Paper: arXiv 2502.17906

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.

Complexity vs Empirical Score

  • Math Complexity: 8.5/10
  • Empirical Rigor: 4.0/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: 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.

Research Flowchart

  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