17 - GARCH processes — some exact results, some difficulties and a suggested remedy

Abstract

This chapter examines both log and levels models of volatility and determines, where possible, the exact properties of the volatility process and the stochastic process itself. Generalized autoregressive conditional heteroscedasticity (GARCH) models can be represented as stochastic recurrence equations. More tractable analysis can be achieved by introducing a simple modification in the GARCH process, which is equivalent to an approximation of the original process. An analysis of some typical UK data using both the regular GARCH model and the suggested alternative does not lead to a clear preference of one specification over the other. The chapter also shows how stochastic volatility models arise naturally in a world with stochastic information flows. It concludes that there is little possibility of deriving analytic results except for moments.