Multivariate quadratic Hawkes processes—part II: non-parametric empirical calibration

Abstract

This is the second part of our work on Multivariate Quadratic Hawkes (MQHawkes) Processes, devoted to the calibration of the model defined and studied analytically in Aubrun, C., Benzaquen, M., & Bouchaud, J. P., Quantitative Finance, 23(5), 741-758 (2023). We propose a non-parametric calibration method based on the general method of moments applied to a coarse-grained version of the MQHawkes model. This allows us to bypass challenges inherent to tick by tick data. Our main methodological innovation is a multi-step calibration procedure, first focusing on ''self'' feedback kernels, and then progressively including cross-effects. Indeed, while cross-effects are significant and interpretable, they are usually one order of magnitude smaller than self-effects, and must therefore be disentangled from noise with care. For numerical stability, we also restrict to pair interactions and only calibrate bi-variate QHawkes, neglecting higher-order interactions. Our main findings are: (a) While cross-Hawkes feedback effects have been empirically studied previously, cross-Zumbach effects are clearly identified here for the first time. The effect of recent trends of the E-Mini futures contract onto the volatility of other futures contracts is especially strong; (b) We have identified a new type of feedback that couples past realized covariance between two assets and future volatility of these two assets, with the pair E-Mini vs TBOND as a case in point; (c) A cross-leverage effect, whereby the sign of the return of one asset impacts the volatility of another asset, is also clearly identified. The cross-leverage effect between the E-Mini and the residual volatility of single stocks is notable, and surprisingly universal across the universe of stocks that we considered.

The authors' abstract, as published at the source. Quantitative Finance, 2026 · DOI ↗