A Comparison of Generalized Hyperbolic Distribution Models for Equity Returns

Abstract

We discuss the calibration of the univariate and multivariate generalized hyperbolic distributions, as well as their hyperbolic, variance gamma, normal inverse Gaussian, and skew Student’s $t$-distribution subclasses for the daily log-returns of seven of the most liquid mining stocks listed on the Johannesburg Stocks Exchange. To estimate the model parameters from historic distributions, we use an expectation maximization based algorithm for the univariate case and a multicycle expectation conditional maximization estimation algorithm for the multivariate case. We assess the goodness of fit statistics using the log-likelihood, the Akaike information criterion, and the Kolmogorov-Smirnov distance. Finally, we inspect the temporal stability of parameters and note implications as criteria for distinguishing between models. To better understand the dependence structure of the stocks, we fit the MGHD and subclasses to both the stock returns and the two leading principal components derived from the price data. While the MGHD could fit both data subsets, we observed that the multivariate normality of the stock return residuals, computed by removing shared components, suggests that the departure from normality can be explained by the structure in the common factors.