Limit Order Book Dynamics and Order Size Modelling Using Compound Hawkes Process

Abstract: Hawkes Process has been used to model Limit Order Book (LOB) dynamics in several ways in the literature however the focus has been limited to capturing the inter-event times while the order size is usually assumed to be constant. We propose a novel methodology of using Compound Hawkes Process for the LOB where each event has an order size sampled from a calibrated distribution. The process is formulated in a novel way such that the spread of the process always remains positive. Further, we condition the model parameters on time of day to support empirical observations. We make use of an enhanced non-parametric method to calibrate the Hawkes kernels and allow for inhibitory cross-excitation kernels. We showcase the results and quality of fits for an equity stock's LOB in the NASDAQ exchange and compare them against several baselines. Finally, we conduct a market impact study of the simulator and show the empirical observation of a concave market impact function is indeed replicated.

• The paper introduces a Compound Hawkes Process that enhances LOB modeling by incorporating empirically calibrated variable order sizes and time-of-day effects.
• The methodology employs a 12-dimensional framework and non-parametric calibration to reliably estimate kernel parameters across various time scales.
• Results demonstrate superior replication of market impact dynamics and spread distributions, setting a new standard in simulating financial market microstructure.

Limit Order Book Dynamics Using Compound Hawkes Process

Introduction

The study entitled "Limit Order Book Dynamics and Order Size Modelling Using Compound Hawkes Process" presents an advanced approach for modeling the dynamics of Limit Order Books (LOB) using a Compound Hawkes Process. Traditional methods, which often employ Poisson processes, fall short in capturing the intricate details of order sizes and event dependencies. This paper addresses these limitations by incorporating variable order sizes drawn from calibrated distributions, thus enhancing the realism of LOB simulations. The model also integrates time-of-day dependencies, capturing the cyclic patterns typical in financial markets.

Methodology

The methodology hinges on the Compound Hawkes Process, a variant of point processes well-suited for financial applications due to its ability to model self-exciting events. The authors propose a 12-dimensional model that captures various types of market events, including limit, market, and cancel orders on both bid and ask sides. A significant innovation is the calibration of order sizes, which are sampled from empirically derived distributions, allowing the model to reflect the true variance seen in market data.

The structure of the model ensures the bid-ask spread remains non-negative by adjusting order intensities based on the current spread. Additionally, the intrinsic seasonality of trading activities is captured through time-of-day conditioned intensities, further aligning the model's predictions with real-world observations.

Calibration Techniques

Calibration is conducted using non-parametric methods adapted to accommodate slow decay kernels characteristic of Hawkes processes. The authors refine existing techniques to improve calibration stability without relying on predetermined kernel shapes. This approach is shown to effectively estimate kernel parameters across varying time scales, providing robustness to the model's predictions.

An integral component of the calibration process involves constructing stationary distributions for order sizes, factoring in empirical observations of traders' preferences for round numbers (e.g., multiples of 100 shares). This nuanced approach allows the model to replicate observed market microstructures with high fidelity.

Results and Discussion

The model demonstrates a strong fit across various empirical measures, such as spread distributions, inter-arrival times, and volatility patterns. The comparison with both traditional Poisson models and existing Hawkes-based frameworks highlights its superior ability to reproduce key market phenomena, including the U-shaped daily volume profile and the endogenous clustering of orders.

Moreover, the study yields critical insights into market impact dynamics. Simulated scenarios show how trading strategies influence price movements, with the Compound Hawkes Process capturing the concave relationship between trade sizes and market impact—an essential consideration for quantitative trading strategies.

Conclusion

This paper extends the frontiers of financial market modeling by incorporating a Compound Hawkes Process that adeptly simulates the stochastic nature of LOB events. The model's ability to incorporate variable order sizes and temporal dependencies positions it as a powerful tool for both academic research and practical application in quantitative finance. Future research directions may explore the model’s application to other asset classes and refine its parameters to account for market liquidity variations. The study thus sets a new standard for the simulation of financial order book dynamics, offering granular insights into the microstructure of trading environments.