The Generalized Hyperbolic Distribution Family: A Review with Applications to 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 SP 500 ETF Trust (SPY) from January 4, 2010, to July 22, 2024. The unrestricted GH distribution and its variance–gamma(VG), normal–inverse Gaussian (NIG), normal reciprocal inverse Gaussian (NRIG), hyperbolic (H), and HA subclasses are estimated by 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.