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

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.

💡 Research Summary

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The paper proposes a parsimonious micro‑structural framework that simultaneously accounts for four well‑documented stylised facts of modern financial markets: (i) long‑range dependence of signed order flow, (ii) roughness of unsigned traded volume, (iii) ultra‑rough volatility, and (iv) the square‑root law of market impact. The authors decompose the aggregate order flow into two conceptual layers. The core flow captures autonomous trading motives (fundamental‑driven strategies, portfolio rebalancing, meta‑order splitting) and is modelled by two independent univariate Hawkes processes for buys and sells, each sharing a baseline intensity ν and a self‑exciting kernel φ₀. The reaction flow represents the market’s endogenous response (liquidity provision, high‑frequency market making, algorithmic adjustments) and is modelled by a bivariate Hawkes process whose intensity is driven both by the core flow and by its own past through a symmetric kernel matrix Φ.

The central theoretical contribution is the derivation of a scaling limit when time is stretched and intensities are appropriately rescaled. In this limit the signed order flow converges to a mixed fractional Brownian motion: a sum of a fractional Brownian motion with Hurst exponent H₀∈(½,1) (inherited from the persistence of the core flow) and a martingale component generated by the reaction flow. This mixed structure explains the empirically observed scale‑dependence of Hurst estimates—high‑frequency data appear diffusive (H≈0.5) because the martingale dominates, while coarser aggregations reveal the long‑memory fractional component.

For unsigned volume, the reaction flow dominates the magnitude, yet the requirement of a non‑degenerate limit forces its dynamics to inherit the same memory parameter H₀. Consequently, cumulative volume converges to the integral of a rough process with Hurst index H₀−½, yielding empirical Hurst values around 0.15–0.35 when H₀≈0.75.

Price formation is linked through a no‑arbitrage condition. In the regime H₀>¾ the mixed fractional Brownian motion admits a semimartingale representation, allowing the construction of a price process that remains a martingale while incorporating the order‑flow driven drift. The drift’s regularity implies that the impact function follows a power law I(x)∝x^{2−2H₀}. When H₀≈0.75 this exponent equals ½, reproducing the celebrated square‑root impact law. Moreover, the same H₀ determines the roughness of volatility: volatility behaves as a rough fractional process with Hurst exponent 2H₀−3/2, which for H₀≈0.75 yields values close to zero, consistent with the ultra‑rough volatility empirics (H≈0.05–0.15).

Empirically, the authors analyse high‑frequency order‑book data for 40 large‑cap stocks over 2021‑2024. Using mixed‑fractional estimation techniques, they obtain stable H₀ estimates in the narrow band 0.75–0.80 across all sampling frequencies. Unsigned volume and volatility Hurst exponents derived from autocovariance‑based GMM methods fall within the ranges predicted by the theory. Impact analysis of large meta‑orders confirms the square‑root scaling.

The paper’s key innovation is the reduction of four distinct phenomena to a single structural parameter H₀, achieved by combining a two‑layer Hawkes representation with a rigorous scaling‑limit analysis. This unification contrasts with prior literature that treats each phenomenon separately (e.g., meta‑order models, rough volatility models, propagator impact models). The framework offers practical benefits: estimating H₀ from order‑flow data suffices to calibrate volume‑forecasting models, rough volatility dynamics, and impact cost functions, thereby providing a coherent tool for risk management, algorithmic execution, and market‑making strategy design.

Limitations include the reliance on symmetric kernels, the omission of order cancellations, multi‑asset cross‑dependencies, and the asymptotic nature of the scaling results, which may not capture extreme market events. Future work could extend the model to asymmetric reaction kernels, incorporate cancellation dynamics, and test robustness under stress‑scenario simulations. Nonetheless, the study delivers a compelling, mathematically grounded theory that bridges micro‑level order dynamics and macro‑level market observables through a single, empirically validated parameter.