Exact Elliptical Distributions for Models of Conditionally Random Financial Volatility

Assuming the time series of random returns to be jointly elliptical, we derive a relationship between its conditional variance and the probability density function of the conditioning set. In the case that such a relationship is linear in a quadratic form for of the conditioning variables, we show that the probability density function of the conditioning variables is multivariate t. This result is then applied to models of conditionally random volatility and used to derive exact results for the GARCH(p,q) class of processes previously thought to be intractable.