Fair Volatility in Financial Markets

• Fair volatility is a reformulation of financial risk, defined as the level of instantaneous return dispersion under efficient market conditions (H = 1/2) using multifractional processes.
• It distinguishes risk from opportunity by incorporating path-dependence and local market inefficiencies that classical volatility measures overlook.
• Empirical implementation involves high-frequency data and wavelet-based estimation of the Hurst–Hölder exponent to detect regime shifts and enhance dynamic risk management.

Fair volatility is a reconceptualization of financial risk that seeks to reconcile volatility as a measure of uncertainty with the requirements of market efficiency, path-dependence, and economic interpretability. Traditional measures—such as standard deviation, realized volatility, and even implied volatility—are blind to the temporal structure and predictability of returns, lack an absolute benchmark, and often fail to distinguish between risk and opportunity, particularly in the context of modern quantitative trading strategies and path-dependent derivatives. Fair volatility, by contrast, is formally defined as the level of instantaneous return dispersion implied under semimartingale (efficient) price dynamics, derived systematically from the local regularity of paths using the Hurst–Hölder exponent measured within the framework of multifractional processes. This framework provides an economically interpretable, efficiency-consistent, and model-independent benchmark for volatility, with deviations serving as quantitative proxies for market inefficiency.

1. Shortcomings of Classical Volatility as a Risk Measure

Historically, volatility has been privileged as the canonical measure of risk within Modern Portfolio Theory and derivative pricing. Its universality, however, is critically limited:

• Path-Independence: Volatility as standard deviation or variance is computed without regard to the ordering of returns or their temporal dependence. It lacks sensitivity to serial correlation, long memory, or regime change (Bianchi et al., 23 Sep 2025).
• No Absolute Benchmark: Traditional volatility is inherently relative; it measures deviations from an evolving sample mean or median but offers no theoretical guidance for what constitutes an efficient or “fair” risk level in the market (Bianchi et al., 23 Sep 2025).
• Collapse in Derivative-Intensive Strategies: For strategies that are explicitly volatility-targeting or derivative-driven, realized volatility can become more a measure of opportunity than risk, further undermining its universal applicability (Bianchi et al., 23 Sep 2025).
• Inefficiency and Market Regimes: In practical settings, volatility can be suppressed by feedback effects (e.g., option market making and program trading), resulting in apparent stability punctuated by abrupt, unpredictable episodes of high risk (Valenti et al., 2017).

These limitations motivate the need for a new framework where “fair volatility” is not merely a descriptive statistic but a theoretically justified, economically interpretable benchmark.

2. Theoretical Foundations: Hurst–Hölder Exponent and Multifractional Processes

A central theoretical advance is the recognition that risk should be measured by unpredictability and path roughness, not merely dispersion. The Multifractional Process with Random Exponent (MPRE) provides a natural probabilistic foundation for this insight (Bianchi et al., 23 Sep 2025, Bianchi et al., 2 Aug 2025).

• Stochastic Structure: In the MPRE, price dynamics are modeled as

$X(t) = \int K(t, s; H(\cdot))\, dW_s,$

where $K$ is a kernel function, $H(\cdot)$ is the (random/time-varying) Hurst–Hölder exponent, and $W_s$ is a Wiener process.

• Local Self-Similarity: At every point $t$, the process looks locally like a fractional Brownian motion with exponent $H(t)$. This regularity governs the scaling of local increments:

$$\operatorname{sd}(X(t+h) - X(t)\,|\,\mathcal{A}_t^{(H,\nu)}) \sim |h|^{H(t)} \cdot \nu(t) \cdot \sqrt{A(H(t))},\qquad h \to 0$$

with

$$A(H) = \frac{\Gamma(H+1/2)^2}{2H\sin(\pi H)\Gamma(2H)}$$

(Bianchi et al., 23 Sep 2025).

• Interpretation of $H(t)$: When $H(t) = 1/2$ (the Brownian semimartingale case), quadratic variation grows linearly, corresponding to pure unpredictability and market efficiency (in the sense of the Efficient Market Hypothesis (EMH)) (Bianchi et al., 2 Aug 2025). Deviations $H(t) \neq 1/2$ encode local market inefficiencies:
  • $H(t) > 1/2$: Persistent (momentum) price dynamics, lower risk than implied by volatility alone.
  • $H(t) < 1/2$: Anti-persistent (mean-reverting), risk higher than standard volatility implies.
  • The function $H(t)$ serves as a localized, scale-invariant measure of market regularity, distinguishing between stochastic environments and market regimes (Bianchi et al., 2 Aug 2025, Bianchi et al., 23 Sep 2025).

3. Formal Definition of Fair Volatility

Fair volatility, denoted $\sigma_{\mathrm{fair}}(t)$, is defined as the volatility level that would materialize under local efficiency, i.e., if $H(t) = 1/2$ throughout:

$$\sigma_{\mathrm{fair}}(t) \equiv \lim_{h \to 0} \frac{\operatorname{sd}(X(t+h)-X(t))}{\sqrt{h}}$$

In the more general multifractional setting, the observed instantaneous volatility is

$$\operatorname{sd}(X(t+h)-X(t)) \sim |h|^{H(t)} \cdot \nu(t) \cdot \sqrt{A(H(t))}$$

and the fair volatility benchmark corresponds to the unique value achieved when $H(t) = 1/2$:

$$\operatorname{sd}_{\mathrm{fair}}(X(t+h)-X(t)) \sim |h|^{1/2} \cdot \nu(t) \cdot \sqrt{A(1/2)}$$

This construction allows any observed volatility level $\sigma_{\text{obs}}(t)$ to be compared against the efficiency-consistent $\sigma_{\text{fair}}(t)$, with the deviation indicating the degree of inefficiency or excess risk embedded in the market at time $t$ (Bianchi et al., 23 Sep 2025, Bianchi et al., 2 Aug 2025).

