EXACT ELLIPTICAL DISTRIBUTIONS FOR MODELS OF CONDITIONALLY RANDOM FINANCIAL VOLATILITY
George A. Christodoulakis
Manchester Business School and Bank of Greece
Stephen E Satchell
Trinity College, University of Cambridge and Bank of Greece
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.
Keywords: Elliptical Distributions, Financial Asset Returns, Conditional Volatility, GARCH.
JEL classification: C22, G11, G12
The authors would like to thank Heather Gibson, Nikitas Pittis, conference participants at HFAA 2005 and Manchester Business School seminar participants for helpful comments. The views expressed in this paper are those of the authors and should in no part be attributed to the Bank of Greece.
George A. Christodoulakis,
Advisor to the Governor,
Bank of Greece, 21, E. Venizelos Ave.,
102 50 Athens, Greece
Tel.: +30 210-320 5006, Fax: +30 210-3205427