June 05, 2026
Decomposing impact as the difference between realized and counterfactual returns and requiring both to be diffusive yields an identity that restricts admissible impact scaling at the level of individual participants. This constraint implies the square-root law in the information-neutral regime and a crossover to linear impact under strong informational coupling, consistent with empirical observations. In the weak-coupling regime, cumulative market impact is itself diffusive – a diagnostic that many propagator and latent liquidity models fail to satisfy.
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A central empirical regularity of financial markets is that return autocorrelations are vanishingly small over time scales spanning more than five orders of magnitude, from sub-minute intervals to weeks [1].
This absence of predictability is generally viewed as structural: arbitrage eliminates persistent correlations. Price diffusivity is therefore adaptive and can coexist with strongly autocorrelated order flow, provided that market impact and liquidity responses compensate these correlations in a precise manner [2]–[4].
Market impact is the causal response of prices to trades – the difference between realized and counterfactual prices that would prevail in the absence of a given trading strategy. Hence diffusivity, as a structural property of both realized and counterfactual prices, imposes strong a priori constraints on impact itself.
This constraint operates at the level of individual market participants, and is therefore more restrictive than earlier balancing arguments based on aggregate order flow [4] or metaorder flow [5], [6]. Under weak, empirically relevant assumptions, the resulting consistency condition essentially determines how market impact scales with time and traded volume. Two regimes emerge. When trading is uncorrelated with underlying price movements – the information-neutral regime – diffusivity forces impact increments to be effectively white, implying square-root scaling of its typical magnitude. When trading is instead aligned with underlying returns, impact grows linearly with volume. This reproduces both the square-root law of metaorder impact [7]–[14] and the recently observed linear-to-sublinear crossover [13], thereby recovering the empirical phenomenology.
The square-root law has often been regarded as anomalous [4], [15]. Our approach suggests rather the opposite: square-root impact is not anomalous but a direct consequence of price diffusivity, independent of specific assumptions about aggregate demand and supply dynamics. More precisely, when trading is uncorrelated with underlying price movements, impact itself must be diffusive – a structural requirement that models based on exogenous, universal kernels generally violate, raising the possibility that many propagator and latent liquidity models mischaracterize market impact.
Consider an investor trading in a market. Let \((x_t)\) denote demeaned returns measured over time bins of size \(\Delta t\). We define the impact of the investor’s trades, \(j_t\), as the difference between the realized return and the counterfactual return \(y_t\) that would have prevailed in the absence of trading: \[\label{eq:decomposition} x_t = y_t + j_t \; .\tag{1}\]
Structural price diffusivity imposes constraints on both \((x_t)\) and \((y_t)\). In the absence of persistent arbitrage opportunities, returns must be uncorrelated. This condition is enforced adaptively: arbitrageurs continuously calibrate to the statistical composition of order flow, restoring diffusivity when participants enter or exit the market. The counterfactual price \((y_t)\), corresponding to the scenario where the investor does not trade, is therefore also expected to be diffusive – a fortiori so when the investor is large. Sec. 6 derives a quantitative detectability criterion that makes this intuition precise.
Accordingly, we impose \[\begin{align} \label{eq:realized} R_{xx}[t] &= \Delta t \, \sigma_X^2 \, \delta[t] \\ R_{yy}[t] &= \Delta t \, \sigma_Y^2 \, \delta[t] \;, \end{align}\tag{2}\] where \(R_{ab}[t] = \mathbb{E}[a_{t+k} b_k]\) for any centered processes \(a,b\), \(\delta\) is the Kronecker delta, and \(\sigma_X, \sigma_Y\) denote the volatilities of \((x_t)\) and \((y_t)\). While \(\sigma_X\) is observable, \(\sigma_Y\) is not.
This constraint holds in an ex-post setting where trading activity is stationary, so that arbitrage can enforce diffusivity. It does not apply to transient interventions or point-in-time optimization (see discussion).
Under these assumptions, Eq. (1 ) implies \[\label{eq:fundamental} R_{yj}[t] + R_{jy}[t] + R_{jj}[t] = \Delta t \left(\sigma_X^2 - \sigma_Y^2\right) \delta[t] \;.\tag{3}\] This constraint operates at the level of individual trading strategies.
