Abstract 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.

中文翻译：

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弱GARCH的连续极限

摘要 GARCH 模型被称为“强”或“弱”，这取决于创新的参数分布假设的存在。对称弱 GARCH(1, 1) 是 GARCH 类中唯一被证明在时间聚合属性下是封闭的模型。这个属性在两个方面是基本的：(a) 时间序列模型对数据频率是不变的；(b) 独特的期权定价模型作为连续时间限制存在。虽然对称弱 GARCH(1, 1) 在时间上精确聚合，因为它没有做参数分布假设，但缺乏这些也使得推导出理论结果变得更加困难。迎接这一挑战，我们证明了它的连续时间限制是一个几何均值回复随机波动过程，其扩散系数受对数收益的时变峰态控制。当对数返回正常时，该限制与 Nelson 的强 GARCH(1, 1) 限制一致。但与强 GARCH 模型不同的是，弱 GARCH(1, 1) 有一个独特的限制，因为它不对模型参数的收敛性做任何假设。每个参数的收敛性由时间聚合属性唯一确定。实证结果表明，附加的时变峰度参数增强了隐含波动率中的期限结构和微笑效应，从而为弱 GARCH 限制提供了更大的灵活性，以适应来自期权价格的现实世界数据。当对数返回正常时，该限制与 Nelson 的强 GARCH(1, 1) 限制一致。但与强 GARCH 模型不同的是，弱 GARCH(1, 1) 有一个独特的限制，因为它不对模型参数的收敛性做任何假设。每个参数的收敛性由时间聚合属性唯一确定。实证结果表明，附加的时变峰度参数增强了隐含波动率中的期限结构和微笑效应，从而为弱 GARCH 限制提供了更大的灵活性，以适应来自期权价格的现实世界数据。当对数返回正常时，该限制与 Nelson 的强 GARCH(1, 1) 限制一致。但与强 GARCH 模型不同的是，弱 GARCH(1, 1) 有一个独特的限制，因为它不对模型参数的收敛性做任何假设。每个参数的收敛性由时间聚合属性唯一确定。实证结果表明，附加的时变峰度参数增强了隐含波动率中的期限结构和微笑效应，从而为弱 GARCH 限制提供了更大的灵活性，以适应来自期权价格的现实世界数据。