The Generalized Hyperbolic Distribution Family: A Unified Review and an Empirical Comparison of Nested Models for Financial Returns

This study provides a unified theoretical and empirical examination of the Generalized Hyperbolic (GH) distribution family. The special and limiting relationships among the GH distribution and its important subclasses are reviewed. The empirical analysis considers daily returns on the SPDR S&P 500 ETF Trust (SPY) from 4 January 2010 to 22 July 2024. The unrestricted GH distribution and its subclasses—Variance Gamma (VG), Normal Inverse Gaussian (NIG), Normal Reciprocal Inverse Gaussian (NRIG), Hyperbolic (H), Reciprocal Hyperbolic (RH) and Hyperbola (HA)—are estimated via the maximum likelihood method. Overall, the results demonstrate that the GH family provides a flexible and effective alternative to the Gaussian benchmark, although the preferred subclass depends on whether predictive adequacy or model parsimony is emphasized.