Revisiting Trade-sign Long-memory and Square-root Law price impact

Chris Angstmann; Tim Gebbie

arxiv: 2606.16269 · v2 · pith:FBB7VI6Enew · submitted 2026-06-15 · 💱 q-fin.TR · math-ph· math.MP

Revisiting Trade-sign Long-memory and Square-root Law price impact

Chris Angstmann , Tim Gebbie This is my paper

Pith reviewed 2026-06-27 02:30 UTC · model grok-4.3

classification 💱 q-fin.TR math-phmath.MP

keywords market microstructuretrade sign long memorysquare root lawmeta-order impactorder bookreaction-diffusionVolterra equationLillo-Mike-Farmer

0 comments

The pith

A reaction-diffusion model of order books derives both the long-memory of trade signs and the square-root law of meta-order impact from the same dynamics.

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

The paper shows that a coupled discrete reaction-diffusion formulation for lit and latent order books, including non-uniform event times and meta-order source terms, produces two well-known market regularities. These are the long-memory in trade signs from Lillo-Mike-Farmer theory and the square-root law for the price impact of meta-orders. The model reduces to a Volterra equation under locally linear order book assumptions and constant participation rates, yielding concave impact trajectories and square-root completion impact. Heavy-tailed Pareto distributions for meta-order lengths then generate the power-law autocorrelations in trade signs. The work reinterprets these as an event-time sign-memory statement for the long-memory and a physical-time viability statement for the square-root law.

Core claim

Starting with a coupled discrete reaction-diffusion formulation for the lit and latent order books with non-uniformly sampled event times and meta-order source terms, the locally linear order book and constant participation-rate execution reduce the dynamics to a Volterra equation. The leading-order solution yields the concave impact trajectory and completion impact proportional to the square root of the meta-order size. Heavy-tailed Pareto meta-order lengths generate power-law trade-sign autocorrelations through the source term in the interface representation.

What carries the argument

The coupled discrete reaction-diffusion model for lit and latent order books with meta-order source terms, reduced to a Volterra equation via locally linear order book and constant participation-rate execution.

If this is right

The LMF long-memory of trade signs is an event-time sign-memory statement.
The square-root law is a physical-time viability statement.
Subordination can alter calendar-time impact trajectories depending on the mappings and interpolation used.
Heavy-tailed meta-order lengths suffice to generate the observed power-law autocorrelations in trade signs.

Where Pith is reading between the lines

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

This approach could connect the two regularities to other order-book phenomena by modifying the diffusion or reaction terms.
Empirical tests separating event-time and physical-time data might distinguish the model's predictions from alternative explanations.
Extending the model to include variable participation rates could reveal how impact laws change under different execution strategies.

Load-bearing premise

The assumption that the order book is locally linear and execution occurs at constant participation rate, allowing reduction to a Volterra equation.

What would settle it

Empirical data showing that meta-order impact completion does not scale as the square root of size, or that Pareto-distributed meta-order lengths fail to produce power-law trade-sign autocorrelations, would falsify the derivations.

read the original abstract

Starting with a coupled discrete reaction--diffusion formulation for the lit and latent order books with non-uniformly sampled event times and meta-order source terms we show how two familiar market-microstructure regularities can emerge from this framework: the long-memory of trade signs associated with the Lillo--Mike--Farmer (LMF) theory and the square-root law (SQRL) of meta-order impact. This uses the locally linear order book and constant participation-rate execution in the front dynamics to reduce the dynamics to a Volterra equation whose leading-order solution then yields the well know result of concave impact trajectory, and a completion impact proportional to the square root of the meta-order size. We then use the interface representation to show how heavy-tailed Pareto meta-order lengths generate power-law trade-sign autocorrelations through the source term. These are familiar derivations, what is slightly different here is that we reinterpret these known derivations to make it clear that LMF law is an event-time sign-memory statement, whereas the square-root law is a physical-time viability statement where subordination can alter the calendar-time impact trajectories depending on the mappings and interpolation used to set continuum operational time.

