitrage condition, there exists an equivalent martingale measure, which requires the (discounted) price process to be a semi-martingale. In the MPRE framework, market efficiency is modeled by the case in which the random exponent yields semi-martingale paths, and fair volatility is defined as the volatility consistent with that case. We will revise the abstract and the relevant theoretical section to include a concise reference to the FTAP and to state the modeling assumption more explicitly. revision: yes

  2. Referee: [Abstract] Abstract (MPRE framework paragraph): the claimed analytical link between volatility and the Hurst-Holder exponent is asserted to yield the fair-volatility definition, yet it is not shown to be independent of the specific probability law chosen for the random exponent or of the local regularity conditions that actually render the MPRE a semi-martingale.

    Authors: The volatility-exponent relationship is derived from the local Hölder regularity of the MPRE paths, which is governed by the pointwise values of the random exponent rather than its global probability measure. Fair volatility corresponds to the parameter value that produces semi-martingale dynamics (local exponent equal to 1/2). This identification holds under the standard measurability and continuity conditions that define the MPRE class, independently of the particular law of the exponent field. We acknowledge that a short clarifying remark or lemma would make the independence explicit and will add such material in the revised manuscript. revision: partial

Circularity Check

1 steps flagged

Fair volatility defined by construction as the level consistent with semi-martingale dynamics under the efficiency assumption in MPRE

specific steps
  1. self definitional [Abstract]
    "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."

    The analytical link is used to define fair volatility as precisely the volatility value that obtains under the semi-martingale dynamics already stipulated as the mathematical content of market efficiency. The resulting measure of inefficiency (deviation from fair volatility) is therefore equivalent by construction to deviation from the semi-martingale property rather than an independent prediction.

full rationale

The paper derives an analytical link between volatility and the random Hurst-Holder exponent in the MPRE framework and then explicitly defines fair volatility as the volatility implied when prices follow semi-martingale dynamics (taken as the signature of market efficiency). This construction makes the central 'fair' benchmark and the associated inefficiency measure (deviations from it) reduce directly to the input identification of efficiency with semi-martingales rather than an independent first-principles result. Empirical application to equity indices supplies some external content, but the definitional step remains load-bearing for the reconceptualization claim. No self-citation chain or fitted-parameter renaming is exhibited in the provided text.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 1 invented entities

The framework rests on the MPRE stochastic process class and the identification of market efficiency with semi-martingale dynamics; no free parameters are explicitly listed in the abstract, but the Hurst-Holder exponent is implicitly estimated from data to compute deviations.

free parameters (1)
  • Hurst-Holder exponent
    Used to link volatility to predictability; its value is determined from observed price paths to compute fair volatility and deviations.
axioms (1)
  • domain assumption Under market efficiency, asset prices follow semi-martingale dynamics
    Invoked in the abstract to anchor the definition of fair volatility as the volatility level implied by that dynamics.
invented entities (1)
  • fair volatility no independent evidence
    purpose: Benchmark volatility level that is consistent with market efficiency
    Newly defined quantity whose independent evidence would require external falsification beyond the efficiency assumption itself

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Randomized Kolmogorov-Smirnov Analysis of Volatility Roughness

    q-fin.MF 2025-09 unverdicted novelty 6.0

    A randomized Kolmogorov-Smirnov estimator for the Hurst exponent applied to VIX implied volatility and S&P 500 realized volatility finds both rougher than Brownian motion with a statistically significant smoothness hierarchy.

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