Reconsidering the continuous time limit of the GARCH(1, 1) process

In this note we reconsider the continuous time limit of the GARCH(1, 1) process. Let Yk and σk2 denote, respectively, the cumulative returns and the volatility processes. We consider the continuous time approximation of the couple (Yk, σk2). 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.