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A unified theory of order flow, market impact, and volatility

Johannes Muhle-Karbe, Youssef Ouazzani Chahd, Mathieu Rosenbaum, Grégoire Szymanski

TL;DR

The paper develops a unified microstructural model for order flow by separating core and reaction components as Hawkes processes. In a scaling-limit regime, a single persistence parameter $H_0$ governs the behavior of signed order flow, unsigned volume, volatility, and the market impact function, linking long-range memory with rough dynamics and a power-law impact $MI(t) \sim t^{2-2H_0}$. Empirically, $H_0$ is estimated around $0.75$--$0.8$, which simultaneously reproduces persistent order flow, rough volume near $H_0-1/2$, and very rough volatility with $2H_0-3/2$, while recovering the square-root market impact at $H_0=3/4$. The framework provides a principled explanation for the joint scaling properties of microstructure and price dynamics and positions markets at the edge of criticality, where prices stay diffusive yet volatility remains ultra-rough.

Abstract

We propose a microstructural model for the order flow in financial markets that distinguishes between {\it core orders} and {\it reaction flow}, both modeled as Hawkes processes. This model has a natural scaling limit that reconciles a number of salient empirical properties: persistent signed order flow, rough trading volume and volatility, and power-law market impact. In our framework, all these quantities are pinned down by a single statistic $H_0$, which measures the persistence of the core flow. Specifically, the signed flow converges to the sum of a fractional process with Hurst index $H_0$ and a martingale, while the limiting traded volume is a rough process with Hurst index $H_0-1/2$. No-arbitrage constraints imply that volatility is rough, with Hurst parameter $2H_0-3/2$, and that the price impact of trades follows a power law with exponent $2-2H_0$. The analysis of signed order flow data yields an estimate $H_0 \approx 3/4$. This is not only consistent with the square-root law of market impact, but also turns out to match estimates for the roughness of traded volumes and volatilities remarkably well.

A unified theory of order flow, market impact, and volatility

TL;DR

The paper develops a unified microstructural model for order flow by separating core and reaction components as Hawkes processes. In a scaling-limit regime, a single persistence parameter governs the behavior of signed order flow, unsigned volume, volatility, and the market impact function, linking long-range memory with rough dynamics and a power-law impact . Empirically, is estimated around --, which simultaneously reproduces persistent order flow, rough volume near , and very rough volatility with , while recovering the square-root market impact at . The framework provides a principled explanation for the joint scaling properties of microstructure and price dynamics and positions markets at the edge of criticality, where prices stay diffusive yet volatility remains ultra-rough.

Abstract

We propose a microstructural model for the order flow in financial markets that distinguishes between {\it core orders} and {\it reaction flow}, both modeled as Hawkes processes. This model has a natural scaling limit that reconciles a number of salient empirical properties: persistent signed order flow, rough trading volume and volatility, and power-law market impact. In our framework, all these quantities are pinned down by a single statistic , which measures the persistence of the core flow. Specifically, the signed flow converges to the sum of a fractional process with Hurst index and a martingale, while the limiting traded volume is a rough process with Hurst index . No-arbitrage constraints imply that volatility is rough, with Hurst parameter , and that the price impact of trades follows a power law with exponent . The analysis of signed order flow data yields an estimate . This is not only consistent with the square-root law of market impact, but also turns out to match estimates for the roughness of traded volumes and volatilities remarkably well.
Paper Structure (21 sections, 11 theorems, 156 equations, 5 figures, 1 table)

This paper contains 21 sections, 11 theorems, 156 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. A two-layer Hawkes model for order flow
  3. Core order flow
  4. Reaction orders
  5. Aggregate order flow
  6. Scaling limits of the order flows
  7. Scaling limit of the core flow
  8. Scaling limit of the reaction orders
  9. Scaling limits for the global order flow
  10. From order flow to market impact and rough volatility
  11. Conclusion
  12. Useful results about Hawkes processes
  13. Proof of the results of Section \ref{['sec:scaling_limits']}
  14. Proof of Theorem \ref{['thm:cnvg_fundamental']}
  15. Proof of Proposition \ref{['prop:holder_fundamental']}
  16. ...and 6 more sections

Key Result

Theorem 3.1

Under Assumptions assumption:jaisson:long_memory:1 and assumption:jaisson:long_memory:2, the process $(\overline F_{t}^{+, T}, \overline F_{t}^{-, T})_{t \in [0,1]}$ is tight for the Skorokhod topology. Furthermore, any limit point $(F_{t}^{+}, F_{t}^{-})$From now on $(F_{t}^{+}, F_{t}^{-})$ denote where $Z^{+}$ and $Z^-$ are two independent continuous martingales with quadratic variations $F^{+}

Figures (5)

  • Figure 1: Cumulative signed order flow of the representative stock LVMH between 2021 and 2024.
  • Figure 2: Daily traded volume of the representative stock LVMH between 2021 and 2024.
  • Figure 3: Average Hurst exponent estimates for signed order flow over 40 stocks for the period 2021--2024, under fractional Brownian motion specification. Note: The data used for the estimations throughout the paper are described in Appendix \ref{['app:data']}.
  • Figure 4: Average Hurst exponent estimates for signed order flow over $40$ stocks for the period 2021--2024, under mixed fractional Brownian motion specification. Note: For each asset and time scale $\Delta$, the Hurst parameter of the fractional Brownian motion component is estimated using quadratic variations computed at the time scales $\Delta$, $2\Delta$, and $4\Delta$. The estimation procedure then follows the methodology developed for mixed fractional processes in CDM22szymanski2026mixed.
  • Figure 5: Average Hurst exponent estimates for unsigned trading volume, averaged over $40$ stocks for the period 2021--2024. Note: For each asset and time scale $\Delta$, the procedure is as follows: the total traded volume is aggregated over bins of size $\Delta$; the intraday seasonal pattern is removed multiplicatively; volume increments are computed and truncated at three times their standard deviation to mitigate the impact of outliers and exclude potential jumps in the volume intensity process; the auto-covariance function is then estimated, and the Hurst exponent is obtained using a GMM-based approach similar to li2016generalized. The methodology closely follows the procedures developed for volatility analysis in chong2022CLTchong2026intraday.

Theorems & Definitions (15)

  • Theorem 3.1
  • Proposition 3.2
  • Proposition 3.3
  • Theorem 3.4
  • Theorem 3.5
  • Theorem 3.6
  • Theorem 4.1
  • Remark 4.2
  • Definition A.1
  • Lemma A.2
  • ...and 5 more