Reconsidering the Continuous Time Limit of the GARCH(1,1) Process.

摘要

In this note we reconsider the continuous time limit of the GARCH(1,1) process. Let Y[subscript k] and sigma[subscript k superscript 2] denote, respectively, the cumulative returns and the volatility processes. We consider the continuous time approximation of the couple (Y[subscript k], sigma[subcript k superscript 2]). We show that, by choosing different parameterizations, as a function of the discrete interval h, we can obtain either a degenerate or a non-degenerate diffusion limit. We then show that GARCH(1,1) processes can be obtained as Euler approximations of degenerate diffusions, while any Euler approximation of a non-degenerate diffusion is a stochastic volatility process.