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

This paper recovers the LMF sign memory and square-root impact from a reaction-diffusion order book model, but the advance is the framing and time-subordination angle rather than new derivations.

read the letter

This paper gives a reaction-diffusion model for the lit and latent order books that recovers both the long-memory trade sign correlations from Lillo-Mike-Farmer and the square-root law for meta-order impact. The twist is framing the first as an event-time statement and the second as a physical-time one that can change under subordination.

It does this by starting with discrete coupled dynamics, adding meta-order source terms, and using non-uniform sampling. The locally linear book and constant participation rate turn the front dynamics into a Volterra equation. The leading solution then gives the concave trajectory and square-root impact at completion. Heavy-tailed Pareto lengths for meta-orders then produce the power-law autocorrelations through the source term.

The model is explicit about its assumptions and the paper is honest that these are familiar derivations. What is different is the reinterpretation in terms of event versus physical time and the use of subordination to connect calendar time impact.

The main limitation is that the contribution sits in the modeling framework rather than in new results. The abstract does not claim independent predictions, and the Volterra step depends on the chosen assumptions holding exactly. If those assumptions are relaxed, it is not clear whether the same functional forms survive without adjustment. The full derivations would need checking for any post-hoc tuning.

This work is aimed at people who build detailed order book models in quantitative finance. A reader interested in how different time scales affect impact laws could get something from the subordination discussion. It is solid enough on its own terms to go to peer review, though the referee should focus on whether the framework adds explanatory power beyond the known results it recovers.

Referee Report

0 major / 1 minor

Summary. The paper starts with a coupled discrete reaction--diffusion formulation for the lit and latent order books with non-uniformly sampled event times and meta-order source terms. It demonstrates the emergence of the Lillo--Mike--Farmer (LMF) long-memory of trade signs from heavy-tailed Pareto meta-order lengths using the interface representation. It also shows the square-root law (SQRL) of meta-order impact by reducing the front dynamics to a Volterra equation under locally linear order book and constant participation-rate execution, leading to concave impact trajectory and square-root completion impact. The work reinterprets these as an event-time sign-memory statement for LMF and a physical-time viability statement for SQRL, with notes on how subordination can alter calendar-time impact trajectories depending on mappings and interpolation.

Significance. If the derivations hold, this work offers a unified reaction-diffusion framework from which both the LMF and SQRL regularities can be recovered. The explicit distinction between event-time and physical-time aspects, along with the discussion of subordination effects, provides a valuable perspective on the temporal robustness of these laws. The paper transparently presents the results as re-derivations of known results with a reinterpretation, so concerns about circular construction do not apply; the contribution lies in the unified modeling and time-scale clarification.

minor comments (1)

[Abstract] Abstract (Volterra reduction paragraph): the description of how the locally linear order book and constant participation-rate assumptions reduce the dynamics to a Volterra equation could include a brief indication of the equation's form or the leading-order analysis steps, to strengthen accessibility of the central SQRL derivation.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive and accurate summary of our work. We are pleased that the unified reaction-diffusion framework, the event-time versus physical-time distinction, and the subordination discussion are viewed as providing a valuable perspective. No major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity; derivations explicitly presented as reinterpretations of known results.

full rationale

The paper states upfront that LMF sign memory and SQRL impact are 'familiar derivations' recovered via standard assumptions (locally linear order book, constant participation rate, Volterra reduction, heavy-tailed meta-order lengths). No step reduces a claimed prediction to a fitted input or self-citation by construction; the framework is offered as a unifying reinterpretation rather than an independent first-principles derivation of new functional forms. No self-citations, ansatzes smuggled via prior work, or uniqueness theorems are invoked in the provided text. This matches the common case of a modeling paper recovering known empirical patterns under explicit assumptions.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 0 invented entities

The central claim rests on three modeling choices that are introduced without independent justification in the abstract: locally linear order book shape, constant participation rate, and heavy-tailed Pareto meta-order lengths. No free parameters or new entities are named.

