Hawkes Process

An Impulse Control Approach to Market Making in a Hawkes LOB Market ArXiv ID: 2510.26438 “View on arXiv” Authors: Konark Jain, Nick Firoozye, Jonathan Kochems, Philip Treleaven Abstract We study the optimal Market Making problem in a Limit Order Book (LOB) market simulated using a high-fidelity, mutually exciting Hawkes process. Departing from traditional Brownian-driven mid-price models, our setup captures key microstructural properties such as queue dynamics, inter-arrival clustering, and endogenous price impact. Recognizing the realistic constraint that market makers cannot update strategies at every LOB event, we formulate the control problem within an impulse control framework, where interventions occur discretely via limit, cancel, or market orders. This leads to a high-dimensional, non-local Hamilton-Jacobi-Bellman Quasi-Variational Inequality (HJB-QVI), whose solution is analytically intractable and computationally expensive due to the curse of dimensionality. To address this, we propose a novel Reinforcement Learning (RL) approximation inspired by auxiliary control formulations. Using a two-network PPO-based architecture with self-imitation learning, we demonstrate strong empirical performance with limited training, achieving Sharpe ratios above 30 in a realistic simulated LOB. In addition to that, we solve the HJB-QVI using a deep learning method inspired by Sirignano and Spiliopoulos 2018 and compare the performance with the RL agent. Our findings highlight the promise of combining impulse control theory with modern deep RL to tackle optimal execution problems in jump-driven microstructural markets. ...

High-Frequency Market Manipulation Detection with a Markov-modulated Hawkes process ArXiv ID: 2502.04027 “View on arXiv” Authors: Unknown Abstract This work focuses on a self-exciting point process defined by a Hawkes-like intensity and a switching mechanism based on a hidden Markov chain. Previous works in such a setting assume constant intensities between consecutive events. We extend the model to general Hawkes excitation kernels that are piecewise constant between events. We develop an expectation-maximization algorithm for the statistical inference of the Hawkes intensities parameters as well as the state transition probabilities. The numerical convergence of the estimators is extensively tested on simulated data. Using high-frequency cryptocurrency data on a top centralized exchange, we apply the model to the detection of anomalous bursts of trades. We benchmark the goodness-of-fit of the model with the Markov-modulated Poisson process and demonstrate the relevance of the model in detecting suspicious activities. ...

Limit Order Book Event Stream Prediction with Diffusion Model ArXiv ID: 2412.09631 “View on arXiv” Authors: Unknown Abstract Limit order book (LOB) is a dynamic, event-driven system that records real-time market demand and supply for a financial asset in a stream flow. Event stream prediction in LOB refers to forecasting both the timing and the type of events. The challenge lies in modeling the time-event distribution to capture the interdependence between time and event type, which has traditionally relied on stochastic point processes. However, modeling complex market dynamics using stochastic processes, e.g., Hawke stochastic process, can be simplistic and struggle to capture the evolution of market dynamics. In this study, we present LOBDIF (LOB event stream prediction with diffusion model), which offers a new paradigm for event stream prediction within the LOB system. LOBDIF learns the complex time-event distribution by leveraging a diffusion model, which decomposes the time-event distribution into sequential steps, with each step represented by a Gaussian distribution. Additionally, we propose a denoising network and a skip-step sampling strategy. The former facilitates effective learning of time-event interdependence, while the latter accelerates the sampling process during inference. By introducing a diffusion model, our approach breaks away from traditional modeling paradigms, offering novel insights and providing an effective and efficient solution for learning the time-event distribution in order streams within the LOB system. Extensive experiments using real-world data from the limit order books of three widely traded assets confirm that LOBDIF significantly outperforms current state-of-the-art methods. ...

No Tick-Size Too Small: A General Method for Modelling Small Tick Limit Order Books ArXiv ID: 2410.08744 “View on arXiv” Authors: Unknown Abstract Tick-sizes not only influence the granularity of the price formation process but also affect market agents’ behavior. We investigate the disparity in the microstructural properties of the Limit Order Book (LOB) across a basket of assets with different relative tick-sizes. A key contribution of this study is the identification of several stylized facts, which are used to differentiate between large, medium, and small-tick assets, along with clear metrics for their measurement. We provide cross-asset visualizations to illustrate how these attributes vary with relative tick-size. Further, we propose a Hawkes Process model that {"\color{black"}not only fits well for large-tick assets, but also accounts for }sparsity, multi-tick level price moves, and the shape of the LOB in small-tick assets. Through simulation studies, we demonstrate the {"\color{black"} versatility} of the model and identify key variables that determine whether a simulated LOB resembles a large-tick or small-tick asset. Our tests show that stylized facts like sparsity, shape, and relative returns distribution can be smoothly transitioned from a large-tick to a small-tick asset using our model. We test this model’s assumptions, showcase its challenges and propose questions for further directions in this area of research. ...

