Multivariate quadratic Hawkes processes—part I: theoretical analysis

Quadratic Hawkes (QHawkes) processes have proved effective at reproducing the statistics of price changes, capturing many of the stylized 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, its tail exhibits power-law behaviour, with an unique exponent that can be exactly computed in some limiting cases.