Multi-kernel property in high-frequency price dynamics under Hawkes model

Paper: arXiv 2302.11822

Authors: Kyungsub Lee

Abstract

This study investigates and uses multi-kernel Hawkes models to describe a high-frequency mid-price process. Each kernel represents a different responsive speed of market participants. Using the conditional Hessian, we examine whether the numerical optimizer effectively finds the global maximum of the log-likelihood function under complicated modeling. Empirical studies that use stock prices in the US equity market show the existence of multi-kernels classified as ultra-high-frequency (UHF), very-high-frequency (VHF), and high-frequency (HF). We estimate the conditional expectations of arrival times and the degree of contribution to the high-frequency activities for each kernel.

Complexity vs Empirical Score

• Math Complexity: 7.5/10
• Empirical Rigor: 7.0/10
• Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper provides a mathematically sophisticated extension of Hawkes processes using multiple kernels to capture distinct participant response speeds. It validates these theoretical insights using high-frequency US equity data and conditional Hessian optimization, though exact code and data availability are limited.

Research Flowchart

  flowchart TD
    A[Goal: Model High-Frequency Mid-Price Dynamics via Multi-Kernel Hawkes Processes] --> B[Input Data: US Equity Market Stock Prices]
    B --> C[Method: Formulate Multi-Kernel Hawkes Model<br>UHF, VHF, HF Speed Kernels]
    C --> D[Optimization & Validation: Maximize Log-Likelihood & Check Conditional Hessian]
    D --> E[Outcome 1: Validated Global Maximum & Verified Multi-Kernel Existence]
    D --> F[Outcome 2: Estimated Arrival Times & Activity Contributions Per Kernel]