On Bonart's interpretation of the Square-Root Impact Law
Abstract
The square-root impact law (SRIL), I = Yσ√(Q/V), bundles two facts that a single mechanism must explain at once: a shape (impact proportional to square-root of traded volume Q) and an amplitude (Y=O(1), independent of the participation rate φ). Bonart has recently proposed an elegant solution: if realized and counterfactual prices are both diffusive, information-neutral impact must have white increments and the SRIL follows without the wart. We reformulate and simplify his argument, and foreground the ingredient he himself regards as essential --- that the market whitens the stationary stream of each participant's metaorders, not any isolated one. The Lillo--Mike--Farmer model satisfies his premises, yet gives an isolated metaorder the super-square-root impact φ(1−γ)/2Q(1+γ)/2, γ∈(0,1), because single metaorders (i.e. not part of a stream) are statistically invisible and thus priced mechanically. We then construct an explicit multi-agent propagator that realizes Bonart's whitening, and show it reproduces the clean SRIL precisely when the market can attribute trades to their issuer. But correct re-attribution of anonymous trades is a formidable task, which requires markets to behave as implausibly efficient signal processing machines. The stream hypothesis makes a sharp, falsifiable prediction --- isolated metaorders, well separated in time, should not obey the SRIL --- that does not seem agree with empirical data known to us. Independently, a volume-conservation argument singles out the square-root law and lays bare the step the diffusivity route must assume: diffusivity fixes the variance of impact --- an additive, stationary, age-blind quantity --- whereas the SRIL is a law for the mean impact, which marginally decays as 1/√(age) and therefore demands an origin of time.
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BibTeX
@misc{bouchaud2026bonarts,
title = {{On Bonart's interpretation of the Square-Root Impact Law}},
author = {J. -P. Bouchaud and I. Mastromatteo and B. Toth},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.10053},
eprint = {2610.10053},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.10053},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - On Bonart's interpretation of the Square-Root Impact Law
AU - J. -P. Bouchaud
AU - I. Mastromatteo
AU - B. Toth
PY - 2026
DA - 2026/10/07
PB - arXiv
DO - 10.48550/arxiv.2610.10053
UR - https://arxiv.org/abs/2610.10053
AB - The square-root impact law (SRIL), I = Yσ√(Q/V), bundles two facts that a single mechanism must explain at once: a shape (impact proportional to square-root of traded volume Q) and an amplitude (Y=O(1), independent of the participation rate φ). Bonart has recently proposed an elegant solution: if realized and counterfactual prices are both diffusive, information-neutral impact must have white increments and the SRIL follows without the wart. We reformulate and simplify his argument, and foreground the ingredient he himself regards as essential --- that the market whitens the stationary stream of each participant's metaorders, not any isolated one. The Lillo--Mike--Farmer model satisfies his premises, yet gives an isolated metaorder the super-square-root impact φ^(1−γ)/2Q^(1+γ)/2, γ∈(0,1), because single metaorders (i.e. not part of a stream) are statistically invisible and thus priced mechanically. We then construct an explicit multi-agent propagator that realizes Bonart's whitening, and show it reproduces the clean SRIL precisely when the market can attribute trades to their issuer. But correct re-attribution of anonymous trades is a formidable task, which requires markets to behave as implausibly efficient signal processing machines. The stream hypothesis makes a sharp, falsifiable prediction --- isolated metaorders, well separated in time, should not obey the SRIL --- that does not seem agree with empirical data known to us. Independently, a volume-conservation argument singles out the square-root law and lays bare the step the diffusivity route must assume: diffusivity fixes the variance of impact --- an additive, stationary, age-blind quantity --- whereas the SRIL is a law for the mean impact, which marginally decays as 1/√(age) and therefore demands an origin of time.
ER -