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

Authors: Johannes Muhle-Karbe, Youssef Ouazzani Chahdi, Mathieu Rosenbaum, Grégoire Szymanski

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.

Complexity vs Empirical Score

Math Complexity: 8.5/10
Empirical Rigor: 8.0/10
Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: This paper presents a breakthrough microstructural model using Hawkes processes that unifies persistent order flow, rough volume, rough volatility, and square-root market impact under a single Hurst parameter H0. The mathematical rigor is high due to scaling limit theorems and mixed fractional Brownian motion representations, complemented by solid empirical validation across a 40-stock dataset.

Research Flowchart

  flowchart TD
    A[Goal: Unify order flow, market impact, & rough volatility] --> B[Model: Dual Hawkes Processes<br>Core Orders & Reaction Flow]
    B --> C[Input: Microstructure Signed Order Flow Data]
    C --> D[Scaling Limit Analysis<br>Pinned by Core Persistence H0]
    D --> E[Empirical Estimation<br>H0 ≈ 3/4]
    E --> F[Outcomes & Predictions]
    F --> F1[Signed Flow: Hurst H0]
    F --> F2[Traded Volume: Rough Hurst H0 - 1/2]
    F --> F3[Volatility: Rough Hurst 2H0 - 3/2]
    F --> F4[Market Impact: Power Law Exponent 2 - 2H0<br>Matches Square-Root Law]

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A nomalous price impact and the critical nature of liquidity in financial markets (2011)
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F ractional F okker– P lanck equation (2015)
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A nonparametric test for rough volatility — → 2407.10659
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Realized variance and market microstructure noise (2006)
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Theory for long memory in supply and demand (2005) — link
Price Impact (2009) — link
The square-root impact law also holds for option markets (2016) — link
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Hawkes processes in finance (2015) — link
Market impacts and the life cycle of investors’ orders in liquid markets (2015) — link
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Bid, Ask and Transaction Prices in a Specialist Market with Heterogeneously Informed Traders (1985) — link
Intraday Volatility Dynamics (2026)
Asymptotic efficiency for mixed fractional B rownian motion (2026)
A theory of passive market impact — → 2412.07461

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