Note that the relevant object is not an individual execution episode – such as a metaorder – but the trading strategy as a stationary stochastic process. Adaptation need not occur within a single metaorder; it will, however, occur over the lifetime of any persistent, stationary strategy, as arbitrageurs respond to its unconditional flow statistics. Stationarity is an empirically relevant idealization: typical investors make only rare or incremental changes to their trading envelope – deployed capital, risk aversion, informational alpha – so that the statistics of their order flow remain stable over macroscopic horizons.
A participant’s informational alpha enters Eq. (3 ) through the coupling \(R_{jy}\). In the information-neutral regime, \[R_{jy}[t]= 0 \,, \qquad \text{for all t}\,.\] A technical subtlety is that both order flow and counterfactual returns may depend on common market-state variables, such as liquidity or volatility. Formally, order flow carries no information about \((y_t)\) if \(q \perp y|\mathfrak h\), where \(\mathfrak h\) denotes all non-directional market states. Since \(j\) is determined by \(q\) and \(\mathfrak h\), this implies \({\mathbb{E}}[j_t|y, \mathfrak h]={\mathbb{E}}[j_t|\mathfrak h]\), which vanishes when \(j\) is centered conditional on \(\mathfrak h\); therefore \({\mathbb{E}}[j_t|y]=0\) and \(R_{jy}=0\).
This regime is relevant in practice, as a substantial fraction of institutional trading – such as risk rebalancing, hedging, or long-horizon allocation – is not or only weakly driven by short-term return forecasts. Eq. (3 ) then reduces to \[\label{eq:fundamental2} R_{jj}[t] = (\sigma_X^2 - \sigma_Y^2) \,\Delta t \, \delta[t]\,.\tag{4}\] Thus, structural diffusivity implies that impact increments are white for every information-neutral participant: the impact mechanism acts as a whitening transformation of individual order flow. The second-order statistics of impact are therefore independent of the temporal fine-structure of the underlying trading strategy and fully characterized by their instantaneous variance. Any mechanical impact model compatible with structural diffusivity must satisfy the requirement Eq. (4 ). Clearly, this requirement cannot, in general, be satisfied by models with universal, strategy-agnostic impact kernels.
Earlier market impact models enforced price diffusivity through a balance between aggregate order-flow correlations and impact decay [16]. Diffusivity was then guaranteed only for the realized price under the flow statistics observed in the market. By contrast, counterfactual diffusivity requires invariance under changes in order flow composition: this constrains impact at the level of individual participants rather than only at the aggregate level. The resulting Brownian scaling of information-neutral impact provides the key mechanism from which the square-root impact law emerges.
Market adaptability is key to this mechanism. Consider a market that is initially diffusive, with returns generated by both fundamentals and the aggregate flow of all participants except the participant under consideration. Let the participant introduce an additional order-flow component \((q_t)\). Let \[j^\text{raw} = G(q)\] denote the immediate mechanical price response to \((q_t)\), determined by the prevailing liquidity environment. In general, \(j^\text{raw}\) need not satisfy Eq. (3 ); indeed, if \((q_t)\) is autocorrelated, it will generally induce predictable returns, \[x_t^\text{raw} = y_t + G(q)\;.\] Such predictability creates statistical arbitrage opportunities. Liquidity providers, market makers, and arbitrageurs therefore adapt their behavior (e. g. modifying quotes) in response to the return patterns generated by \((G(q)_t)\). We denote the resulting equilibrium adjustments by \((K(G(q))_t)\), so that realized returns become: \[x_t = y_t + G(q)_t + K(G(q))_t\;.\] Both \((G(q)_t)\) and \((K(G(q))_t)\) are causal consequences of the participant’s trading activity: the former is the direct mechanical response, while the latter arises indirectly through the adaptive reaction of the rest of the market. Their sum therefore constitutes the total impact, \[j_t = G(q)_t + K(G(q))_t \;.\] Since the background process \((y_t)\) is already diffusive, any effect of adaptation to a new strategy is marginal to that strategy. This need not happen because market participants explicitly detect individual participants, but because competitive arbitrage continuously removes the return predictability that such flows would otherwise generate. Sec. 6 derives an explicit detectability criterion.
