arxiv: 2604.15519 · v1 · submitted 2026-04-16 · 💱 q-fin.ST

Recognition: unknown

Broken Symmetry, Conservation Law, and Scaling in Accumulated Stock Returns -- a Modified Jones-Faddy Skew t-Distribution Perspective

Arshia Ghasemi, R. A. Serota, Siqi Shao

Pith reviewed 2026-05-10 09:11 UTC · model grok-4.3

classification 💱 q-fin.ST

keywords stock returnsvariance scalingskew t-distributionsymmetry breakingS&P500accumulated returnslinear scalingmean scaling

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The pith

Variance and mean of stock returns scale linearly with accumulation time despite broken symmetry between gains and losses.

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

The paper examines daily to ten-day accumulated returns from the S&P500 index. It finds that even with the distribution showing positive average returns and negative skew due to more frequent small gains than large losses, the variance still increases linearly with the number of days. The mean return also scales linearly in a near-perfect way. Using a modified version of the Jones-Faddy skew t-distribution, the authors derive and confirm this behavior both mathematically and through data. This suggests an underlying scaling property that holds regardless of the asymmetry.

Core claim

Analysis of historic S&P500 multi-day returns reveals that despite symmetry breaking between gains and losses producing a positive mean and negative skew in the return distribution, the realized variance maintains a remarkably good linear dependence on the accumulation period. The mean itself exhibits near-perfect linear scaling with time. These properties are established both analytically and numerically through the use of a modified Jones-Faddy skew t-distribution.

What carries the argument

The modified Jones-Faddy skew t-distribution, which incorporates skewness and heavy tails to model the asymmetric return distribution while preserving the linear scaling of moments.

If this is right

Multi-period volatility can be obtained by simple multiplication of daily variance.
Long-term expected returns can be reliably scaled from short-term observations.
Risk assessment models need not adjust for asymmetry when computing accumulated variance.
The scaling persists across different accumulation lengths up to at least ten days.

Where Pith is reading between the lines

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

This scaling may indicate a conservation-like principle in return processes that survives asymmetry.
It could be tested by applying the same analysis to individual stocks or other indices.
The result might allow simpler forecasting of long-horizon portfolio risks without full distributional simulation.
Extensions to higher moments or intraday data could reveal if the linearity breaks at other scales.

Load-bearing premise

That the modified Jones-Faddy skew t-distribution fits the data well enough that the linear scaling of variance and mean is a genuine property rather than a result of how the parameters are chosen or the data is selected.

What would settle it

A clear deviation from linear scaling in variance when returns are accumulated over periods longer than ten days or in different market regimes would falsify the claim.

Figures

Figures reproduced from arXiv: 2604.15519 by Arshia Ghasemi, R. A. Serota, Siqi Shao.

**Figure 1.** Figure 1: S&P500. Left: rt = log(St/S0), St is price on day t, t changes in daily increments (τ = 1 in text); Right:S&P500; xt = rt − µ1t where index in µ1 reflects daily increments of t (τ = 1 in text). From [1]. 0 50 100 150 200 0 0.005 0.01 0.015 0.02 0.025 τ RV 0 10 20 30 40 50 0 1 2 3 4 5 x 10-3 τ RV [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗

**Figure 2.** Figure 2: Realized variance (RV) of S&P500 as a function of th [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗

**Figure 3.** Figure 3: The variance of the distribution of returns as a fun [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗

**Figure 4.** Figure 4: αg, αl and θ from [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗

**Figure 5.** Figure 5: (δ/α) 2 and (δ/θ) 2 vs. τ; δ = αg − αl . 1 1 τ [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗

**Figure 6.** Figure 6: Location parameter µ from [PITH_FULL_IMAGE:figures/full_fig_p007_6.png] view at source ↗

**Figure 7.** Figure 7: Mean m1(τ) of S&P500 and mJF1 fit and linear fit of S&P500. Bottom insert: m1(τ)/τ. Top insert: Positive and negative terms in (12) and resulting m1(τ) – notice two orders of magnitude difference between central plot and insert. 4. Summary and Discussion Empirical evidence very strongly suggests that realized variance (squared realized volatility) of major stock indices S&P500 and DJIA scales linearly with… view at source ↗

**Figure 8.** Figure 8: Variance m2(τ) of S&P500 and its linear fit. Top insert: m2(τ) of mJF1 and its linear fit. Bottom insert: m2(τ)/τ of S&P500 and mJF1 and θ(τ). 5. Data Availability We obtained S&P500 data at Yahoo! Finance. Our datasets are available upon request. 6. Acknowledgments We used MathWorks Matlab for numerical work and Wolfram Mathematica for analytical calculations. Siqi Shao acknowledges support in part by The… view at source ↗

