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

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.