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

TL;DR

This paper explains why financial prices follow Brownian motion despite predictable order flow by modeling nonlinear price impact as a Le9vy-walk, showing that the square-root law ensures diffusive prices.

Contribution

It unifies the Lillo-Mike-Farmer model with nonlinear impact dynamics, providing an exact solution that explains the diffusive nature of prices despite persistent order flow.

Findings

Price dynamics remain diffusive under the square-root law.

The model maps to an exactly solvable Le9vy-walk.

Large metaorders are mitigated, supporting efficient market hypothesis.

Abstract

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 L\'evy-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.