Modeling of correlations by Hawkes processes for pure jump asset movements

Abstract

The aim of this paper is to develop correlated pure jump asset dynamics by the Hawkes processes incorporating market microstructure noise and trade clustering and to compute the variances, covariance and correlation between the asset price returns. As well as the simple Hawkes process which has a constant jump size, the marked Hawkes process which has a random jump size is considered. In the marked Hawkes model, we assume that impact of the mark to the future jump intensity is linear. Without the assumptions for any speci？c distribution of the mark size, we derive the variance, covariance and correlation formulas for the price returns based on stochastic calculus and statistic methods. The simulation results for the generalized marked Hawkes model are consistent with the theoretic results when the marks are independent and identically distributed. The empirical results show that the slope parameter of the impact function is positive in general, which means that the large jump a？ects the large impact in the future intensities. The trends of the variances and covariance are similar to the two time scales realized variances and covariance, respectively. In the perspective of the variances and covariance estimation, the results with i.i.d. marks and those with non-assuming i.i.d. marks are similar. Thus for practice, it may be su？cient to regard that marks are i.i.d.. The generalized marked Hawkes model is a useful tool to compute the intraday variances and covariance of the two asset price returns.