The Pitfalls of Continuous Heavy-Tailed Distributions in High-Frequency Data Analysis

ArXiv ID: 2510.09785 “View on arXiv”

Authors: Vladimír Holý

Abstract

We address the challenges of modeling high-frequency integer price changes in financial markets using continuous distributions, particularly the Student’s t-distribution. We demonstrate that traditional GARCH models, which rely on continuous distributions, are ill-suited for high-frequency data due to the discreteness of price changes. We propose a modification to the maximum likelihood estimation procedure that accounts for the discrete nature of observations while still using continuous distributions. Our approach involves modeling the log-likelihood in terms of intervals corresponding to the rounding of continuous price changes to the nearest integer. The findings highlight the importance of adjusting for discreteness in volatility analysis and provide a framework for incroporating any continuous distribution for modeling high-frequency prices.

Keywords: Discrete price changes, High-frequency data, Maximum likelihood estimation, Volatility modeling, GARCH, General (Financial Markets)

Complexity vs Empirical Score

• Math Complexity: 6.5/10
• Empirical Rigor: 7.0/10
• Quadrant: Holy Grail
• Why: The paper presents advanced statistical derivations and model modifications (e.g., interval likelihood for discrete rounding) with moderate math density, and is heavily data-driven, using extensive high-frequency tick data (15M observations), multiple aggregation frequencies, and out-of-sample validation across different stocks and software packages.

  flowchart TD
    A["Research Goal:<br>Modeling High-Frequency<br>Integer Price Changes"] --> B{"Data/Inputs:<br>High-Frequency<br>Discrete Data"}
    B --> C["Methodology:<br>Continuous Distributions<br>Student's t/GARCH"]
    C --> D["Computational Process:<br>Modified MLE<br>Interval-Based Log-Likelihood"]
    D --> E["Key Findings:<br>1. Traditional GARCH<br>Ill-Suited for Discrete Data<br>2. Proposed Adjustment<br>Essential for Accuracy<br>3. Framework for<br>General Continuous Distributions"]