axioms (3)

domain assumption The order book is locally linear near the mid-price
Invoked to reduce the reaction-diffusion system to a Volterra equation (abstract).
domain assumption Constant participation-rate execution governs the front dynamics
Required for the leading-order concave impact trajectory (abstract).
domain assumption Meta-order lengths follow a heavy-tailed Pareto distribution
Used to generate power-law trade-sign autocorrelations via the source term (abstract).

pith-pipeline@v0.9.1-grok · 5732 in / 1519 out tokens · 56819 ms · 2026-06-27T02:30:06.874161+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

25 extracted references · 19 canonical work pages · 3 internal anchors

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work page doi:10.1016/j.jcp.2014 2015
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doi:10.1080/14697688.2018.1504167

Benzaquen, M., Bouchaud, J.P., 2018. Market im- pact with multi-timescale liquidity. Quantitative Fi- nance 18, 1781–1790. doi:10.1080/14697688.2018. 1444279

work page doi:10.1080/14697688.2018 2018
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2013
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work page doi:10.1017/9781316659335
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work page internal anchor Pith review Pith/arXiv arXiv doi:10.1088/1469-7688/ 2004
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Re- visiting the epps effect using volume time averag- ing: An exercise in r

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work page doi:10.48550/arxiv.1912.02416 2020
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Anomalous diffusion and price impact in the fluid-limit of an order book

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A fully consistent, minimal model for non- linear market impact

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work page doi:10.1080/14697680903373692
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The impact of asyn- chronous trading on epps effect

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work page doi:10.7494/manage.2016.17.1.59 2016
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Agent-based models for latent liquidity and concave price impact

Mastromatteo, I., Tóth, B., Bouchaud, J.P., 2014. Agent-based models for latent liquidity and concave price impact. Physical Review E 89, 042805. doi:10. 1103/PhysRevE.89.042805

2014
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Stochastic Models for Fractional Calculus

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2012
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Market impact and trading profile of large trading orders in stock markets

Moro, E., Vicente, J., Moyano, L.G., Gerig, A., Farmer, J.D., Vaglica, G., Lillo, F., Mantegna, R.N., 2009. Market impact and trading profile of hidden orders in stock markets. Physical Re- view E 80, 066102. URL:https://doi.org/10. 1103/PhysRevE.80.066102, doi:10.1103/PhysRevE. 80.066102,arXiv:0908.0202

work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physreve 2009
[23]

A simple microstructural expla- nation of the concavity of price impact

Nadtochiy, S., 2022. A simple microstructural expla- nation of the concavity of price impact. Mathemati- cal Finance 32, 78–113. URL:https://doi.org/10. 1111/mafi.12314, doi:10.1111/mafi.12314

work page doi:10.1111/mafi.12314 2022
[24]

Inferring micro- scopic financial information from the long memory in market-order flow: A quantitative test of the lillo– mike–farmer model

Sato, Y., Kanazawa, K., 2023. Inferring micro- scopic financial information from the long memory in market-order flow: A quantitative test of the lillo– mike–farmer model. Physical Review Letters 131, 197401. doi:10.1103/PhysRevLett.131.197401

work page doi:10.1103/physrevlett.131.197401 2023
[25]

Exact solution to a generalised lillo–mike–farmer model with heteroge- neous order-splitting strategies

Sato, Y., Kanazawa, K., 2024. Exact solution to a generalised lillo–mike–farmer model with heteroge- neous order-splitting strategies. Journal of Statistical Physics 191. doi:10.1007/s10955-024-03264-1

work page doi:10.1007/s10955-024-03264-1 2024
[26]

Anomalous price impact and the critical nature of liquidity in financial markets

Tóth, B., Lempérière, Y., Deremble, C., de Latail- lade, J., Kockelkoren, J., Bouchaud, J.P., 2011. Anomalous price impact and the critical nature of liquidity in financial markets. Physical Review X 1, 021006. doi:10.1103/PhysRevX.1.021006. 13

work page doi:10.1103/physrevx.1.021006 2011

[1] [1]

doi:10.1111/0022-1082.00286

Ané, T., Geman, H., 2000. Order flow, transac- tion clock, and normality of asset returns. The Journal of Finance 55, 2259–2284. URL:https:// doi.org/10.1111/0022-1082.00286, doi:10.1111/ 0022-1082.00286

work page doi:10.1111/0022-1082.00286 2000

[2] [2]

