CLOSING THE GARCH GAP - CONTINUOUS-TIME GARCH MODELING

It is the purpose of this paper to build a bridge between continuous t ime models, which are central in the modern finance literature, and (w eak) GARCH processes in discrete time, which often provide parsimoniou s descriptions of the observed data. The properties of continuous time processes which exhibit GARCH-type behavior at all discrete frequenci es will be discussed. Several examples of such processes illustrate th e general theory. The class of continuous time GARCH models can be div ided into two subclasses. In the first group (GARCH diffusions) the sa mple paths are smooth and in the other group (GARCH jump-diffusions) t he sample paths are erratic. A simple, complete characterization of bo th types is given in terms of the kurtosis of the observed discrete ti me 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, knowledg e of the discrete time GARCH parameters at only one frequency complete ly determines the continuous time coefficients of the GARCH process. S o, in estimating a continuous time GARCH process it suffices to estima te the discrete time GARCH parameters for the available data frequency . The analysis carries over to models with an autoregressive component .