Eq. (4 ) does not imply that impact simply equals order flow surprise as, e. g., in [17]–[19]. Assume \((q_t)\) is generated from white noise \(\hat{\eta}\) through a causal, time-translational invariant (TTI) filter \(L\), \[q_t = (L\cdot \hat{\eta})_t \;, \qquad \hat{\eta} \text{ standardized white Gaussian}\;,\] where \(\cdot\) denotes convolution. Then, Eq. (4 ) only implies that \[\label{eq:jU} j_t \propto (U\cdot\hat{\eta})_t\,, \qquad \sum_{h\ge0}U[h]U[t+h] = \delta[t] \, \text{ for all t\ge0}\,.\tag{5}\] The causal all-pass (orthogonal) filter \(U\) is a degree of freedom not determined by structural price diffusivity. A simple example is \[\label{eq:allpass} U[t] = u_0 \delta[t] - (1-u_0^2) u_0^{t-1} 1_{t \ge 1} \,, \qquad 0<u_0<1\,,\tag{6}\] i. e. an exponential propagator with a certain additional instantaneous component. Therefore, impact parametrized by Eq. (5 ) whitens the flow and at the same time couples flow surprises in a nontrivial way, thereby allowing for transient effects such as impact decay – see Sec. 5.0.0.2.
We now examine the scaling of the zero-lag volatility factor \(\sigma_X^2 - \sigma_Y^2\).
Modern execution brokers determine single order placement and size as a function of locally available liquidity. Taking liquidity selectively implies that single-trade volume is only weakly correlated with target participation ratio, and single-trade impact only weakly correlated with size [1]. Since price is close to a random walk, an empirical relationship [1], [16] between the number of market transactions and volatility is therefore \[\label{eq:sigmadiff} \sigma_X^2 \simeq \frac{\lambda_X^2 N}{\Delta t}\,,\tag{7}\] where \(\lambda_X\) the volatility per trade (of the order of the bid-ask spread) and \(N\) the number of market transactions within the interval \(\Delta t\). This relationship is very robust both on small-tick [20] and large-tick assets, provided a continuous proxy of the true equilibrium price is used [1], [21].
Set \(\Delta t=1\) without loss of generality. Consider a strategy with a participation rate \(\mu\), i.e. \(\Delta N = \mu N\) trades in the market are due to the strategy. Counterfactual price diffusivity then implies \[\sigma_Y^2 \simeq \lambda_Y^2 \, (N-\Delta N)\,.\] Write \[\lambda_X=\lambda(0)\,,\qquad \lambda_Y=\lambda(\mu) \,.\] Taylor expand around small \(\mu\): \[\label{eq:impactvol} \sigma_X^2 - \sigma_Y^2 \simeq \left(\lambda^2(0) - \frac{\partial\lambda^2}{\partial \mu}(0) \right) \Delta N = \lambda_\text{imp}^2 N \mu = \mathcal{O}(N\mu) \,,\tag{8}\] where \(\lambda_\text{imp}\) measures the typical single-trade impact of the strategy. Thus, \(\sigma_X^2 - \sigma_Y^2\) scales linearly in the participation rate (at fixed \(N\)).
This result can be further understood as follows: assume price dynamics are driven by many heterogeneous strategies, so total order flow can be decomposed into many orthogonal sub-flows with individual \(\mu^{(r)}\), each contributing independently to total volatility with \(\sigma_r^2\). Therefore, \[\lambda_X^2 N = \sum_r \sigma_r^2 = \lambda_X^2 \sum_r (N \mu_r)\,.\] Each orthogonal sub-flow only ‘sees’ the aggregate rest, so \(\sigma_r^2 = \sigma_r^2\left(\mu_r, 1-\mu_r\right)\) which is thus a function of \(\mu_r\) only. Write \(\sigma_r^2 = S(\mu_r)\). Since prices should be structurally diffusive irrespective of the decomposition \(\{\mu_r\}\), \(S\) must be linear.
