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

ArXiv ID: 2411.13965 “View on arXiv”

Authors: Unknown

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.

Keywords: square-root law (SRL), price impact, transaction volume, microscopic dataset, universal hypothesis, Equities

Complexity vs Empirical Score

• Math Complexity: 7.5/10
• Empirical Rigor: 9.0/10
• Quadrant: Holy Grail
• Why: The paper employs advanced statistical physics concepts and power-law analysis, requiring a solid mathematical background (≥5). It demonstrates exceptional empirical rigor by analyzing a unique, high-precision microscopic dataset of an entire stock exchange over eight years, with careful error estimation and model rejection, making it highly backtest-ready (≥5).

  flowchart TD
    A["Research Goal<br/>Test universality of the<br/>Square-Root Law SRL"] --> B["Methodology<br/>Complete survey of all accounts<br/>on Tokyo Stock Exchange TSE"]
    
    B --> C["Data Input<br/>8-year microscopic dataset<br/>All liquid stocks & traders"]
    
    C --> D["Computational Process<br/>High-precision measurement<br/>of price impact exponent δ"]
    
    D --> E{"Statistical Analysis<br/>Is δ ≈ 1/2?"}
    
    E -- Yes --> F["Key Finding 1<br/>SRL confirmed as universal<br/>δ = 1/2 within errors"]
    E -- No --> G["Key Finding 2<br/>Non-universal models rejected<br/>Gabaix & Farmer models refuted"]
    
    F --> H["Outcome<br/>Conclusive evidence for<br/>universality hypothesis"]
    G --> H