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 Y_k and σ²_k denote, respectively, the cumulative returns and the volatility processes. We consider the continuous time approximation of the couple (Y_k, σ²_k). 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.