Square-Root Law of Market Impact

• Square-root law is a scaling relation showing that the average price impact of a metaorder scales with the square root of its volume, establishing a nonlinear, concave effect.
• Empirical evidence from diverse asset classes supports this law, with microstructural models like latent liquidity and propagator frameworks providing theoretical justification.
• This universal law is crucial for optimizing trade execution, managing risk, and understanding how liquidity replenishment governs the decay of market impact.

The square-root law of market impact is an empirically robust, theoretically justified scaling relation describing how the average price impact $I(Q)$ of a large order (metaorder) depends on its executed volume $Q$. Specifically, it asserts that $I(Q) \propto Q^{1/2}$ across a wide range of financial instruments, asset classes, and market conditions. This nonlinear, concave functional form plays a central role in reconciling persistent order flow with the diffusive nature of asset prices and underpins modern models of market microstructure and optimal execution.

1. Empirical Observation and Definition

Across equity, futures, options, and cryptocurrency markets, strong evidence indicates that the average realized market impact of large metaorders (sequential slices of institutionally motivated trades) follows a square-root scaling. The canonical form is

$I(Q) = Y\,\sigma\,\sqrt{\frac{Q}{V}}$

where $I(Q)$ is the impact (e.g., in units of volatility), $Q$ is the traded volume, $V$ is a normalization scale (such as daily traded volume), $\sigma$ is daily volatility, and $Y$ is an order-unity prefactor. Extensive datasets—e.g., millions of metaorders across the Tokyo Stock Exchange—show that the impact exponent $\delta$ for $I(Q) \propto Q^\delta$ is universally very close to $1/2$, at both the stock and individual trader level, with statistical error bars below $0.01$ (Sato et al., 21 Nov 2024).

This universality holds across all liquid stocks, over eight years, and is robust to the method of metaorder reconstruction. The law persists in a variety of market structures, including anonymous and non-anonymous settings, and remains valid for real and synthetic metaorders (Maitrier et al., 22 Feb 2025).

2. Theoretical Foundations and Mechanisms

The square-root law arises naturally from several microstructural models that emphasize the role of liquidity, order splitting, and market efficiency:

• Latent Liquidity Models: The locally linear order book (LLOB) and reaction-diffusion frameworks assume liquidity is hidden and grows (linearly or with a power-law) away from the current price. A metaorder must consume increasingly deeper layers of liquidity, yielding a concave, often square-root, impact (Donier et al., 2014, Mastromatteo et al., 2013, Benzaquen et al., 2017, Benzaquen et al., 2017).
• Propagator and Agent-Based Models: In these, each trade generates a decaying price response ("propagator"). Long-range correlations in order flow, together with appropriately slow decay of impact, ensure price diffusion while respecting the square-root law (Jaisson, 2014, Donier et al., 2014).
• Bayesian Market Making: When market makers update beliefs about hidden order flows (metaorders) using Bayesian inference under microstructure noise, the posterior expectation resums to a square-root function of the accumulated traded volume, as in the error function expansion (Saddier et al., 2023).
• Dimensional Analysis: Arguments based on financial invariance and dimensional consistency single out the square-root relation as the unique scaling law for impact when only volatility, order size, and traded volume are relevant variables (Pohl et al., 2017).

The concavity of the law ($\delta<1$) acts as a mitigating mechanism: even persistent, autocorrelated order flows arising from metaorder splitting do not induce superdiffusive price trends, thus preserving market efficiency (Sato et al., 25 Feb 2025).

3. Microstructural and Mechanical Origin

High-resolution transaction data with trader IDs on the TSE (Maitrier et al., 22 Feb 2025) demonstrate that the square-root scaling is already present at the level of individual child orders. Once the market has "digested" a child order (i.e., sufficient time for liquidity providers to re-equilibrate), the impact scales as $\sqrt{q}$ in child order size $q$. The cumulative impact of a metaorder arises from a "double" square-root effect: individual child order impact is $\sqrt{q}$, while the aggregate impact via temporal summation and relaxation exhibits an inverse square-root decay over the metaorder's life.

The law holds identically for actual and synthetic metaorders (constructed by scrambling issuer identities), implying that impact is fundamentally mechanical: a consequence of supply and demand structure and order flow, not the information content of the trades.

4. Universality, Limitations, and Competing Hypotheses

The comprehensive TSE analysis (Sato et al., 21 Nov 2024) rejects competing models positing non-universal exponents—such as those in the Gabaix-Gopikrishnan-Plerou-Stanley (linking to metaorder size distribution exponents) or the Farmer-Gerig-Lillo-Waelbroeck frameworks (linking to metaorder length exponent). No correlation between these exponents and the observed $\delta$ was found.

Logarithmic and other more concave forms have sometimes been suggested to fit data over exceptionally wide dynamic ranges (Zarinelli et al., 2014), but across practical scales and high-precision datasets, the exponent $\delta$ remains tightly centered on $1/2$.

The law is observed in markets with widely divergent microstructure characteristics—including options (in terms of vega traded and volatility-of-volatility as scaling variables) (Toth et al., 2016), Bitcoin (where statistical arbitrage is largely absent) (Donier et al., 2014), and traditional equities (Donier et al., 2014, Mastromatteo et al., 2013).

5. Macroscopic Implications: Price Diffusion and Excess Volatility

Despite the long-range correlations in metaorder signs (persistent order flow), the square-root law guarantees that the cumulative mean-squared price displacement remains diffusive (i.e., grows linearly with time), resolving the "diffusivity puzzle" (Sato et al., 25 Feb 2025, Benzaquen et al., 2017). In a Lévy-walk mapping, the mean-square displacement $\langle[\Delta m(t)]^2\rangle$ transitions from superdiffusive to diffusive as soon as the impact exponent $\delta=1/2$ and the metaorder size tail exponent $\alpha > 1$ (with values observed in data). Thus, markets retain their Brownian (random-walk) character at macroscopic scales.

This reconciles efficient price formation (unpredictable prices) with predictable, autocorrelated order flow and heavy-tailed metaorder size distributions.

6. Impact Decay, Permanent Impact, and Volatility Decomposition

After the completion of a metaorder, impact typically decays (often as an inverse square root in time). In fair-pricing equilibria (Farmer et al., 2011), only a fraction (often about $2/3$) of the peak impact remains permanently. Decomposition into transient (mechanical) and permanent (informational) components is tractable in propagator and LLOB models (Donier et al., 2014).

Simulations and empirical studies show that price volatility can be fully accounted for by the aggregation of many correlated, square-root-impacting metaorders ("order-driven" volatility theory) (Maitrier et al., 9 Jun 2025, Maitrier et al., 5 Sep 2025), without requiring a dominant "fundamental" component. The scaling structure of the correlation between generalized order flow and returns, and the time decay of impact, are quantitatively replicated in synthetic markets that obey the square-root law.

7. Practical and Theoretical Significance

The strict universality of the square-root law simplifies the estimation of execution costs for institutional traders: linear models dramatically underestimate costs associated with large metaorders. The law forms the backbone of optimal execution algorithms and is critical for macroprudential risk management and regulatory policy.

Furthermore, the underlying mechanical (liquidity-driven) basis, validated at the microscopic (child order) level, implies that the law is robust to changes in market microstructure, trader identity, and even execution protocols.

Empirically and theoretically, the square-root law of market impact is now established as a universal, equilibrium feature that arises from the interplay of liquidity, strategic order-splitting, and market efficiency. Its existence is central to understanding market stability, price formation rules, and the scaling structure of return volatility in modern electronic markets.