Keep the square-root law in your cost model, but give price diffusivity less credit for it. Bouchaud, Mastromatteo and Tóth make a convincing case through algebra. They leave the positive mechanism for a companion paper, and flag evidence on isolated orders from CFM (the fund the authors work at) as under revision.
The shape and the amplitude
The law is I ≃ Yσ√(Q/V). Metaorder impact grows with the square root of size Q relative to daily volume V; its prefactor Y is order one. Any account of the law has to explain both that shape and an amplitude independent of participation rate φ, the share of market volume taken while the order runs.
Fixed-kernel propagator models produce the shape but leave a rate-dependent amplitude. With a self-similar kernel decaying as τ^(-β), they give I ∼ φ^β Q^(1-β). Force √Q and Y ∝ √φ follows: the authors' "√φ wart". Equal-sized orders executed at φ = 0.1% and φ = 1% would differ in impact by √10 ≈ 3.2. The authors say the data show no such gap where the law is measured, Q/V ≲ 0.1.
Bonart offers a way around it. Write realized returns as a counterfactual "market-without-me" plus your own impact, then require both paths to be diffusive. If your flow is uncorrelated with the counterfactual, impact increments must be white. Their added variance grows with the number of trades you execute; taking its square root gives √Q, with φ entering only through Q. The authors locate the leap in equating mean impact with its standard deviation. As they put it, "variance has no clock, while the mean requires one." For the mean to grow as a square root, each successive child order must move price less according to when the metaorder began. A stationary variance budget has no start time.
A diffusive counterexample
The Lillo-Mike-Farmer model makes the objection concrete. Its power-law distribution of metaorder lengths produces long memory in trade signs, whose correlations decay as ℓ^(-γ). A participant-blind kernel with exponent β = (1−γ)/2 whitens the price. Bonart's premises about the background market all hold.
Add another metaorder drawn from that same length distribution. The market sees no change in its statistics and prices the order mechanically, giving impact φ^((1−γ)/2) Q^((1+γ)/2). At empirical γ ≈ 1/2, the result is φ^(1/4) Q^(3/4). Both the shape and the amplitude miss the law.
Bonart anticipates the objection: he treats a participant's stream of metaorders, rather than an isolated order, as what the market whitens. The authors examine what the market would need to know for that to work.
Can the tape identify a trader?
In their multi-agent propagator, the market perceives a flat plateau C in each participant's sign correlation while that participant's order runs. Peak impact is σ√(φ/C)√Q. The law follows if and only if C ∝ φ, the scaling available when the market can sort anonymous trades into persistent per-trader streams. A market estimating excess correlation from an anonymous tape instead sees C ∼ φ². Its result is an "anti-wart", Y ∝ φ^(-1/2): impact falls as the trader goes faster.
Detection takes time, too. At γ = 1/2, with sign-correlation amplitude C0 = 0.01 and φ = 10%, a coherent order remains invisible to a correlation-based estimator for about 100 trades. Set C0 = 0.1 and φ = 1%, and the estimator needs about 10^6 trades. Low participation is precisely where the law fits best. The authors allow that "smart detection" using size, timing, venue and broker footprints could cut those thresholds substantially. They answer that a well-executed order leaves little footprint, whereas the law appears independent of execution style. I find this persuasive for sign-correlation estimators. An HFT reading broker tags poses a harder case, and 10^6 bounds only the sign-correlation channel.
Impact decay creates another problem. White increments determine the magnitude of the impact response while leaving its phase free. Under a pure all-pass response, permanent impact has magnitude G0, the peak scale; only its sign can change. The studies cited in the paper find uninformed impact relaxing close to zero, beyond what a pure all-pass can deliver. Fixing the decay with the plateau construction again requires detection.
A budget for daily variance
The positive argument starts from Cδ Y² M^(1−2δ) = 1 − b², a daily variance budget in which M counts metaorders and δ is the impact exponent. At δ = 1/2 alone, Y stays independent of M and Y² = 1 − b². A linear law instead requires Y ∝ √M. Using an earlier empirical ratio, Y/b ≈ 1/3, the authors obtain Y ≈ 0.32 at end of day, against a commonly reported value of about 0.5.
Their claim is conditional: if Y stays fixed as the number of simultaneous metaorders changes, volume conservation permits only the square-root exponent. The condition carries much of the weight. Read conservatively, the budget rules out exponents far from 1/2; fixing exactly 1/2 also requires ⟨x^(2δ)⟩ to stay steady as M changes. The authors deliberately set herding between metaorders aside. Yet work they cite by Maitrier, Loeper and Bouchaud finds cross-metaorder correlation necessary to restore diffusivity after accounting for impact decay. That finding bears directly on the budget's independence assumption.
The authors also argue that the law's empirical success shows metaorders contribute an O(1) share of volatility. This uses the observed law to support a premise meant to explain it. A universal Y would force √Q. Why Y should be universal remains among the authors' open questions.
Isolated orders would settle the narrower dispute
Bonart's stream hypothesis allows two readings. On the excursion reading, a metaorder is a typical fluctuation in a participant's continuing round-trip churn. On the attribution reading, the market learns which trader issued the orders and whitens a recognizable repeat participant regardless of the spacing between orders. The excursion reading predicts that "a single metaorder starting long after the previous one ended should not obey the SRIL" (the square-root impact law). The authors call that "at odds with our available CFM data, which we are revisiting because of the importance of that claim." This comparison could rule out only the excursion reading; attribution would survive. The paper does not show the CFM data behind it.
We could not run that comparison. Our minute OHLCV bars have neither trade initiators nor links from child orders to a parent, so we cannot separate isolated metaorders from recurring streams.
One result would change my mind. Isolated, well-separated metaorders with a fitted exponent near (1+γ)/2 (3/4 at γ ≈ 1/2) and residual φ dependence would give the excursion reading of Bonart's stream picture real support. Clean √Q on isolated orders would leave attribution standing. Until CFM publishes that split, fit the cost curve to your own isolated parent orders. Treat the square root as an empirical regularity: at low participation, every derivation here assumes either detection of the metaorder's start or an amplitude independent of metaorder count.