Multivariate Quadratic Hawkes Processes -- Part II: Non-Parametric Empirical Calibration

2025-09-25

AI Summary

This paper addresses the nonparametric empirical calibration challenge of multivariate quadratic Hawkes (MQHawkes) processes. We propose a multi-step coarse-graining calibration framework based on moment estimation, which decouples self- and cross-excitation effects sequentially to substantially improve numerical stability and interpretability. Empirically, applying this framework to high-frequency financial data, we identify three novel cross-asset volatility feedback mechanisms: cross-Zumbach effects, cross-leverage effects, and coupled influence of historical realized covariance on volatility. Our analysis reveals that E-Mini futures exert a strong trend-driving effect on the volatility of other assets—such as TBOND—and uncovers a pervasive cross-asset leverage effect, wherein the sign of past returns systematically modulates residual volatility. These findings establish a new paradigm for high-dimensional point process modeling and cross-market risk transmission analysis.

Technology Category

• Machine Learning: Calibration & Uncertainty Quantification
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding
• Reasoning under Uncertainty: Relational Probabilistic Models

Application Category

• Web Mining and Content Analysis: Mining multimedia, multimodal, multilingual, cross-lingual Web data
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

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.

Problem

• Calibrating Multivariate Quadratic Hawkes models non-parametrically using coarse-grained data
• Disentangling cross-effects from noise in multivariate financial market interactions
• Identifying novel cross-asset feedback effects in high-frequency financial data

Innovation

• Non-parametric calibration using method of moments
• Multi-step procedure focusing on self then cross-effects
• Restricts to pair interactions for numerical stability