What is the best Lévy model for stock indices? A comparative study with a view to time consistency

Lévy models are frequently used for asset log-returns. An important criterion is the distributional assumption on the increments. Candidates include, for example, the generalized hyperbolic, the normal inverse Gaussian, and the (skew) Student–Lévy process. We perform a comparative study for multiple equity indices of different countries using different Lévy models to determine the best fit using the Kolmogorov–Smirnov statistic, the Anderson–Darling statistic, and the Bayesian information criterion as goodness-of-fit measures. We fit Lévy models both to daily and to hourly log-returns. To date, the literature has paid little attention to the question of whether these Lévy models for daily returns also fit well at higher frequencies, that is, intraday returns, and vice versa. Eberlein and Özkan (Quant Finance 3(1):40–50, 2003) called this “time consistency.” Our key finding is that there are time inconsistencies. This means that some models that fit well for daily returns, for example, the variance gamma model, fit poorly for hourly returns. We find that the Student–Lévy process is a more appropriate alternative.