Non-Parametric Estimation of Multi-dimensional Marked Hawkes Processes ArXiv ID: 2402.04740 “View on arXiv” Authors: Unknown Abstract An extension of the Hawkes process, the Marked Hawkes process distinguishes itself by featuring variable jump size across each event, in contrast to the constant jump size observed in a Hawkes process without marks. While extensive literature has been dedicated to the non-parametric estimation of both the linear and non-linear Hawkes process, there remains a significant gap in the literature regarding the marked Hawkes process. In response to this, we propose a methodology for estimating the conditional intensity of the marked Hawkes process. We introduce two distinct models: \textit{“Shallow Neural Hawkes with marks”}- for Hawkes processes with excitatory kernels and \textit{“Neural Network for Non-Linear Hawkes with Marks”}- for non-linear Hawkes processes. Both these approaches take the past arrival times and their corresponding marks as the input to obtain the arrival intensity. This approach is entirely non-parametric, preserving the interpretability associated with the marked Hawkes process. To validate the efficacy of our method, we subject the method to synthetic datasets with known ground truth. Additionally, we apply our method to model cryptocurrency order book data, demonstrating its applicability to real-world scenarios. ...

Self and mutually exciting point process embedding flexible residuals and intensity with discretely Markovian dynamics ArXiv ID: 2401.13890 “View on arXiv” Authors: Unknown Abstract This work introduces a self and mutually exciting point process that embeds flexible residuals and intensity with discretely Markovian dynamics. By allowing the integration of diverse residual distributions, this model serves as an extension of the Hawkes process, facilitating intensity modeling. This model’s nature enables a filtered historical simulation that more accurately incorporates the properties of the original time series. Furthermore, the process extends to multivariate models with manageable estimation and simulation implementations. We investigate the impact of a flexible residual distribution on the estimation of high-frequency financial data, comparing it with the Hawkes process. ...

Hawkes-based cryptocurrency forecasting via Limit Order Book data ArXiv ID: 2312.16190 “View on arXiv” Authors: Unknown Abstract Accurately forecasting the direction of financial returns poses a formidable challenge, given the inherent unpredictability of financial time series. The task becomes even more arduous when applied to cryptocurrency returns, given the chaotic and intricately complex nature of crypto markets. In this study, we present a novel prediction algorithm using limit order book (LOB) data rooted in the Hawkes model, a category of point processes. Coupled with a continuous output error (COE) model, our approach offers a precise forecast of return signs by leveraging predictions of future financial interactions. Capitalizing on the non-uniformly sampled structure of the original time series, our strategy surpasses benchmark models in both prediction accuracy and cumulative profit when implemented in a trading environment. The efficacy of our approach is validated through Monte Carlo simulations across 50 scenarios. The research draws on LOB measurements from a centralized cryptocurrency exchange where the stablecoin Tether is exchanged against the U.S. dollar. ...

Limit Order Book Dynamics and Order Size Modelling Using Compound Hawkes Process ArXiv ID: 2312.08927 “View on arXiv” Authors: Unknown 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. ...

Estimation of an Order Book Dependent Hawkes Process for Large Datasets ArXiv ID: 2307.09077 “View on arXiv” Authors: Unknown Abstract A point process for event arrivals in high frequency trading is presented. The intensity is the product of a Hawkes process and high dimensional functions of covariates derived from the order book. Conditions for stationarity of the process are stated. An algorithm is presented to estimate the model even in the presence of billions of data points, possibly mapping covariates into a high dimensional space. The large sample size can be common for high frequency data applications using multiple liquid instruments. Convergence of the algorithm is shown, consistency results under weak conditions is established, and a test statistic to assess out of sample performance of different model specifications is suggested. The methodology is applied to the study of four stocks that trade on the New York Stock Exchange (NYSE). The out of sample testing procedure suggests that capturing the nonlinearity of the order book information adds value to the self exciting nature of high frequency trading events. ...