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

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.

• The paper demonstrates that market impact follows a double square-root law, with immediate and temporal effects scaling with the square root of traded volume.
• It employs a granular dataset with trader IDs to dissect metaorders into child orders, providing precise empirical evidence that contests traditional linear liquidity models.
• The findings suggest a mechanical origin for market impact, challenging information-driven theories and offering insights for developing adaptive trading strategies.

The "Double" Square-Root Law: Mechanical Origins of Market Impact Utilizing Tokyo Stock Exchange Data

Introduction

This essay encapsulates the findings of "The 'double' square-root law: Evidence for the mechanical origin of market impact using Tokyo Stock Exchange data" in which the authors explore the origins of market impact using an ID-resolved dataset from the Tokyo Stock Exchange (TSE). The study reveals the mechanical roots of the square-root law for market impact, challenging the conventional information-driven theories.

Square-Root Law in Market Impact

Fundamentally, the paper addresses the square-root law, which posits that the average price impact of a metaorder is proportional to the square root of the traded volume relative to daily volume. The well-documented square-root impact regime deviates from classical economic models like Kyle’s model, which predict a linear relationship, thus questioning the liquidity assumptions made in such models. Canonical examples include the linear model’s failure to describe liquidity deficits for small trade volumes accurately.

Detailed Data Analysis

The research utilizes a comprehensive dataset from the TSE, unique for its inclusion of trader IDs. This dataset allows the authors to define metaorders meticulously and to examine the microscopic origins of the square-root law. The dataset’s granularity permits a meticulous disaggregation of trades into child orders and examination of the impact these orders have both individually and collectively.

Empirical Evidence of the Double Square-Root Law

The paper’s cornerstone is the demonstration of a "double" square-root effect. Precisely, it reveals two emergent phenomena: the immediate impact of child orders correlating with the square root of their volume and a temporal inverse square-root relaxation subsequent to the child orders' execution.

Figure 1: Average price profile during the execution of a metaorder, illustrating the cumulative and relaxed impact behavior consistent with the double square-root law.

Surprisingly, the profound results show that the square-root impact law persists at the granularity of individual market orders. Notably, these results underline that the impact of single market orders follows a square-root function of volume when accounted for over time, a finding confirmed even after shuffling trader IDs, thereby nullifying individual trading strategy influence.

Mechanical Character of Market Impact

The authors argue for a "mechanical" interpretation of market impact by demonstrating that market orders—once processed by the market makers—result in a latent liquidity landscape that aligns with theoretical linear order book models. Herein lies the insight: once an order is processed, the liquidity finds an equilibrium, implying that individual market orders exhibit consistent impact profiles regardless of the originating trading strategy, be it informed or not.

Figure 2: Impact of public market orders illustrating consistent square-root behavior over volume time equal to order size.

Implications and Future Directions

These findings challenge information-centric theories of market impact and strongly support the idea of a mechanically-driven impact process. This mechanical interpretation could lead to novel perspectives in reinterpreting market liquidity models and might serve as a springboard for adaptive trading strategies that minimize market impact in operational execution.

Additionally, this methodology paves the way for future exploration of synthetic metaorders constructed from public tape alone, potentially broadening empirical research avenues in financial markets.

Conclusion

The research delivers compelling evidence in favor of a mechanical origin for the square-root law of price impact. By dissecting the market impact across both macro and micro scales, the study provides foundational insights into liquidity dynamics, urging a re-evaluation of traditional economic interpretations of market behavior. The findings reassure that the understanding of market mechanics could benefit from an emphasis on market structure and liquidity processes rather than solely the informational aspects of trading strategies.