Eq. (3 ), Eq. (4 ) and Eq. (8 ) impose non-trivial constraints on how impact scales with trading activity. Under mild and empirically relevant assumptions, these constraints imply a diffusive impact in an information-neutral setting, a square-root scaling in duration, a linear-to-square-root crossover of impact as a function of trading rate and informational alpha and, consequently, the empirical square-root law of metaorder impact.
Denote the impact increment over a horizon \(h\) by \[I_t = \sum_{k=t-h/\Delta t+1}^t j_{k}\;.\] Write \[I_t = I_t(h, \mu)\] to make the dependence on both horizon \(h\) and participation rate \(\mu\) explicit. In the hydrodynamic limit, \(\Delta t\to 0\), Eq. (4 ) and Eq. (8 ) imply \[\label{eq:scalingY} I_t(h, \mu) \sim \sqrt{h \mu} \, I_{t}(1, 1)\; \text{ in distribution}\,, \qquad {\mathrm{Std}}[I_{t}(1, 1)] = \lambda_\text{imp}\sqrt{N}\;.\tag{9}\]
Assuming that metaorders are typical signed excursions of stationary trading activity, conditioning on their direction and trading rate \(\mu\) over the interval \([t-h,t]\) fixes the sign and prefactor but leaves the scaling exponent unchanged (see below): \[\label{eq:scalingYm} I_t(h, \mu) | \text{metaorder} \propto \sqrt{h \mu} \, I_t(1,1) | \text{metaorder}\;,\tag{10}\] where the proportionality constant does not depend on \(\mu\) and \(h\). We discuss metaorder conditioning more rigorously below.
\(I_t(1,1)| \text{metaorder}\) is positive for a buy and negative for a sell metaorder. Matching to single-trade impact, Eq. (9 ) implies \[I_t(h, \mu) | \text{metaorder} \propto \text{sign}(Q)\, \lambda_\text{imp} \, \sqrt{\frac{|Q|}{\nu}}\;,\] where \(\nu\) is the typical trade size and \(Q=h\nu\mu N\) the total volume of the metaorder. Note that \(|Q|/\nu\) is the number of executions from a metaorder; the square-root law in duration and trading rate is therefore equivalent to a square-root law in number of child orders. Daily volatility scales as \(\sigma_X \sim \lambda_X\sqrt{N}\), daily volume as \(V \sim \nu N\), and \(\lambda_\text{imp} \approx \lambda_X\) if strategy-executions are market typical. Then, \[I_t(h, \mu) | \text{metaorder} \propto \text{sign}(Q) \, \sigma_X \, \sqrt{\frac{|Q|}{V}}\;.\] This is the canonical square-root law [1].
This scaling is valid until (unmodeled) restoring forces set in. We cannot pin down the precise horizon when this happens; trend and mean-reversion effects appear insufficient, over horizons shorter than several weeks, to invalidate the conclusions drawn from structural price diffusivity. To quantify this intuition, consider a trader executing \(1-2\%\) of daily volume. This corresponds to a weekly impact level of the order of \(25\%\) of volatility, and a quarterly impact level around \(100\%\), since \(\sqrt{63} \times \sqrt{0.02} \approx 1\). The result is plausible: the mechanical mispricing caused by this trading strategy can reach \(50-200\) basis points for a typical stock (\(2\%\) daily volatility) until major deviations from theory are expected.
Sec. 6 derives a high-frequency bound on the regime of validity.
The transition from Eq. (9 ) to Eq. (10 ) requires that metaorders generate the typical fluctuations of impact.
Consider a strategy consisting of metaorders with fixed duration and trading rate \((h,\mu)\), and choose the bin length \(\Delta t = h\). Assume metaorder signs are generated by a ternary discretization of an autocorrelated latent Gaussian process \((\eta_t)\) via a causal TTI filter, \[\eta_t = (\tilde{L} \cdot \hat{\eta})_t \;, \qquad \hat{\eta}_t \sim \text{ standardized white Gaussian} \,,\] so that \[\epsilon_t = \begin{cases} +1 \,, & \eta_t > a \\ -1 \,,& \eta_t < -a \\ 0 \,,& \text{else} \end{cases}\;,\] where \(a>0\) is some threshold. Impact is diffusive when \[j_t = A(h, \mu)\,(\tilde{U}\cdot \hat{\eta})_t\,,\] where \(\tilde{U}\) is again a causal all-pass, and \[A(h, \mu) \sim \sqrt{\mu h}\] is the impact amplitude fixed by impact diffusivity.
