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arxiv: 2509.20015 · v4 · submitted 2025-09-24 · 💱 q-fin.MF · q-fin.CP· stat.ME

Randomized Kolmogorov-Smirnov Analysis of Volatility Roughness

Sergio Bianchi , Daniele Angelini This is my paper

Pith reviewed 2026-05-18 15:12 UTC · model grok-4.3

classification 💱 q-fin.MF q-fin.CPstat.ME
keywords volatility roughnessHurst exponentKolmogorov-Smirnov statisticrandom permutationimplied volatilityrealized volatilityrough volatilityscaling behavior
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The pith

A Kolmogorov-Smirnov estimator applied after random permutation shows implied volatility is smoother than realized volatility, yet both have Hurst exponents well below one half.

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

The paper introduces an estimator for the roughness of log-volatility that examines how full distributions of increments scale with time rather than relying on individual moments. To apply this to serially dependent financial data, it uses random permutations that break temporal correlations while keeping the shape of the distributions intact. This setup yields an estimator with known asymptotic variance for statistical inference. When applied to the VIX index and five-minute realized volatility of the S&P 500, the method detects a clear ordering in which implied volatility appears less rough than realized volatility. Both series nevertheless display Hurst parameters substantially lower than the classical value of one half.

Core claim

The authors establish a distribution-based Hurst estimator that employs the Kolmogorov-Smirnov statistic on the scaling of empirical distribution functions of log-volatility increments. They introduce a random permutation step that removes serial dependence while preserving marginal distributions, thereby justifying the use of the KS framework on dependent observations. They derive the estimator's asymptotic variance and demonstrate that derivative-free optimizers such as Brent's method and Nelder-Mead deliver accurate results at lower computational cost than exhaustive search. Empirical application to CBOE VIX and S&P 500 realized volatility data produces statistically significant evidence:

What carries the argument

The randomized Kolmogorov-Smirnov estimator, which compares the empirical cumulative distribution of rescaled increments to a reference distribution after random permutation to remove dependence.

If this is right

  • The estimator supplies a distribution-level alternative to moment-based roughness measures that can be used on any dependent time series whose marginals are preserved under permutation.
  • The observed hierarchy implies that market-implied volatility smooths some of the local roughness present in high-frequency realized paths.
  • Both volatility measures lying well below Hurst one-half supports the use of fractional models with low Hurst exponents for pricing and risk applications.
  • The asymptotic variance formula enables construction of confidence intervals and formal tests for differences in roughness between series.

Where Pith is reading between the lines

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

  • The method could be tested on other high-frequency financial series such as exchange rates or commodity prices to check whether the roughness hierarchy is universal.
  • Further work could combine the estimator with multi-scale analysis to separate genuine local roughness from any long-memory component in the same data.
  • If the low-Hurst finding holds, option-pricing routines that assume Hurst one-half would systematically misprice volatility derivatives.
  • The computational efficiency gains from Brent and Nelder-Mead optimizers suggest the estimator can be run routinely on large intraday datasets.

Load-bearing premise

Random permutation removes serial correlation from the volatility series while leaving the scaling behavior of its marginal distributions unchanged.

What would settle it

If the Hurst values recovered by this KS procedure on the same VIX and realized-volatility series differ materially from those obtained by well-established moment-based or wavelet estimators, the new method's consistency would be called into question.

Figures

Figures reproduced from arXiv: 2509.20015 by Daniele Angelini, Sergio Bianchi.

Figure 1
Figure 1. Figure 1: Top panel: Computational time (in seconds) distributions required by the seven optimization [PITH_FULL_IMAGE:figures/full_fig_p016_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Estimates of the Hurst exponent Hˆ (eq. (4)) and the Kolmogorov-Smirnov statistics ˆδ (eq. (3)) for the VIX and 5-minute realized volatility (RV5). The top-left panel reports the estimates of Hˆ V IX with corresponding 95%-confidence intervals, while the top-right panel shows the analogous results for HˆRV 5. The theoretical normal distributions used for comparison are displayed in the side of each panels … view at source ↗
read the original abstract

We introduce a novel distribution-based estimator for the Hurst parameter of log-volatility, leveraging the Kolmogorov-Smirnov statistic to assess the scaling behavior of entire distributions rather than individual moments. To address the temporal dependence of financial volatility, we propose a random permutation procedure that effectively removes serial correlation while preserving marginal distributions, enabling the rigorous application of the KS framework to dependent data. We establish the asymptotic variance of the estimator, useful for inference and confidence interval construction. From a computational standpoint, we show that derivative-free optimization methods, particularly Brent's method and the Nelder-Mead simplex, achieve substantial efficiency gains relative to grid search while maintaining estimation accuracy. Empirical analysis of the CBOE VIX index and the 5-minute realized volatility of the S&P 500 reveals a statistically significant hierarchy of roughness, with implied volatility smoother than realized volatility. Both measures, however, exhibit Hurst exponents well below one-half, reinforcing the rough volatility paradigm and highlighting the open challenge of disentangling local roughness from long-memory effects in fractional modeling.

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

3 major / 2 minor

Summary. The manuscript introduces a distribution-based estimator for the Hurst exponent of log-volatility that uses the Kolmogorov-Smirnov statistic to compare scaling of entire distributions rather than moments. A random permutation step is proposed to remove serial correlation in financial time series while preserving marginals and scaling properties, allowing application of the KS framework. The paper derives the asymptotic variance of the resulting estimator, compares derivative-free optimizers (Brent, Nelder-Mead) against grid search for computational efficiency, and applies the method to CBOE VIX and 5-minute S&P 500 realized volatility, reporting Hurst exponents well below 1/2 with a statistically significant roughness hierarchy (implied volatility smoother than realized).

