Square-Root Law of Market Impact

The square-root law (SRL) is the empirically observed regularity that the price impact of a metaorder — a large parent order split into many child orders — scales approximately as the square root of the order size. In dimensionless form, impact ≈ Y · σ · √(Q/V), where Q is the metaorder size, V is the average daily volume, σ is volatility, and Y is a calibrated prefactor of order 0.5-1.0. Equivalently, I(Q) ∝ Q^δ with the exponent δ ≈ 1/2. It captures the central fact that trading cost per unit traded rises sublinearly but relentlessly with order size relative to liquidity — a small order is cheap per share, a large order is expensive per share — which is why a flat basis-point cost assumption understates the cost of large or fast trades and silently inflates backtests.

The empirical case for the law has become very strong. The headline 2024 study in this round — Sato & Kanazawa’s complete survey of the Tokyo Stock Exchange — measured δ across every trading account for all liquid stocks over eight years (2,299 stock-level data points, ~30 million metaorders), controlling estimation error to order 0.06. They found δ = 1/2 within statistical error at both the stock level (mean ⟨δ⟩ = 0.489, σ = 0.071) and the individual trader level (mean ⟨δ⟩ = 0.493) — and showed the residual dispersion is almost entirely a finite-sample artefact. They argue this establishes strict universality: the exponent is exactly 1/2, like a critical exponent in physics, independent of the stock, the country, or the trader’s strategy. The study directly rejects the two prominent competing models (Gabaix-Gopikrishnan-Plerou-Stanley; Farmer-Gerig-Lillo-Waelbroeck) that predicted a stock-specific exponent. A companion Tokyo study (the “double square-root law”, arXiv 2502.16246) finds the law has microscopic roots already at the single-child-order level, supporting a mechanical rather than informational origin for impact.

Square-Root Law of Market Impact [defines] Transaction Costs and Slippage Square-Root Law of Market Impact [supports] Optimal Execution

This concept matters to the vault because it is the load-bearing reason Transaction Costs and Slippage cannot be modelled as a constant fee. Impact is nonlinear and convex-in-the-right-way: it rises with √(Q/V), so doubling order size raises total impact cost roughly 2.8-fold, not 2-fold. The law also has a more subtle counterpart in the empirical record: Almgren Thum Hauptmann Li 2005, fitting a permanent-plus-temporary impact model to Citigroup US equity-desk data, actually preferred a 3/5 power law for temporary impact over the pure √ form across their order-size range. The two findings are not in flat contradiction — the √ law is best supported for peak metaorder impact while temporary-impact decay and order-size range matter for the exponent — but they jointly establish that the exponent is concave and well below 1, and the √ form remains the standard reduced-form approximation used in optimal-execution theory and in RL backtesting (e.g. Abbade and Reali Costa 2026 calibrate their Almgren-Chriss cost environment to it).

Almgren Thum Hauptmann Li 2005 [contradicts] Square-Root Law of Market Impact Square-Root Law of Market Impact [relates] Optimal Execution

For the vault’s profitability question the implication is sharp. The SRL is the mechanism that converts turnover into cost: because impact scales with order size relative to liquidity, a high-turnover Markov strategy — one that trades larger or faster to act on a fresh regime signal — pays a disproportionate, nonlinear penalty. A regime-switching backtest that assumes a flat 10 bp cost is not merely optimistic; it is using the wrong functional form, and the error grows precisely where the strategy is most active. The square-root law is therefore not an alpha source — it is a cost law — but it is the empirical foundation that makes Optimal Execution a well-posed problem and that sets the realistic bar every Markov trading strategy in this vault must clear.

Connections

• Transaction Costs and Slippage — defines, the law is why a flat fee is structurally wrong, source: https://arxiv.org/html/2411.13965v3
• Optimal Execution — supports, the impact law drives the speed-vs-cost trade-off, source: https://arxiv.org/html/2411.13965v3
• Almgren Chriss 2000 — supports, supplies the empirical impact function the model assumes, source: https://www.simtrade.fr/blog_simtrade/understanding-almgren-chriss-model-for-optimal-trade-execution/
• Almgren Thum Hauptmann Li 2005 — contradicts, empirically prefers a 3/5 power law for temporary impact, source: https://www.cis.upenn.edu/~mkearns/finread/costestim.pdf
• Abbade and Reali Costa 2026 — relates, uses the law to calibrate RL backtest cost environments, source: https://arxiv.org/html/2603.29086

Sources

• Sato, Y. & Kanazawa, K. (2024), “Strict universality of the square-root law in price impact across stocks: a complete survey of the Tokyo stock exchange” (arXiv 2411.13965) — https://arxiv.org/html/2411.13965v3
• “The ‘double’ square-root law: Evidence for the mechanical origin of market impact using Tokyo Stock Exchange data” (arXiv 2502.16246) — https://arxiv.org/abs/2502.16246
• Almgren, Thum, Hauptmann & Li (2005), “Direct Estimation of Equity Market Impact”, Risk — https://www.cis.upenn.edu/~mkearns/finread/costestim.pdf
• Abbade & Reali Costa (2026), “Realistic Market Impact Modeling for Reinforcement Learning Trading Environments” (arXiv 2603.29086) — https://arxiv.org/html/2603.29086