[論文レビュー] A unified theory of order flow, market impact, and volatility

Johannes Muhle‐Karbe, Youssef Ouazzani Chahd+2 more|arXiv (Cornell University)|Jan 30, 2026

Stochastic processes and financial applications被引用数 0

ひとこと要約

The paper builds a two-layer Hawkes-order-flow model (core and reaction) whose scaling limits yield a mixed fractional Brownian motion for signed order flow. This links persistent order flow, rough volume and volatility, and a square-root market impact via a single persistence parameter H0.

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.

研究の動機と目的

• Explain persistent, multi-scale order flow observed in markets
• Derive scaling limits for core and reaction order flows
• Show how a single parameter H0 governs order flow, volume, volatility, and market impact
• Provide a parsimonious, microstructure-consistent model that reconciles empirical stylized facts

提案手法

• Two-layer Hawkes model for order flow with core (F+ , F−) and reaction (N+, N−) processes
• Assumptions A–E to formalize nearly-unstable, heavy-tailed Hawkes scaling and non-degenerate limits
• Derivation of scaling limits: core flow converges to mixed fractional Brownian-like process; reaction flow yields rough volume; aggregate flow combines these effects
• Finite-dimensional convergence to rough Heston-type dynamics for reaction-driven components
• Replacement of the limiting signed flow with a mixed fractional Brownian motion (Brownian + fractional Brownian with H0) to study market impact and volatility
• Linking H0 to market impact exponent (2-2H0) and rough volatility parameter (2H0-3/2)
• Discussion of scale-dependent H estimates and empirical support for H0 ≈ 0.75

実験結果

リサーチクエスチョン

• RQ1 Can a two-layer Hawkes model reproduce both persistent signed order flow and rough unsigned volume?
• RQ2 What is the scaling limit of core versus reaction order flows, and how do they jointly determine market impact and volatility?
• RQ3 Does a single persistence parameter H0 govern the observed relations between order flow, volume, volatility, and price impact?
• RQ4 How does mixed fractional structure explain scale-dependent estimators of Hurst exponents in practice?

主な発見

• The core order flow converges to a mixed fractional process with H0 > 1/2, explaining persistence.
• The unsigned volume converges to a rough process with H0 − 1/2, accounting for rough volume dynamics.
• Under no-arbitrage and scaling, volatility becomes rough with H = 2H0 − 3/2 and price impact follows a power law with exponent 2 − 2H0.
• In the empirically relevant regime H0 ≈ 3/4, the model reproduces persistent order flow, rough volume, rough volatility, and the square-root market impact law.
• The mixed fractional structure explains scale-dependent H estimates and aligns with empirical estimates H0 ≈ 0.75–0.80 across sampling scales.