Applying the Gaussian regression formula, we find \[{\mathbb{E}}[j_t|\eta_t] = \eta_t\,\frac{\mathrm{Cov}(j_t, \eta_t)}{\mathrm{Var}(\eta_t)} = \eta_t\,A(h, \mu)\,\frac{\sum_{k\ge 0} \tilde{U}[k]\,\tilde{L}[k]}{\sum_{k\ge 0} \tilde{L}[k]^2}\,,\] and, using \({\mathbb{E}}[\eta_t|\epsilon_t] \propto \epsilon_t\), we recover the square-root scaling of average metaorder impact, \[{\mathbb{E}}[j_t|\epsilon_t] \propto \epsilon_t \, \sqrt{\mu h}\,.\] Note that the prefactor depends on the undetermined all-pass filter \(\tilde{U}\) and on the parametrization \((a, \tilde{L})\) of the metaorder sign process, but not on \(\mu\) or \(h\).
A nontrivial \(\tilde{U} \ne I\) encodes a departure from a pure surprise mechanism, in which impact responds only to the contemporaneous order flow innovation. It therefore allows for nontrivial impact decay. For simplicity, consider a flow of uncorrelated metaorders. The lag-\(1\) post-order response then satisfies \[{\mathbb{E}}[j_t | \epsilon_t = 0, \epsilon_{t-1}] \sim \sqrt{\mu h}\, \tilde{U}[1]\, \epsilon_{t-1}\] Since slippage is positive, \(\tilde{U}[0]>0\), but \(\tilde{U}[1]\) may be negative, as is the case for the simple exponential all-pass filter in Eq. (6 ).
While structural price diffusivity alone does not determine \(\tilde{U}\), it remains fully compatible with the empirically observed relaxation of informationless impact [18], [22]–[24], despite its diffusivity.
\(R_{yj}\) encodes the coupling between impact and counterfactual returns: positive lags generally correspond to forward predictability and negative lags to endogeneity – trades induced by price moves, e. g. passive executions.
Mechanical impact dominates informational alpha in the regime \[R_{jj}[0] \gg 2R_{jy}[0]\;.\] Eq. (8 ) then implies \[R_{jj}[0] =\mathcal{O}(\mu) \qquad \text{strong impact, weak information.}\] Thus, the typical impact magnitude scales as \(\mathcal{O}(\sqrt{\mu})\).
Conversely, in the weak impact regime \[R_{jj}[0] \ll 2R_{jy}[0]\] we have \[R_{yj}[0] =\mathcal{O}(\mu) \qquad \text{weak impact with information.}\] Since \(j\) is a causal functional of volume and assuming a fixed correlation structure with \(y\), the typical impact magnitude must scale as \(\mathcal{O}(\mu)\).
Altogether, therefore, \[\label{eq:impactscaling} j \sim \varphi(\mu) \,, \qquad \varphi(\mu) = \begin{cases} \mathcal{O}(\mu) & \text{weak impact with information} \\ \mathcal{O}(\sqrt{\mu}) & \text{strong impact, weak information} \end{cases}\tag{11}\] The crossover occurs when \(j \sim \hat{y}\) in magnitude, where \(\hat{y}\) is the contemporaneous informational alpha, i.e. the \(y\)-component that is correlated with the strategy. Thus, impact is more linear for strategies that trade on relatively stronger short-term alpha. This linear departure from the square-root law is indeed consistent with empirical findings [11], [13].
