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Weak Diffusion Limits of Two Real-Time GARCH-type Models

We derive the diffusion limits of two Real-Time GARCH (RT-GARCH)-type models. We show the diffusion limit of the original RT-GARCH model fails to guarantee with probability one positive volatility unless it is degenerate. Consequently, we show a novel model that builds upon the idea of RT-GARCH proposed by Ding (2021) has the usual GARCH diffusion limit. As a result, we call for caution when using RT-GARCH since it lacks compatibility with existing asset pricing theories. On the contrary, Ding’s (2021) model combines the advantages of both RT-GARCH and GARCH models.