**Figure 9.** Figure 9: First Pearson coefficient of skewness ζ1 from [PITH_FULL_IMAGE:figures/full_fig_p012_9.png] view at source ↗

**Figure 10.** Figure 10: Second Pearson coefficient of skewness ζ2 from [PITH_FULL_IMAGE:figures/full_fig_p012_10.png] view at source ↗

**Figure 11.** Figure 11: Rescaled PDF of mJF1. 1 1 1 1 1 1 11 y √ μ 1 [PITH_FULL_IMAGE:figures/full_fig_p013_11.png] view at source ↗

**Figure 12.** Figure 12: Tails of rescaled PDF of mJF1. 13 [PITH_FULL_IMAGE:figures/full_fig_p013_12.png] view at source ↗

read the original abstract

We analyze historic S&P500 multi-day returns: from daily returns to those accumulated over up to ten days. Despite symmetry breaking between gains and losses in the distribution of returns, resulting in its positive mean and negative skew, realized variance (volatility squared) exhibits remarkably good linear dependence on the number of days of accumulation. Mean of the distribution also shows near perfect linear dependence as well. We analyze this phenomenon both analytically and numerically using a modified Jones-Faddy skew t-distribution.

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.

Desk Editor's Note private letter to a colleague

The paper shows linear scaling of mean and variance in 1-10 day S&P500 accumulated returns under a skew-t model, but the result looks incremental and the fitting details need scrutiny.

read the letter

The main point is that accumulated S&P 500 returns keep linear scaling in both mean and variance over short horizons even though the distribution is asymmetric, with positive mean and negative skew from gains and losses. The authors model this with a modified Jones-Faddy skew t and check it analytically plus numerically on historic data. That combination is the useful part: it gives a concrete example that standard scaling assumptions still hold when asymmetry is built in, which matters for anyone doing multi-period risk calculations in finance. The analytic treatment of the moments under this distribution is a plus, as is the direct use of real market returns rather than pure simulation. It is not a big conceptual leap, since linear variance scaling is already standard in random-walk style models, but the explicit check with this particular skew family adds a small empirical data point. The soft spots are real but not fatal. The abstract calls the linearity 'remarkably good' and 'near perfect' without any numbers, so the manuscript has to supply actual fit statistics, error bars, or at least R-squared values to make the claim credible. The bigger issue is the fitting procedure itself. If the four parameters are re-optimized independently for each accumulation length, the scale parameter can simply be set proportional to sqrt(n) and the linear variance result follows by construction; the skew-t part then adds little explanatory power. The paper needs to state clearly whether parameters are held fixed across horizons or whether the analytic section derives the scaling without refitting. Data handling details like exact sample period, exclusion rules, and robustness to outliers are also missing from the summary. This is for quantitative finance people who care about return distributions and short-horizon aggregation. A reader working on volatility modeling or multi-period forecasts could pick up the plots and moment expressions, but it is not required reading for the broader field. I would send it to peer review. The empirical observation is worth documenting with proper methods, and the analytic-numeric mix is solid enough to justify referee time even if revisions are needed on the fitting and statistics.

Referee Report

3 major / 2 minor

Summary. The paper analyzes historical S&P 500 returns accumulated over horizons from 1 to 10 days. It reports that, despite symmetry breaking between gains and losses (producing positive mean and negative skew), both realized variance and the mean exhibit remarkably good linear dependence on accumulation length. The phenomenon is examined analytically and numerically via a modified Jones-Faddy skew t-distribution.

Significance. If the linear scaling of variance and mean is shown to arise independently of per-horizon parameter re-optimization, the result would indicate a robust scaling property in equity returns that survives distributional asymmetry. The analytic use of the skew-t family could then supply closed-form moment expressions that explain why variance remains linear while higher moments reflect the skew. Credit is due for attempting both analytic derivation and numerical verification on real data.

major comments (3)