A nontrivial \[A[t] = R_{yj}[t] + R_{jy}[t] + R_{jj}[t]\] must be detectable for diffusivity to be restored by arbitrageurs. Applying the Kolmogorov-Szegő formula [25] for small perturbations, the optimal predictor under the given covariance \(R_{xx}[t] = \sigma_Y^2\Delta t\delta[t] + A[t]\) satisfies \[{\mathrm{Std}}[\hat{x}] \simeq \frac{1}{\sigma_X \sqrt{\Delta t}} \left(\sum_{h\ge 1} A[h]^2\right)^{1/2}\,, \qquad \frac{{\mathrm{Std}}[\hat{x}]}{\sigma_X\sqrt{\Delta t}} \ll 1\,.\] Choose the bin length equal to the horizon, measured in days, over which impact remains aligned: \[\Delta t = \frac{H}{\mu N}\,,\] where \(H\) the number of strategy-executions over which impact remains order-one correlated and \(N\) is the number of market transactions per day. Then \[A[t] \sim \left(\lambda^2 + \lambda\alpha\right)H^2 \qquad t \le 1\] and \(A[t] \approx 0\) for larger lags. Here, \(\lambda\) denotes the impact scale of a typical trade and \(\alpha\) the coupling to underlying price moves. Using again the empirical scaling \[\sigma_X \sim \lambda \sqrt{N},\] this simplifies to \[{\mathrm{Std}}[\hat{x}] \sim (\lambda + \alpha) \sqrt{\mu H^3} \,.\]
Empirically, \(\lambda\) is an order of magnitude smaller than the bid-ask spread [16], denoted \(S\), which is also the threshold beyond which arbitrage, liquidity response and impact decay start to blunt predictable patterns in the order flow. Mid-price diffusivity sets in when the volatility exceeds the spread, and continuous proxies of the equilibrium price that smooth high-frequency bid-ask bounces are diffusive much earlier, as demonstrated by accurate predictions of the Madhavan-Richardson-Roomans model [26] on large-tick assets [1], [21].
Thus, impact autocorrelations are suppressed beyond \[H_* \sim C\mu^{-1/3} \qquad C = \left(\frac{S}{\lambda+\alpha}\right)^{2/3}\] and structural price diffusivity, Eq. (3 ), operates for bin sizes \(\gtrsim H_*\), measured in execution-clock. With the choice \(C \sim 1\), we find \(H_* \approx 5\) executions for a moderate participation ratio \(\mu = 1\%\), and \(H_* \approx 2\) for \(\mu = 10\%\). More specifically, we therefore expect impact of a \(1\%\) strategy executed on a stock with (say) \(20,000\) daily transactions to be whitened after ten minutes. In other words, structural diffusivity is an accurate description of the impact dynamics induced by typical multiday metaorders.
Structural price diffusivity suggests that much of the optimal execution literature neglects a central ingredient: the adaptive nature of impact. A persistent change in trading behavior should induce a persistent change in impact itself, potentially offsetting gains predicted by tuning trading style to minimize impact under an ex-ante precalibrated kernel.
On the other hand, our framework also suggests that the historical impact of stationary strategies should not be extrapolated to point-in-time optimization scenarios involving previously unexplored execution regimes.
The theory further predicts:
Stronger alignment between order flow and informational alpha reduces the concavity of impact. In particular, conditioning metaorder impact on the concurrent residual market imbalance should produce a linear departure from square-root scaling in trading rate.
Any stationary sequence of trades should generate square-root impact in the information-neutral regime, irrespective of its origin. Consistently, recent work finds square-root scaling even for “synthetic” metaorders constructed from market order flow via a stationary selection procedure [27].
A rationale for, and generalization of, Market Microstructure Invariance. Decompose order flow into independent “bets”, each contributing additively a signed volume \(Q_b\) and variance \(\sigma_b^2\), so that \[\label{eq:mmi} \sigma V = \sigma_b Q_b N^{3/2}\,,\tag{12}\] where \(\sigma\) denotes volatility, \(V\) the traded volume measured over a given interval, and \(N\) the number of independent bets. Market Microstructure Invariance is the hypothesis that the exchanged “bet risk” \(\sigma_b Q_b\) is invariant across assets and time horizons [28]. Early empirical studies reported good agreement with this hypothesis [28], [29], whereas later work [30], [31] documented substantial deviations, consistent with the observation that \(\sigma_b Q_b\) scales with the impact cost of a bet.
Equation Eq. (12 ) is tautological under the abstract notion of independent bets, but becomes nontrivial when applied to metaorders or individual transactions, which are manifestly correlated.