The relationship can be made explicit: | Regime | $H(t)$ | Local Scaling Law | Relation to $\sigma_{\mathrm{fair}}$ | |-------------------------------|----------------|-----------------------|----------------------------------------------| | Efficient (semimartingale) | $= 1/2$ | $\sim \sqrt{h}$ | $\sigma_{\text{obs}} = \sigma_{\mathrm{fair}}$ | | Momentum (persistent) | $> 1/2$ | $\gg \sqrt{h}$ | $\sigma_{\text{obs}} < \sigma_{\mathrm{fair}}$ | | Mean-reverting (anti-persistent)| $< 1/2$ | $\ll \sqrt{h}$ | $\sigma_{\text{obs}} > \sigma_{\mathrm{fair}}$ |

4. Empirical Implementation and Measurement

Empirically, estimating fair volatility requires extracting both the observed local volatility and the time-varying Hurst–Hölder exponent. The following methodologies are applied (Bianchi et al., 23 Sep 2025, Bianchi et al., 2 Aug 2025):

• Estimation of $H(t)$: High-frequency price data is analyzed using local variation statistics and wavelet-based estimators to produce a time series for $H(t)$. The estimates are robust to microstructure noise and capture both low- and high-frequency path characteristics (Mouti, 2023, Gatheral et al., 2014).
• Scaling Law and Transformation: The empirically observed $H(t)$ and volatility are combined using the analytical relationship between $H(t)$ and standard deviation to reconstruct the “implied fair volatility.” Deviations between observed and fair levels are interpreted as local inefficiency signals.
• Interpretation: When $H(t)$ is stably close to 1/2, realized volatility is a sufficient statistic for risk. Under persistent deviations, classical volatility metrics underestimate (for $H>1/2$) or overestimate (for $H<1/2$) the true risk.
• Market Regime Detection: The framework distinguishes between momentum-driven, mean-reverting, and efficient regimes in equity indices and other assets, revealing that market inefficiency is dynamically reflected in the evolution of $H(t)$.

5. Consequences and Practical Applications

The fair volatility framework provides several practical and theoretical benefits:

• Absolute Risk Benchmarking: Unlike rolling-window volatility or scale-invariant wavelet-based measures, the fair volatility derived via $H(t)$ admits a universal, model-based, and scale-calibrated benchmark (Bianchi et al., 23 Sep 2025, Bianchi et al., 2 Aug 2025).
• Market Efficiency Assessment: By quantifying the local divergence between $H(t)$ and 1/2, one obtains a continuous, model-independent measure of efficiency that is statistically robust and economically interpretable. This provides a formal bridge between rational finance and behavioral phenomena, as departures from $H=1/2$ correspond to pathologies such as herding (momentum) or overreaction (mean reversion).
• Dynamic Risk Management: Risk managers and quantitative strategists can calibrate hedging, leverage, and stop-loss strategies in real time by monitoring whether the market regime is “too rough” or “too smooth” compared to the fair benchmark, in addition to the absolute volatility level.
• Trading and Regime Forecasting: Significant deviations from the fair volatility level can serve as early-warning indicators of regime shifts, impending crashes, or bubbles, since such deviations quantitatively reflect market inefficiencies and increased predictability (Bianchi et al., 23 Sep 2025, Bianchi et al., 2 Aug 2025).

6. Implications for Financial Theory and Future Directions

The reconceptualization of risk via fair volatility has implications for the foundations and empirical practice of financial economics:

• Reconciling Risk and Predictability: Fair volatility enables an overview of market microstructure, long-memory models, and multifactor stochastic calculus models by directly linking risk to predictability, as opposed to mere random fluctuation.
• Absolute vs. Relative Risk: This framework moves volatility from being merely a relative performance measure to an absolute efficiency-consistent standard.
• Integration with Existing Frameworks: While fair volatility is most naturally equipped to supplement realized and implied volatility measures in regime detection and risk control, it is also directly compatible with stochastic volatility models and multifractional volatility paradigms. It provides a theoretical underpinning for when and why volatility surface calibration or tail risk adjustments are necessary (Vazquez, 2014).
• Further Research: Open problems include precise statistical testing for persistent (in)efficiency, integration with high-frequency order book models, and production-grade implementations for portfolio risk management.

7. Comparative Summary: Fair Volatility vs. Classical Volatility

Dimension	Classical Volatility	Fair Volatility (MPRE/Hurst–Hölder)
Benchmark	Relative, rolling window	Absolute, efficiency-based ($H=1/2$)
Path Dependence	Ignored (path-independent)	Fully incorporated (via $H(t)$)
Model Assumptions	Implied i.i.d. returns	Accommodates non-stationarity, memory effects
Risk-Opportunity Distinction	Blurred	Distinguished via regime classification
Economic Interpretability	Weak	Robust (linked to EMH and price predictability)

Fair volatility provides a principled, mathematically derivable, and empirically robust metric for financial risk, grounded in local path regularity. It offers a direct link between observed market fluctuations and the level of predictability tolerated under efficiency, yielding a risk measure that is both theoretically consistent and operationally interpretable in the context of market dynamics (Bianchi et al., 23 Sep 2025, Bianchi et al., 2 Aug 2025).