Closing the GARCH gap: Continuous time GARCH modeling

Abstract

It is the purpose of this paper to build a bridge between continuous time models, which are central in the modern finance literature, and (weak) GARCH processes in discrete time, which often provide parsimonious descriptions of the observed data. The properties of continuous time processes which exhibit GARCH-type behavior at all discrete frequencies will be discussed. Several examples of such processes illustrate the general theory. The class of continuous time GARCH models can be divided into two subclasses. In the first group (GARCH diffusions) the sample paths are smooth and in the other group (GARCH jump-diffusions) the sample paths are erratic. A simple, complete characterization of both types is given in terms of the kurtosis of the observed discrete time data. These two groups of GARCH processes can be described by three and four coefficients, respectively. Explicit formulas of all implied discrete time weak GARCH parameters are available. Moreover, knowledge of the discrete time GARCH parameters at only one frequency completely determines the continuous time coefficients of the GARCH process. So, in estimating a continuous time GARCH process it suffices to estimate the discrete time GARCH parameters for the available data frequency. The analysis carries over to models with an autoregressive component.