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

Paper: arXiv 2509.21244

Authors: Cecilia Aubrun, Michael Benzaquen, Jean-Philippe Bouchaud

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.

Complexity vs Empirical Score

• Math Complexity: 9.0/10
• Empirical Rigor: 7.5/10
• Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper introduces a highly advanced mathematical framework involving Multivariate Quadratic Hawkes processes, kernel calibrations, and moment method approaches, indicating high math complexity. It also demonstrates strong empirical rigor through non-parametric calibration on real high-frequency market data, identifying new cross-effects with explicit trading universe analysis.

Research Flowchart

  flowchart TD
  A["Research Goal<br>Calibrate MQHawkes Model<br>Non-Parametrically"] --> B["Data & Input<br>Tick-by-Tick Futures Data<br>E-Mini, TBOND, Stocks"]
  
  B --> C["Key Methodology<br>Coarse-Grained Calibration<br>General Method of Moments"]
  
  C --> D{"Computational Process<br>Multi-Step Calibration"}
  
  D --> E["Step 1: Self-Effects<br>Stabilize Kernels"]
  D --> F["Step 2: Cross-Effects<br>Disentangle from Noise"]
  
  E --> G
  F --> G["Pairwise Interactions<br>Bi-Variate QHawkes Only"]
  
  G --> H["Key Findings & Outcomes<br>1. Cross-Zumbach Effects<br>2. Cross-Covariance Feedback<br>3. Cross-Leverage Effect"]

Cited References (63)

1. Crises de liquidité endogènes dans les marchés financiers (2020)
2. A nested factor model for non-linear dependencies in stock returns (2015)
3. Quadratic ARCH models (1995)
4. Hawkes processes and their applications to finance: a review (2018)
5. Volatility is (mostly) path-dependent (2023)
6. Limit theorems for nearly unstable Hawkes processes (2015)
7. Quadratic Hawkes processes for financial prices (2017)
8. Trades, quotes and prices: financial markets under the microscope (2018)

1 of 63 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.