Skip to content
All library documents

Square-Root Price Impact and Diffusive Prices Under Predictable Order Flow

Article arXiv papers · Author: Yuki Sato et al.
Read the original

Summary

This paper addresses why prices can behave diffusively over long horizons even when market-order flow is predictable because of long-range correlations. Such predictability, combined with the generally positive price impact of orders, appears inconsistent with Brownian-like price dynamics.

The authors extend the Lillo–Mike–Farmer model by incorporating a nonlinear square-root price-impact law. They map the resulting time-series models to Lévy walks, a class of non-Markovian stochastic processes with exact solutions, and prove that prices remain diffusive at long times under the stated assumptions. The result offers a theoretical explanation for how predictable order flow can coexist with diffusive prices; the document does not describe empirical validation or specify how broadly the assumed impact law applies.

Key ideas

  • Long-range correlations make market-order flow predictable.
  • Predictable order flow with positive impact creates an apparent tension with diffusive prices.
  • The model extends the Lillo–Mike–Farmer framework using square-root price impact.
  • A mapping to Lévy walks supports an exact analysis of the model.
  • The authors prove long-run diffusion under the model's assumptions.

Tags

Market microstructureStatisticsExecution
Full text
# Exactly solvable model for the diffusive price-dynamics paradox under long-range correlated market-order flow


# Exactly solvable model for the diffusive price-dynamics paradox under long-range correlated market-order flow









We develop an exactly solvable nonlinear time-series model by incorporating the square-root price-impact law into the Lillo--Mike--Farmer (LMF) model to resolve the diffusive price-dynamics paradox under predictable market-order flow. In financial market microstructure, it is well established that the price dynamics are approximately described by Brownian motion at long times. However, it is also well-known that market-order flow is clearly predictable due to long-range correlations, as mathematically formulated by the LMF model. Since market orders have a positive price impact in general, predictable market-order flow seems to contradict Brownian price dynamics. In this work, we resolve this diffusive price-dynamics paradox by developing nonlinear time-series models that generalize the LMF model based on the square-root price-impact law. Our time-series models can be mathematically mapped onto the Lévy-walk framework---an exactly solvable class of non-Markovian stochastic processes developed in statistical physics. We prove that the price dynamics are diffusive at long times under the square-root law even under predictable market-order flow. Our work highlights the crucial practical importance of the square-root law in understanding the microstructural foundation of the Efficient Market Hypothesis.

Shown in full with attribution under the source's licence. Licence: abstract CC0

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.