Strict universality of the square-root law in price impact across stocks: a complete survey of the Tokyo stock exchange

Universal power laws have been scrutinised in physics and beyond, and a long-standing debate exists in econophysics regarding the strict universality of the nonlinear price impact, commonly referred to as the square-root law (SRL). The SRL posits that the average price impact $I$ follows a power law with respect to transaction volume $Q$, such that $I(Q) \propto Q^δ$ with $δ\approx 1/2$. Some researchers argue that the exponent $δ$ should be system-specific, without universality. Conversely, others contend that $δ$ should be exactly $1/2$ for all stocks across all countries, implying universality. However, resolving this debate requires high-precision measurements of $δ$ with errors of around $0.1$ across hundreds of stocks, which has been extremely challenging due to the scarcity of large microscopic datasets – those that enable tracking the trading behaviour of all individual accounts. Here we conclusively support the universality hypothesis of the SRL by a complete survey of all trading accounts for all liquid stocks on the Tokyo Stock Exchange (TSE) over eight years. Using this comprehensive microscopic dataset, we show that the exponent $δ$ is equal to $1/2$ within statistical errors at both the individual stock level and the individual trader level. Additionally, we rejected two prominent models supporting the nonuniversality hypothesis: the Gabaix-Gopikrishnan-Plerou-Stanley and the Farmer-Gerig-Lillo-Waelbroeck models (Nature 2003, QJE 2006, and Quant. Finance 2013). Our work provides exceptionally high-precision evidence for the universality hypothesis in social science and could prove useful in evaluating the price impact by large investors – an important topic even among practitioners.

💡 Research Summary

The paper tackles a long‑standing debate in econophysics: whether the square‑root law (SRL) of price impact, (I(Q)\propto Q^{\delta}) with (\delta\approx ½), is a universal feature of financial markets or a stock‑specific, non‑universal phenomenon. To resolve this, the authors exploit an unprecedented microscopic dataset from the Japan Exchange Group covering every order submission for all liquid stocks on the Tokyo Stock Exchange (TSE) over eight years (2017‑2024). Crucially, the data contain virtual‑server identifiers that allow the reconstruction of individual trader (desk) activity, enabling analysis at both the stock and trader levels.

Methodology
A meta‑order is defined as a sequence of consecutive market orders with the same sign (buy or sell). The total signed volume (Q) of a meta‑order is normalized by daily traded volume (V_D) and the price change (\Delta p) is normalized by daily volatility (\sigma_D), yielding dimensionless variables (Q) and (I). For each stock with more than (10^5) meta‑orders (≈2,000 stocks), the authors fit the relation (I=c,Q^{\delta}) using nonlinear relative least squares on binned averages. The same procedure is applied to each active trader (≥10 000 meta‑orders, 1,293 traders).

Because the observations are time‑series with serial correlation, standard IID error estimates would be severely biased. The authors therefore construct a Monte‑Carlo statistical model that (i) randomizes meta‑order signs while preserving timestamps, (ii) enforces the exact SRL with (\delta=½) and adds realistic noise, and (iii) repeats the simulation 100 times. This yields a robust estimate of the sampling error, (\langle!\langle\sigma_{\delta}\rangle!\rangle\approx0.063), which matches the empirical cross‑sectional dispersion.

Results – Stock Level
The cross‑sectional average exponent is (\langle\delta\rangle=0.489) with a standard error of 0.0015 and a dispersion (\sigma_{\delta}=0.071). The prefactor averages (\langle c\rangle=0.842). Both the mean and the dispersion are fully compatible with the theoretical value (\delta=½) within the estimated error bars.

Results – Trader Level
For individual traders, the average exponent is (\langle\delta_i\rangle=0.493\pm0.0050) with a dispersion (\sigma_{\delta_i}=0.177). Monte‑Carlo simulations of the trader‑level model produce (\langle!\langle\delta_i\rangle!\rangle=0.521\pm0.0048) and (\langle!\langle\sigma_{\delta_i}\rangle!\rangle=0.169), confirming that the observed variability is again attributable to finite‑sample effects rather than genuine heterogeneity.

Testing Non‑Universality Models
The paper explicitly confronts two prominent micro‑foundations that predict non‑universal exponents: the Gabaix‑Gopikrishnan‑Plerou‑Stanley (GGPS) model and the Farmer‑Gerig‑Lillo‑Waelbroeck (FGLW) model. Both link (\delta) to the tail exponents of the meta‑order volume distribution ((\beta)) and the number‑of‑child‑orders distribution ((\alpha)) via (\delta=\beta-1) or (\delta=\alpha-1). Using the same TSE data, the authors estimate (\beta) and (\alpha) for each stock and find them scattered around 1.5, but the corresponding (\delta) values remain tightly clustered near 0.5, contradicting the model predictions. Hence, the empirical evidence rejects the non‑universality hypothesis of these models.

Implications and Limitations
The study provides the first high‑precision, large‑scale confirmation that the SRL holds universally across more than two thousand stocks and over a thousand active traders in a major market. By achieving an exponent measurement error of roughly 0.06, the authors meet the precision threshold previously deemed unattainable in social‑science data. The findings support theoretical frameworks that derive (\delta=½) from latent order‑book dynamics and challenge micro‑economic models that predict stock‑specific exponents.

Nevertheless, the reconstruction of meta‑orders relies on the assumption that consecutive same‑sign market orders belong to the same hidden order, which may not capture more sophisticated execution algorithms. Moreover, the analysis normalizes by daily volume and volatility; extending the approach to intraday normalizations or to markets with different microstructure (e.g., fragmented US markets) would test the robustness of the universality claim. Finally, the dataset, while exhaustive for TSE, does not include off‑exchange venues, which could affect the measured impact for large institutional traders.

Conclusion
By leveraging a complete, eight‑year microscopic dataset of the Tokyo Stock Exchange, the authors deliver compelling, high‑precision evidence that the square‑root law of price impact is strictly universal across stocks and traders. The work settles a decade‑long debate, validates the (\delta=½) exponent as a genuine universal law in finance, and provides a benchmark for future theoretical and empirical investigations of market impact.