Structural price diffusivity provides a natural rationale. Rewriting Eq. (12 ) yields \[\label{eq:mmi2} \sigma_b = \sigma\frac{\sqrt{Q_b}}{\sqrt{V}}\;.\tag{13}\] Under structural diffusivity, impact increments generated by any stationary strategy must be asymptotically uncorrelated, so their induced variances add additively, even if the bets themselves remain correlated. Consequently, Eq. (13 ) should hold across a broad range of horizons for any trading strategy sufficiently orthogonal to the rest of the market, and a fortiori for the market as a whole.
Let us retrace the main argument: if both the observed and counterfactual prices are diffusive, their difference – market impact – satisfies a structural constraint which, unlike earlier balancing arguments applied to aggregate flow, operates at the level of each individual participant.
This constraint nearly fixes the admissible scaling structure of market impact. In the absence of informational coupling, impact increments must be effectively uncorrelated, independently of aggregate supply and demand dynamics. Information-neutral impact is therefore diffusive. Conversely, a square-root-to-linear crossover is predicted to occur when information content starts to dominate mechanical impact. Our framework thus recovers the empirical phenomenology. It further indicates that square-root impact is not anomalous and need not reflect market instability due to critically low liquidity, but instead arises as a generic consequence of adaptive price formation.
Consider the entry of a new trading strategy whose activity induces predictable price patterns. As long as this activity is stationary and persistent, arbitrage restores diffusivity. But the arbitrage trades themselves generate mechanical impact, leading to a cascade across time scales. The process stabilizes when the new equilibrium price recovers diffusivity, reflecting market impact that is specific to the new strategy, yet does not require explicit detection of the participant: only the aggregate statistical constraint must be satisfied. In fact, any stationary sequence of trades should generate square-root impact in the information-neutral regime, regardless of its origin.
Under this interpretation, impact is not merely the response to aggregate flow. Rather, it is the equilibrium response of an adaptive market to the introduction of the flow component of a particular strategy into an already diffusive environment. Since diffusivity constrains the marginal contribution of each strategy, equilibrium adaptation is generally path-dependent: introducing two strategies sequentially, allowing the market to re-equilibrate after each addition, need not lead to the same impact decomposition as introducing their aggregate flow at once. Consequently, the impact of an aggregate strategy need not equal the sum of the impacts of its constituents, even when the latter are traded independently. Different histories by which the same aggregate flow is composed can therefore correspond to distinct, yet equally diffusive, market equilibria.
Impact diffusivity also implies that the impact magnitude at the single-trade level matches the volatility of impact fluctuations over larger horizons. Does this imply that long-horizon impact is set by microstructural scales? Probably not: the microstructural MRR model [26] yields accurate predictions about the equilibrium price even on sub-tick scales [21]. In other words, even substantial price discretization – despite its mechanical constraints – does not break diffusivity. Furthermore, profound changes in market structure – e. g. the rise of high-frequency trading, tighter spreads, much lower latency, and increased volume – have had only limited effect on long-run volatility and market stability. This suggests a top-down mechanism whereby long-term volatility sets microstructural scales such as single-trade impact. The key implication remains: impact must be diffusive in the informationless regime.
Crucially, all these results hold for stationary trading activity and should be understood as an unconditional, ex-post property. They do not apply to abrupt interventions or point-in-time optimal execution problems, where deviations from diffusivity may persist over finite horizons.
Finally, the framework provides a diagnostic for mechanical impact models. In short, any model that fails to whiten order flow violates structural diffusivity. When used to infer de-impacted prices for simulation purposes, such models produce autocorrelated return trajectories, indicating a systematic inconsistency. For example, the latent liquidity model – to which the author has contributed in the past [32] – does not generally satisfy this constraint. This restrictiveness was not anticipated. Yet many models imply that scaling a strategy or removing it altogether would lead to inefficient, non-diffusive price dynamics; a conclusion difficult to reconcile with expectations and the empirical observation that prices quickly revert to diffusivity following major market events, structural changes, or regime shifts.
The author thanks Nicholas Westray, Xavier Gabaix and Natascha Hey for their critical reading of the draft, and Jean-Philippe Bouchaud, Iacopo Mastromatteo and Bence Toth for the interesting and helpful comments.