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

TL;DR

The paper addresses whether the square-root price-impact law with exponent $\delta\approx 1/2$ is universal across assets. It advances by performing a complete eight-year survey of all liquid stocks on the Tokyo Stock Exchange, defining trader units via virtual server IDs, and non-dimensionalizing $\mathcal{Q}=Q/V_D$ and $\mathcal{I}(\mathcal{Q})=I(Q)/\sigma_D$, then fitting $\mathcal{I}(\mathcal{Q}) \approx c \mathcal{Q}^{\delta}$ for stocks and traders with robust error estimation. The main findings show $\delta=1/2$ within statistical errors for all stocks (mean $\delta\approx 0.489$, SEM $\approx 0.0015$) and all active traders (mean $\delta^{(i)}\approx 0.493$, SEM $\approx 0.0050$); the trader- and stock-level distributions peak near $0.5$, and the predicted correlations in the GGPS and FGLW nonuniversality models are absent, leading to their rejection. These results provide exceptionally precise evidence for the universality of the SRL and have implications for modeling liquidity and the price impact of large investors.

Abstract

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.

Figures (6)

• Figure 1: Square-root price impact law. Practitioners typically split a large volume of market orders (called a metaorder, with size $Q$) into a long sequence of child orders (with length $L$). Price impact $I(Q):= \langle \epsilon \Delta p \mid Q\rangle$ is the average mid-price change $\Delta p$ between the beginning and the end of a buy (sell) metaorder with $\epsilon=+1$ ($\epsilon=-1$) and follows $I(Q)\propto Q^{\delta}$ with $\delta\approx 1/2$.
• Figure 2: Stock-level price impact. (a) Average price-impact profile for four stocks (red: Toyota Motor Corp., blue: Nippon Telegraph and Telephone (NTT) Corp., green: SoftBank Group Corp., orange: Takeda Pharmaceutical Company Limited). Errorbars represent the standard errors of the means (SEMs). (b) Aggregate scaling plot, visualising the average scaling profiles across all stocks, confirms the square-root law for all stocks. (c) Histogram of the exponent $\delta$ across all stocks, exhibiting a sharp peak around $0.5$ with $\langle\delta\rangle=0.489$ with its SEM of $0.0015$ and $\sigma_\delta:= \sqrt{\langle (\delta- \langle\delta\rangle)^2\rangle}=0.071$. (d) Histogram of the prefactor $c$ across all stocks, with $\langle c\rangle=0.842$.
• Figure 3: Errorbar estimation based on our statistical model. (a) $\delta$'s histgram across all stocks for one Monte Carlo iteration, consistent with the $\delta$'s histgram \ref{['fig:DeltaStatistics']}(c) for our TSE dataset. (b) Histgram of $\delta$'s errorbars $\langle\!\langle \sigma_{\delta}\rangle\!\rangle$ for individual stocks by the repeated Monte Carlo simulation. Each errorbar was estimated with $100$ iterations.
• Figure 4: Trader-level price impact. (a) Trader-level aggregate scaling plot confirms the square-root law for all traders. (b) Histogram of the exponent $\delta^{(i)}$ across all traders for $0 \leq \delta^{(i)} \leq 1$ shows a sharp peak around $0.5$, with $\langle \delta^{(i)} \rangle = 0.493$ with its SEM of $0.0050$ and $\sigma_{\delta^{(i)}} := \sqrt{\langle(\delta^{(i)} - \langle\delta^{(i)}\rangle)^2\rangle} = 0.177$. A few outliers were observed outside this region due to fitting errors. (c) Histogram of the prefactor $c^{(i)}$ across all traders, with $\langle c^{(i)}\rangle=1.501$.
• Figure 5: Quantitative test of (a) the GGPS prediction $\delta=\beta-1$ and for (b) the FGLW prediction $\delta=\alpha-1$. No correlation was found and, thus, both models were rejected.