Non-unique time and market incompleteness

Angstmann, C., Gebbie, T., 2026. Non-unique time and market incompleteness. URL:https:// arxiv.org/abs/2604.23608, doi:10.48550/arXiv. 2604.23608,arXiv:2604.23608

work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv 2026

[3] [3]

A discrete time random walk model for anomalous diffusion

Angstmann, C.N., Donnelly, I.C., Henry, B.I., Nichols, J.A., 2015. A discrete time random walk model for anomalous diffusion. Journal of Computa- tional Physics 293, 53–69. doi:10.1016/j.jcp.2014. 08.003

work page doi:10.1016/j.jcp.2014 2015

[4] [4]

doi:10.1080/14697688.2018.1504167

Benzaquen, M., Bouchaud, J.P., 2018. Market im- pact with multi-timescale liquidity. Quantitative Fi- nance 18, 1781–1790. doi:10.1080/14697688.2018. 1444279

work page doi:10.1080/14697688.2018 2018

[5] [5]

The non-linear market impact of large trades: Evidence from buy- side order flow

Bershova, N., Rakhlin, D., 2013. The non-linear market impact of large trades: Evidence from buy- side order flow. Quantitative Finance 13, 1759–

2013

[6] [6]

Gould, Mason A

URL:https://doi.org/10.1080/14697688. 2013.861076, doi:10.1080/14697688.2013.861076

work page doi:10.1080/14697688 2013

[7] [7]

Bouchaud, J.P., Bonart, J., Donier, J., Gould, M.D.,

[8] [8]

Cambridge University Press, Cambridge

Trades, Quotes and Prices: Financial Markets Under the Microscope. Cambridge University Press, Cambridge. doi:10.1017/9781316659335

work page doi:10.1017/9781316659335

[9] [9]

How markets slowly digest changes in supply and demand, in: Handbook of Financial Markets: Dynamics and Evolution

Bouchaud, J.P., Farmer, J.D., Lillo, F., 2009. How markets slowly digest changes in supply and demand, in: Handbook of Financial Markets: Dynamics and Evolution. Elsevier, pp. 57–160

2009

[10] [10]

Fluctuations and response in financial markets: the subtle nature of `random' price changes

Bouchaud, J.P., Gefen, Y., Potters, M., Wyart, M., 2004. Fluctuations and response in fi- nancial markets: The subtle nature of random price changes. Quantitative Finance 4, 176–190. URL:https://doi.org/10.1088/1469-7688/ 4/2/007, doi:10.1088/1469-7688/4/2/007, arXiv:cond-mat/0307332

work page internal anchor Pith review Pith/arXiv arXiv doi:10.1088/1469-7688/ 2004

[11] [11]

Re- visiting the epps effect using volume time averag- ing: An exercise in r

Chang, P., Bukuru, R., Gebbie, T., 2020. Re- visiting the epps effect using volume time averag- ing: An exercise in r. URL:https://arxiv.org/ abs/1912.02416, doi:10.48550/arXiv.1912.02416, arXiv:1912.02416

work page doi:10.48550/arxiv.1912.02416 2020

[12] [12]

A subordinated stochastic pro- cess model with finite variance for speculative prices

Clark, P.K., 1973. A subordinated stochastic pro- cess model with finite variance for speculative prices. Econometrica 41, 135–155. URL:https://doi.org/ 10.2307/1913889, doi:10.2307/1913889

work page doi:10.2307/1913889 1973

[13] [13]

Anomalous diffusion and price impact in the fluid-limit of an order book

Diana, D., Gebbie, T., 2024. Anomalous diffusion and price impact in the fluid-limit of an order book. Journal of Computational and Applied Mathematics 445, 116202. doi:10.1016/j.cam.2024.116202

work page doi:10.1016/j.cam.2024.116202 2024

[14] [14]

