Modeling High Frequency Data Using Hawkes Processes with Power-law Kernels1

Those empirical properties exhibited by high frequency financial data, such as time-varying intensities and self-exciting features, make it a challenge to model appropriately the dynamics associated with, for instance, order arrival. To capture the microscopic structures pertaining to limit order books, this paper focuses on modeling high frequency financial data using Hawkes processes. Specifically, the model with power-law kernels is compared with the counterpart with exponential kernels, on the goodness of fit to the empirical data, based on a number of proposed quantities for statistical tests. Based on one-trading-day data of one representative stock, it is shown that Hawkes processes with power-law kernels are able to reproduce the intensity of jumps in the price processes more accurately, which suggests that they could serve as a realistic model for high frequency data on the level of microstructure.