The double square-root law: Evidence for the mechanical origin of market impact using Tokyo Stock Exchange data

Paper: arXiv 2502.16246

Abstract

Understanding the impact of trades on prices is a crucial question for both academic research and industry practice. It is well established that impact follows a square-root impact as a function of traded volume. However, the microscopic origin of such a law remains elusive: empirical studies are particularly challenging due to the anonymity of orders in public data. Indeed, there is ongoing debate about whether price impact has a mechanical origin or whether it is primarily driven by information, as suggested by many economic theories. In this paper, we revisit this question using a very detailed dataset provided by the Japanese stock exchange, containing the trader IDs for all orders sent to the exchange between 2012 and 2018. Our central result is that such a law has in fact microscopic roots and applies already at the level of single child orders, provided one waits long enough for the market to “digest” them. The mesoscopic impact of metaorders arises from a “double” square-root effect: square-root in volume of individual impact, followed by an inverse square-root decay as a function of time. Since market orders are anonymous, we expect and indeed find that these results apply to any market orders, and the impact of synthetic metaorders, reconstructed by scrambling the identity of the issuers, is described by the very same square-root impact law. We conclude that price impact is essentially mechanical, at odds with theories that emphasize the information content of such trades to explain the square-root impact law.

Complexity vs Empirical Score

• Math Complexity: 7.0/10
• Empirical Rigor: 6.0/10 (capped from 8.5: transaction costs not modelled; no out-of-sample test; see penalty flags)
• Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper presents advanced mathematical models (non-linear propagator, power-law distributions) and derivations typical of econophysics, while being heavily grounded in a unique, high-frequency dataset with detailed trader IDs and robust backtest-ready methodologies.

Research Flowchart

  flowchart TD
  A["Research Goal:<br>What is the microscopic origin<br>of the square-root impact law?"] --> B["Data Source:<br>Tokyo Stock Exchange Data 2012-2018<br>(Unique: Trader IDs)"]
  B --> C["Methodology:<br>Analyze single child orders &<br>reconstruct synthetic metaorders<br>(Scrambling trader IDs)"]
  C --> D["Computation:<br>Measure individual impact decay<br>(Inverse square-root in time)"]
  D --> E["Computation:<br>Aggregate impact of volume<br>(Square-root in volume)"]
  E --> F["Key Finding:<br>'Double' square-root law<br>(Volume + Time)"]
  F --> G["Conclusion:<br>Price impact is mechanical,<br>not information-driven"]

Cited References (48)

1. Inferring microscopic financial information from the long memory in market-order flow: A quantitative test of the Lillo-Mike-Farmer model (2023)
2. In search of the origins of financial fluctuations: The inelastic markets hypothesis (2021)
3. The inelastic market hypothesis: a microstructural interpretation (2022)
4. Empirical market microstructure: The institutions, economics, and econometrics of securities trading (2007)
5. A fully consistent, minimal model for non-linear market impact (2015)
6. Anomalous price impact and the critical nature of liquidity in financial markets (2011)
7. The square-root impact law also holds for option markets (2016) — link
8. Generating realistic metaorders from public data (2025) — link

1 of 48 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.