Multivariate Quadratic Hawkes Processes – Part I: Theoretical Analysis

22 Pages Posted: 6 Jul 2022

See all articles by Cecilia Aubrun

Cecilia Aubrun

Polytech'Savoie

Michael Benzaquen

Ecole Polytechnique, Palaiseau; Capital Fund Management

Jean-Philippe Bouchaud

Université Paris VI Pierre et Marie Curie - College de France

Date Written: June 21, 2022

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.

Suggested Citation

Aubrun, Cecilia and Benzaquen, Michael and Benzaquen, Michael and Bouchaud, Jean-Philippe, Multivariate Quadratic Hawkes Processes – Part I: Theoretical Analysis (June 21, 2022). Available at SSRN: https://ssrn.com/abstract=4142521 or http://dx.doi.org/10.2139/ssrn.4142521

Cecilia Aubrun (Contact Author)

Polytech'Savoie ( email )

BP 80439
74944 ANNECY LE VIEUX Cedex
France

Michael Benzaquen

Capital Fund Management ( email )

23 rue de l'Université
Paris, 75007
France

Ecole Polytechnique, Palaiseau ( email )

Route de Saclay
Palaiseau, 91128
France

Jean-Philippe Bouchaud

Université Paris VI Pierre et Marie Curie - College de France ( email )

11 Place Marcellin Berthelot
Paris, 75005
France

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