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

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  • Yuki Sato
  • Kiyoshi Kanazawa

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.

Suggested Citation

  • Yuki Sato & Kiyoshi Kanazawa, 2025. "Why do financial prices exhibit Brownian motion despite predictable order flow?," Papers 2502.17906, arXiv.org, revised May 2025.
  • Handle: RePEc:arx:papers:2502.17906
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