A fully consistent, minimal model for non- linear market impact

Donier, J., Bonart, J., Mastromatteo, I., Bouchaud, J.P., 2015. A fully consistent, minimal model for non- linear market impact. Quantitative Finance 15, 1109–

2015

[15] [15]

doi:10.1080/14697688.2015.1040056. 12

work page doi:10.1080/14697688.2015.1040056 2015

[16] [17]

No-dynamic-arbitrage and mar- ket impact

Gatheral, J., 2010b. No-dynamic-arbitrage and mar- ket impact. Quantitative Finance 10, 749–759. doi:10.1080/14697680903373692

work page doi:10.1080/14697680903373692

[17] [18]

The impact of asyn- chronous trading on epps effect

Gurgul, H., Machno, A., 2016. The impact of asyn- chronous trading on epps effect. comparative study on warsaw stock exchange and vienna stock exchange. Managerial Economics 17, 59–75. URL:https: //doi.org/10.7494/manage.2016.17.1.59, doi:10. 7494/manage.2016.17.1.59

work page doi:10.7494/manage.2016.17.1.59 2016

[18] [19]

Theoryforlong memory in supply and demand

Lillo, F., Mike, S., Farmer, J.D., 2005. Theoryforlong memory in supply and demand. Physical Review E 71, 066122. doi:10.1103/PhysRevE.71.066122

work page doi:10.1103/physreve.71.066122 2005

[19] [20]

Agent-based models for latent liquidity and concave price impact

Mastromatteo, I., Tóth, B., Bouchaud, J.P., 2014. Agent-based models for latent liquidity and concave price impact. Physical Review E 89, 042805. doi:10. 1103/PhysRevE.89.042805

2014

[20] [21]

Stochastic Models for Fractional Calculus

Meerschaert, M.M., Sikorskii, A., 2012. Stochastic Models for Fractional Calculus. De Gruyter, Berlin

2012

[21] [22]

Market impact and trading profile of large trading orders in stock markets

Moro, E., Vicente, J., Moyano, L.G., Gerig, A., Farmer, J.D., Vaglica, G., Lillo, F., Mantegna, R.N., 2009. Market impact and trading profile of hidden orders in stock markets. Physical Re- view E 80, 066102. URL:https://doi.org/10. 1103/PhysRevE.80.066102, doi:10.1103/PhysRevE. 80.066102,arXiv:0908.0202

work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physreve 2009

[22] [23]

A simple microstructural expla- nation of the concavity of price impact

Nadtochiy, S., 2022. A simple microstructural expla- nation of the concavity of price impact. Mathemati- cal Finance 32, 78–113. URL:https://doi.org/10. 1111/mafi.12314, doi:10.1111/mafi.12314

work page doi:10.1111/mafi.12314 2022

[23] [24]

Inferring micro- scopic financial information from the long memory in market-order flow: A quantitative test of the lillo– mike–farmer model

Sato, Y., Kanazawa, K., 2023. Inferring micro- scopic financial information from the long memory in market-order flow: A quantitative test of the lillo– mike–farmer model. Physical Review Letters 131, 197401. doi:10.1103/PhysRevLett.131.197401

work page doi:10.1103/physrevlett.131.197401 2023

[24] [25]

Exact solution to a generalised lillo–mike–farmer model with heteroge- neous order-splitting strategies

Sato, Y., Kanazawa, K., 2024. Exact solution to a generalised lillo–mike–farmer model with heteroge- neous order-splitting strategies. Journal of Statistical Physics 191. doi:10.1007/s10955-024-03264-1

work page doi:10.1007/s10955-024-03264-1 2024

[25] [26]

Anomalous price impact and the critical nature of liquidity in financial markets

Tóth, B., Lempérière, Y., Deremble, C., de Latail- lade, J., Kockelkoren, J., Bouchaud, J.P., 2011. Anomalous price impact and the critical nature of liquidity in financial markets. Physical Review X 1, 021006. doi:10.1103/PhysRevX.1.021006. 13

work page doi:10.1103/physrevx.1.021006 2011