How Square-Root Impact Can Preserve Diffusive Prices

Summary

The document addresses a market microstructure puzzle: order flow can persist because institutions split large orders, while prices still appear diffusive and hard to predict. It also considers the empirical square-root relation between a metaorder’s size and its price impact.

The authors generalize the Lillo-Mike-Farmer model to nonlinear price impact and map it to a solvable Lévy-walk model. Their solution finds that square-root impact can keep price dynamics diffusive despite persistent order flow, with the impact function limiting the movement caused by large orders. The evidence described is theoretical and model-based; the excerpt gives no empirical tests or implementation details. Its conclusion therefore depends on the model’s assumptions and does not establish that all markets or time horizons behave this way.

Key ideas

• Institutional order splitting can produce persistent, predictable order flow.
• The document links square-root metaorder impact to diffusive price behavior.
• A generalized order-flow model is mapped to a solvable Lévy-walk model.
• The stated result is theoretical and does not establish behavior across all markets.

Tags

# Why do financial prices exhibit Brownian motion despite predictable order flow?


# Why do financial prices exhibit Brownian motion despite predictable order flow?









In financial market microstructure, there are two enigmatic empirical laws: (i) the market-order flow has predictable persistence due to metaorder splitters by institutional investors, well formulated as the Lillo-Mike-Farmer model. However, this phenomenon seems paradoxical given the diffusive and unpredictable price dynamics; (ii) the price impact $I(Q)$ of a large metaorder $Q$ follows the square-root law, $I(Q)\propto \sqrt{Q}$. Here we theoretically reveal why price dynamics follows Brownian motion despite predictable order flow by unifying these enigmas. We generalize the Lillo-Mike-Farmer model to nonlinear price-impact dynamics, which is mapped to an exactly solvable Lévy-walk model. Our exact solution shows that the price dynamics remains diffusive under the square-root law, even under persistent order flow. This work illustrates the crucial role of the square-root law in mitigating large price movements by large metaorders, thereby leading to the Brownian price dynamics, consistently with the efficient market hypothesis over long timescales.

Shown in full with attribution under the source's licence. Licence: abstract CC0

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.