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

ArXiv ID: 2502.17906 “View on arXiv”

Authors: Unknown

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 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.

Keywords: Market microstructure, Price impact (square-root law), Order flow persistence, Lévy-walk model, Brownian motion, Equities

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 4.0/10
• Quadrant: Lab Rats
• Why: The paper is densely packed with advanced mathematical physics concepts, including power-law distributions, exact solutions to Lévy-walk models, and derivations of diffusion exponents, which yields a high math complexity score. Empirical rigor is low because the work is purely theoretical, presenting a generalized model and its analytical solution without any backtesting on historical data, statistical validation against new datasets, or implementation-focused code; the empirical claims are derived from prior studies (like the LMF model and square-root law) rather than direct new analysis.

  flowchart TD
    A["Research Question<br>Why do prices exhibit Brownian motion<br>despite predictable order flow?"] --> B

    subgraph B ["Methodology"]
        B1["Empirical Law 1<br>Lillo-Mike-Farmer Model<br>Persistent Order Flow"] --> B3
        B2["Empirical Law 2<br>Square-Root Law<br>Price Impact I(Q) ∝ √Q"] --> B3
        B3["Generalized Lillo-Mike-Farmer Model<br>Nonlinear Dynamics"]
    end

    B --> C["Theoretical Analysis<br>Mapping to Lévy-Walk Model"]
    C --> D{"Exact Solution Analysis"}

    D --> E["Outcome 1<br>Price dynamics remain diffusive<br>despite flow persistence"]
    D --> F["Outcome 2<br>Square-root law mitigates large price movements<br>from large metaorders"]

    E --> G["Conclusion<br>Unified Explanation: Why Brownian motion<br>emerges despite predictable order flow"]
    F --> G