Application of Spread constrained Limit Order Book Hawkes Process to financial high frequency data modelling

In this work, we adopt Spread constrained Limit Order Book Hawkes Process (ScLOBHP) introduced by Zheng, Roueff and Abergel (2012) to model the financial high frequency data. The best limit prices in a limit order book are modelled using ScLOBHP since the spread plays a constraint in the transition probability of itself. For large stocks, we show that most of the best limit price changes is only change of one tick. We propose a basic ScLOBHP considering only the best limit prices' one tick change events and calibrate the parameters by maximum likelihood estimation. The cross-excitation relationship between different events are proven to be stable over a long period. However, the immigrant intensity and the exponential decaying fertility rate are likely to be determined by the liquidity. Finally, we propose a more sophisticated ScLOBHP to model extreme events such as multi-ticks price changes in order to provide a view on the impact of extreme events. We highlight that two-ticks price change events have an impact on one-tick price change events, nevertheless, few impact is detected in reverse.