A Class of Generalized Hyperbolic Continuous Time Integrated Stochastic Volatility Likelihood Models

Abstract

This paper discusses and analyzes a class of likelihood models which are based on two distributional innovations in financial models for stock returns. That is, the notion that the marginal distribution of aggregate returns of log-stock prices are well approximated by generalized hyperbolic distributions, and that volatility clustering can be handled by specifying the integrated volatility as a random process such as that proposed in a recent series of papers by Barndorff-Nielsen and Shephard (BNS). The BNS models produce likelihoods for aggregate returns which can be viewed as a subclass of latent regression models where one has n conditionally independent Normal random variables whose mean and variance are representable as linear functionals of a common unobserved Poisson random measure. James (2005b) recently obtains an exact analysis for such models yielding expressions of the likelihood in terms of quite tractable Fourier-Cosine integrals. Here, our idea is to analyze a class of likelihoods, which can be used for similar purposes, but where the latent regression models are based on n conditionally independent models with distributions belonging to a subclass of the generalized hyperbolic distributions and whose corresponding parameters are representable as linear functionals of a common unobserved Poisson random measure. Our models are perhaps most closely related to the Normal inverse Gaussian/GARCH/A-PARCH models of Brandorff-Nielsen (1997) and Jensen and Lunde (2001), where in our case the GARCH component is replaced by quantities such as INT-OU processes. It is seen that, importantly, such likelihood models exhibit quite different features structurally. One nice feature of the model is that it allows for more flexibility in terms of modelling of external regression parameters.