A modulated renewal Hawkes process and its application to modelling extreme mid-price drops on cryptocurrencies
The Hawkes process was first proposed by Alan G. Hawkes in which the arrival of events exhibits a self-exciting behaviour. One extension of the classical Hawkes process is the renewal Hawkes process, which allows the underlying process for background events to be a renewal process, rather than the homogeneous Poisson process in the classical Hawkes process. The renewal Hawkes process is stationary in nature, so it is not suitable in situations where there are systematic trends in event occurrence rate. Therefore, in this thesis, we propose a renewal Hawkes process in which a trend function is employed to account for the systematic patterns in the event occurrence rate. We term the process the modulated renewal Hawkes process. Due to the lack of an explicit expression for the intensity process, likelihood evaluation for the modulated renewal Hawkes process model is not trivial. However, by modifying the likelihood evaluation algorithm for renewal Hawkes process in Chen & Stindl (2018), we are able to propose an algorithm to evaluate the exact likelihood of the modulated renewal Hawkes process model. The evaluated likelihood can then be maximised to obtain the maximum likelihood estimator (MLE) of the model parameters. We also propose a method to obtain fast and accurate approximations to the likelihood. In the case where a suitable parametric form of the trend function is not available, we approximate the trend function using B-spline functions. We also derive the Rosenblatt residuals of the modulated renewal Hawkes process, which can serve as a basis for assessing the goodness-of-fit of the model. Simulation experiments were conducted to assess the performance of the MLE of the modulated renewal Hawkes process with either exact or approximate likelihood evaluation, both in the parametric model and in the semiparametric model with an unspecified trend function. We also present an application of the modulated renewal Hawkes process model to the analysis of cryptocurrency data. The modulated renewal Hawkes process model with a B-spline trend function is applied to model extreme intraday negative returns on several cryptocurrencies. The estimated trend function suggests an inverse U-shaped trend in the intraday occurrence times of extreme negative returns on cryptocurrencies. We also compared the model fitting results with several simpler models, such as the nonstationary Hawkes process and the renewal Hawkes process. On most of the cryptocurrency data sets considered in this work, the modulated renewal Hawkes process was found to provide the best fit both by the Rosenblatt residuals based goodness-of-fit check and by the Akaike Information Criterion.
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