Significance. If the consistency of the randomized KS estimator after permutation is rigorously established, the work supplies a practical, distribution-based alternative for Hurst estimation in rough-volatility settings together with usable asymptotic inference and efficient numerical implementation. The empirical finding of sub-0.5 exponents and the implied-versus-realized hierarchy would add concrete support to the rough-volatility paradigm and highlight the modeling challenge of separating local roughness from long memory.

major comments (3)
  1. [§3 (Random Permutation Procedure)] §3 (Random Permutation Procedure): the claim that random permutation removes serial correlation while preserving the marginal scaling behavior required for the KS statistic to recover the original Hurst parameter is load-bearing for all empirical conclusions. For a process whose increments satisfy Var(ΔX_t) ~ Δt^{2H}, shuffling destroys temporal ordering; it is not shown that the resulting collection of marginal distributions still encodes the H-dependent scaling in a manner that the KS distance to a reference family consistently identifies the pre-permutation H. A formal consistency argument or targeted simulation study is required.
  2. [Theoretical results section (asymptotic variance)] Theoretical results section (asymptotic variance): the abstract states that the asymptotic variance is established and is used for inference, yet the derivation, regularity conditions, and explicit expression are not supplied in sufficient detail to verify the reported confidence intervals or the statistical significance of the roughness hierarchy. Without this, the empirical claims rest on an unverified step.
  3. [Empirical section (data handling)] Empirical section (data handling): the manuscript does not specify data-exclusion rules, exact sample periods, or how error bars are constructed for the CBOE VIX and 5-minute realized-volatility series. These details are necessary to assess robustness of the reported Hurst values and the claimed hierarchy.
minor comments (2)
  1. [Notation] Notation for the KS statistic on permuted samples should be made fully explicit (including the precise reference distribution family) to facilitate reproducibility.
  2. [Figures] Figures displaying estimated Hurst values should include the asymptotic confidence bands derived from the established variance so that visual assessment of significance is immediate.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive and detailed comments, which have helped us identify areas for improvement in the manuscript. We address each major comment below and outline the revisions we intend to make in the next version.

read point-by-point responses

  1. Referee: [§3 (Random Permutation Procedure)] §3 (Random Permutation Procedure): the claim that random permutation removes serial correlation while preserving the marginal scaling behavior required for the KS statistic to recover the original Hurst parameter is load-bearing for all empirical conclusions. For a process whose increments satisfy Var(ΔX_t) ~ Δt^{2H}, shuffling destroys temporal ordering; it is not shown that the resulting collection of marginal distributions still encodes the H-dependent scaling in a manner that the KS distance to a reference family consistently identifies the pre-permutation H. A formal consistency argument or targeted simulation study is required.

    Authors: We agree that additional justification is needed to confirm that the random permutation preserves the H-dependent scaling properties in a way that allows consistent recovery of the original Hurst parameter. In the revised manuscript, we will add a targeted simulation study: we will simulate paths from fractional Brownian motion (and related rough volatility processes) with known Hurst exponents, apply the random permutation, and verify that the randomized KS estimator recovers the true H with accuracy comparable to the non-permuted case. This will provide concrete empirical support for the procedure's validity in the relevant setting. revision: yes

  2. Referee: [Theoretical results section (asymptotic variance)] Theoretical results section (asymptotic variance): the abstract states that the asymptotic variance is established and is used for inference, yet the derivation, regularity conditions, and explicit expression are not supplied in sufficient detail to verify the reported confidence intervals or the statistical significance of the roughness hierarchy. Without this, the empirical claims rest on an unverified step.

    Authors: We acknowledge that the current presentation of the asymptotic variance derivation lacks sufficient detail. In the revision, we will expand the theoretical results section to include the full derivation, the required regularity conditions, and the explicit expression for the asymptotic variance. This will enable direct verification of the reported confidence intervals and the statistical significance of the implied-versus-realized roughness hierarchy. revision: yes

  3. Referee: [Empirical section (data handling)] Empirical section (data handling): the manuscript does not specify data-exclusion rules, exact sample periods, or how error bars are constructed for the CBOE VIX and 5-minute realized-volatility series. These details are necessary to assess robustness of the reported Hurst values and the claimed hierarchy.

    Authors: We thank the referee for highlighting these omissions. We will revise the empirical section to specify the exact sample periods for both the CBOE VIX and the 5-minute S&P 500 realized volatility series, detail any data-exclusion rules (e.g., handling of missing observations or outliers), and provide a clear description of how the error bars are constructed using the derived asymptotic variance. revision: yes

Circularity Check

0 steps flagged

No significant circularity; estimator and empirical claims are self-contained

full rationale

The paper defines a new distribution-based Hurst estimator by optimizing the KS statistic after random permutation of the series to remove serial correlation while preserving marginals. It separately derives the asymptotic variance of this estimator and applies the procedure to VIX and realized-volatility data to obtain the reported hierarchy and sub-0.5 values. No equation or step reduces the output Hurst parameter to a fitted input by construction, nor does any central claim rest on a self-citation chain or imported uniqueness theorem. The derivation chain therefore remains independent of the target empirical results.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard properties of the Kolmogorov-Smirnov statistic under permutation and on the assumption that the permutation leaves the scaling of distributions intact; no new physical entities are introduced.

free parameters (1)
  • Hurst exponent H
    Target parameter obtained by optimizing the randomized KS statistic against data; its value is not known a priori and is fitted per series.
axioms (2)
  • domain assumption Random permutation of the series removes serial dependence while preserving marginal distributions and their scaling properties.
    Invoked to justify applying the classical KS framework to temporally dependent volatility data.
  • standard math The KS statistic on scaled distributions converges to a limit that identifies the Hurst parameter.
    Background result from extreme-value or empirical-process theory used to motivate the estimator.

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