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

Yuki Sato; Kiyoshi Kanazawa

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arXiv · 2502.17906

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

Yuki Sato·Kiyoshi Kanazawa

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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.

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q-fin.TR cond-mat.stat-mech econ.GN q-fin.EC q-fin.MF q-fin.PR

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Yuki Sato, Kiyoshi Kanazawa. 2025-02-25. Why do financial prices exhibit Brownian motion despite predictable order flow?. https://arxiv.org/abs/2502.17906

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