[Abstract] Abstract: the claims of 'remarkably good' and 'near perfect' linear dependence supply no quantitative measures (R², slope standard errors, p-values, or data-exclusion rules). Without these, it is impossible to determine whether the reported linearity exceeds what any four-parameter family with finite second moment would produce under independent fits.
[Analysis section] Analysis section (presumably §3–4): the manuscript does not state whether the four parameters of the modified Jones-Faddy skew t (location, scale, skewness, degrees of freedom) are optimized independently for each accumulation horizon or held fixed (or functionally dependent on n) across horizons. If the former, the linear scaling of variance is recovered by construction via the scale parameter and adds no explanatory power beyond the model choice.
[Numerical results] Numerical results (presumably §5): no table or figure reports the fitted parameter values versus horizon, nor a direct comparison against a null model of independent daily fits rescaled by sqrt(n). Such a comparison is required to rule out that the observed linearity is an artifact of unconstrained re-fitting rather than a property of the returns.

minor comments (2)

[Model definition] Notation for the modified Jones-Faddy density should be stated explicitly (including any re-parameterization of the original Jones-Faddy form) so that the analytic moment derivations can be reproduced.
[Figures] Figure captions should include the exact sample period, number of observations, and any filtering applied to the S&P 500 series.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive and detailed comments. We address each major point below and indicate the revisions we will make to strengthen the manuscript.

read point-by-point responses

Referee: [Abstract] Abstract: the claims of 'remarkably good' and 'near perfect' linear dependence supply no quantitative measures (R², slope standard errors, p-values, or data-exclusion rules). Without these, it is impossible to determine whether the reported linearity exceeds what any four-parameter family with finite second moment would produce under independent fits.

Authors: We agree that quantitative support is needed to substantiate the descriptive claims. In the revised manuscript we will add R² values, slope coefficients with standard errors, and p-values for the linear regressions of both mean and variance on accumulation horizon. We will also state the precise data-exclusion rules applied (e.g., handling of non-trading days and any minimum observation thresholds). These additions will allow readers to judge whether the observed linearity is stronger than would be expected from the model family under independent per-horizon fits. revision: yes
Referee: [Analysis section] Analysis section (presumably §3–4): the manuscript does not state whether the four parameters of the modified Jones-Faddy skew t (location, scale, skewness, degrees of freedom) are optimized independently for each accumulation horizon or held fixed (or functionally dependent on n) across horizons. If the former, the linear scaling of variance is recovered by construction via the scale parameter and adds no explanatory power beyond the model choice.

Authors: We will explicitly state in the revised analysis section that the four parameters are optimized independently for each horizon. However, the linearity is not automatic for an arbitrary four-parameter family. The modified Jones-Faddy skew-t supplies closed-form moment expressions; our analytic derivation shows that the empirical scaling of the first two moments is preserved by the fitted parameters while the higher moments continue to reflect the observed asymmetry. This is a property of the return process captured by the model rather than an artifact of the fitting procedure itself. We will expand the text to distinguish the result from what would be obtained with a generic four-parameter distribution lacking the same moment structure. revision: partial
Referee: [Numerical results] Numerical results (presumably §5): no table or figure reports the fitted parameter values versus horizon, nor a direct comparison against a null model of independent daily fits rescaled by sqrt(n). Such a comparison is required to rule out that the observed linearity is an artifact of unconstrained re-fitting rather than a property of the returns.

Authors: We will add a table (or supplementary figure) reporting the four fitted parameters for each horizon from 1 to 10 days. We will also include a direct comparison to the null model in which daily parameters are held fixed and the scale is rescaled by sqrt(n) under an independence assumption. The revised numerical section will show that the actual multi-horizon fits produce variance that remains linear while the skew and kurtosis evolve differently from the rescaled null, thereby demonstrating that the linearity is not solely an artifact of unconstrained re-fitting. revision: yes

Circularity Check

0 steps flagged

No significant circularity; empirical scaling observed independently of model fits

full rationale

The paper reports an empirical observation that realized variance and mean of accumulated S&P 500 returns scale linearly with accumulation horizon (1-10 days) despite distributional asymmetry. It then applies a modified Jones-Faddy skew-t distribution to model and analyze this scaling both analytically and numerically. No equations or steps in the provided abstract reduce the claimed scaling to a fitted parameter by construction, nor does the text rename a fit as a prediction or rely on self-citation for a uniqueness theorem. The scaling is presented as a data feature that the model is used to interpret, not generated tautologically from the model's own parameters or prior author work.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Based solely on the abstract; the modified Jones-Faddy skew t-distribution is invoked but its exact form, any added parameters, and the justification for linearity are not specified. No independent evidence or derivations are described.

pith-pipeline@v0.9.0 · 5384 in / 1288 out tokens · 35830 ms · 2026-05-10T09:11:28.718461+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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2024