Modelling the dynamics of the limit order book in financial markets

Abstract

This thesis develops models and methods for the statistical properties of the limit order book for financial markets, a complex dynamical system of orders and cancellations, in continuous time and at multiple price levels. Initially, the heavy tailed features of limit order book volumes, aggregated to short, evenly spaced time intervals are investigated. These are found to require heavy tailed distributional models to adequately capture their statistical features.

A novel process to transform the physically operating order book into data suitable for analysis is presented. Limitations for point process modelling, such as events frequently occurring at the same time, are established. The marked Hawkes process is identified as a suitable model for event clustering. This overcomes many data constraints by aggregating events, allowing additional information potentially impacting the intensity, to be incorporated in marks attached to these events. Marks are identified by empirical research, which is guided by available literature.

A detailed description, methods of simulation, and parameter estimation of the univariate Hawkes process with multivariate marks is presented. This incorporates dependence features via copula models, with heavy tailed marginal distributions and requires substantial MATLAB implementation. Joint estimation via maximum likelihood, with the number and complexity of identified marks, necessitates the development of a method for screening marks that is computationally straightforward to implement. This new approach is based on the score test, which only requires the single fitting of the unboosted Hawkes process to the sequence of observed event times, together with the estimates of the moments of the functions of marks under assessment. The moments can be obtained parametrically, or non-parametrically. The test has an asymptotic chi-squared distribution under the null hypothesis that the marks do not impact the intensity. Extensive simulations confirm the power and utility of the test under realistic models and sample sizes. Application of the score test is made to futures data, and the identified serial dependence of the marks, leads to the new decoupled approximate method of likelihood estimation. This reduces model assumptions on statistical properties of the marks and leads to good performance of Hawkes process parameter estimation.