Multivariate Quadratic Hawkes Processes -- Part I: Theoretical Analysis

TL;DR

This paper extends quadratic Hawkes processes to a multivariate setting to better model financial market co-jumps, deriving calibration methods and analyzing volatility distribution with power-law behavior.

Contribution

It introduces a multivariate quadratic Hawkes framework, derives calibration equations, and analyzes the process's volatility distribution, advancing modeling of complex asset interactions.

Findings

Derivation of Yule-Walker equations for calibration.

Identification of power-law volatility distribution.

Conditions for stationarity and endogeneity ratios.

Abstract

Quadratic Hawkes (QHawkes) processes have proved effective at reproducing the statistics of price changes, capturing many of the stylised facts of financial markets. Motivated by the recently reported strong occurrence of endogenous co-jumps (simultaneous price jumps of several assets) we extend QHawkes to a multivariate framework (MQHawkes), that is considering several financial assets and their interactions. Assuming that quadratic kernels write as the sum of a time-diagonal component and a rank one (trend) contribution, we investigate endogeneity ratios and the resulting stationarity conditions. We then derive the so-called Yule-Walker equations relating covariances and feedback kernels, which are essential to calibrate the MQHawkes process on empirical data. Finally, we investigate the volatility distribution of the process and find that, as in the univariate case, it exhibits power-law behavior, with an exponent that can be exactly computed in some limiting cases.