Why Order Splitting and Liquidity Replenishment Produce Square-Root Market Impact

Summary

The document tests explanations for the square-root law of market impact, which describes how impact grows with metaorder size relative to traded volume. It compares three proposed predictions with results from a minimal limit-order-book model containing heterogeneous interacting agents. The model is calibrated against a Tokyo Stock Exchange benchmark and simulated across 2,000 independently parameterized stocks; its average impact exponent is close to the benchmark value.

Counterfactual ablations indicate that order splitting and market-maker liquidity replenishment are both important within this model: removing either lowers the estimated exponent. Changes to momentum trading, price limits, the splitting rule, or background liquidity have smaller effects. The authors conclude that neither the metaorder-size tail nor visible book shape alone accounts for the simulated law. These findings are model-based causal evidence, not proof that the same mechanisms explain market impact in every real market; the document provides no further details on empirical validation or implementation costs.

Key ideas

• The study compares three proposed explanations for the square-root market-impact law on the same simulated data.
• The limit-order-book model reproduces an average impact exponent near the stated Tokyo Stock Exchange benchmark.
• Suppressing order splitting substantially lowers the simulated impact exponent.
• Removing market-maker liquidity replenishment also lowers the exponent, supporting a joint role for both mechanisms.
• Other tested changes have smaller effects, and the result is specific to the model described.

Tags

# Order Splitting and Liquidity Replenishment Are Jointly Necessary for the Square-Root Law of Market Impact:


# Order Splitting and Liquidity Replenishment Are Jointly Necessary for the Square-Root Law of Market Impact:









Three quantitative predictions have been advanced for the square-root law (SRL) of market impact, $I/σ_D = c\,(Q/V_D)^δ$ with $δ\approx 0.5$: GGPS ($δ=β-1$), FGLW ($δ=α-1$), and LOB walking ($δ=1/(1+γ)$). Using a minimal limit-order-book model populated by heterogeneous interacting agents and calibrated against the Tokyo Stock Exchange benchmark ($\langleδ\rangle = 0.489$~\citep{satoStrictUniversalitySquareRoot2025}), we test all three on identical simulated data and find that none matches the per-stock measured $δ$: GGPS and FGLW over-predict by factors of two and four respectively, while LOB walking under-predicts. The model reproduces $\langleδ\rangle = 0.539\pm 0.048$ across 2000 independently parameterised stocks. To identify which mechanisms are causally responsible, we perform counterfactual ablation by selectively suppressing each component. Removing order splitting collapses $δ$ from $0.549$ to $0.324$; removing liquidity replenishment by market makers drops it to $0.386$; perturbations that leave both intact (momentum trading, price limits, splitting rule, background liquidity) move $δ$ by less than $10\%$. Order splitting and liquidity replenishment are thus jointly identified as the necessary mechanisms for the SRL within this model, with the simulated SRL depending on neither the metaorder size tail nor the visible book shape in isolation.

Shown in full with attribution under the source's licence. Licence: abstract CC0

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