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State-Dependent Hawkes Models for Limit Order Book Volatility

Article arXiv papers · Author: Akitoshi Kimura
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Summary

This paper introduces an Extended State-Dependent Hawkes Process for modeling limit order book activity. The model allows states to disappear, a feature intended to represent high-frequency market behavior. The authors use Karush–Kuhn–Tucker conditions to show that maximum likelihood estimation remains separable, supporting an efficient two-step estimation procedure.

The empirical analysis uses three months of high-frequency data for Mitsubishi UFJ Financial Group. The model reproduces the upward slope of a volatility signature plot, attributing it to locally elevated event intensity during market disequilibrium. Aggressive market orders trigger transitions away from equilibrium, while marketable limit orders contribute to liquidity depletion as spreads widen. The authors report that standard state-dependent Hawkes models can have unstable simulations and explosive spectral radii when physical constraints are absent. The evidence is limited to the studied stock and sample; the findings do not establish that the model will generalize to other markets or periods.

Key ideas

  • The model allows order book states to disappear, representing transitions seen in high-frequency trading.
  • KKT conditions support separable maximum likelihood estimation and a two-step procedure.
  • Aggressive market orders trigger disequilibrium, while marketable limit orders contribute to liquidity depletion.
  • The model reproduces an upward volatility signature plot in the studied stock data.
  • The authors report stability problems in unconstrained comparison models.

Tags

EquitiesHigh-frequency tradingMarket microstructureVolatilityStatistics
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# 2604.23961


# Extended State-dependent Hawkes Process for Limit Order Books: Mathematical Foundation and the Reproduction of Volatility Signature Plots









This paper proposes an Extended State-Dependent Hawkes Process (ExsdHawkes) to model the intricate dynamics of Limit Order Books (LOBs). Our theoretical contribution lies in relaxing traditional constraints by allowing for state disappearances---a phenomenon frequently observed in high-frequency trading. We mathematically prove, using Karush--Kuhn--Tucker (KKT) conditions, that the maximum likelihood estimation remains separable, justifying an efficient two-step procedure. In the empirical section, we apply our model to three months of high-frequency tick data of Mitsubishi UFJ Financial Group (8306). We demonstrate that ExsdHawkes successfully replicates the characteristic upward slope of the volatility signature plot by capturing the ``local super-criticality'' triggered during disequilibrium states. Crucially, we clarify that the transition out of equilibrium is deterministically triggered by Aggressive Market Orders (AMS/AMB), while Marketable Limit Orders (MLO) function as a critical liquidity-depletion catalyst within the expanded spread. Comparative analysis reveals that models lacking physical constraints (e.g., standard SD-Hawkes) suffer from explosive spectral radii and fail to maintain simulation stability. Our findings suggest that physical consistency is not merely a mathematical nicety, but a prerequisite for accurately modeling macro-level volatility. By enforcing the physical geometry to `pause' the residual accumulation during inadmissible periods, ExsdHawkes maintains statistical integrity where unconstrained models succumb to structural bias and simulation instability.

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