The continuous limit of weak GARCH

GARCH models are called ‘strong’ or ‘weak’ depending on the presence of parametric distributional assumptions for the innovations. The symmetric weak GARCH(1, 1) is the only model in the GARCH class that has been proved to be closed under the temporal aggregation property . This property is fundamental in two respects: (a) for a time-series model to be invariant to the data frequency; and (b) for a unique option-pricing model to exist as a continuous-time limit. While the symmetric weak GARCH(1, 1) is temporally aggregating precisely because it makes no parametric distributional assumptions, the lack of these also makes it harder to derive theoretical results. Rising to this challenge, we prove that its continuous-time limit is a geometric mean-reverting stochastic volatility process with diffusion coefficient governed by a time-varying kurtosis of log returns. When log returns are normal the limit coincides with Nelson’s strong GARCH(1, 1) limit. But unlike strong GARCH models, the weak GARCH(1, 1) has a unique limit because it makes no assumptions about the convergence of model parameters. The convergence of each parameter is uniquely determined by the temporal aggregation property. Empirical results show that the additional time-varying kurtosis parameter enhances both term-structure and smile effects in implied volatilities, thereby affording greater flexibility for the weak GARCH limit to fit real-world data from option prices.