🤖 AI Summary
This study investigates the origin of the exponent δ ≈ 0.5 in the square-root law (SRL) of market impact and evaluates the validity of competing theoretical frameworks, including GGPS, FGLW, and limit order book (LOB) walking models. Leveraging a heterogeneous agent-based LOB model calibrated to empirical data and validated through large-scale simulations across 2,000 independent stocks, the authors employ controlled experiments and counterfactual ablation analyses to demonstrate that order splitting by informed traders and liquidity replenishment by market makers constitute two necessary and synergistic mechanisms underpinning the SRL. The model reproduces δ = 0.539 ± 0.048, closely matching the empirically observed value of 0.489 from the Tokyo Stock Exchange; removing either mechanism causes δ to deviate significantly from 0.5, whereas other perturbations induce deviations of less than 10%.
📝 Abstract
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.