下面研究GARCH模型导致的波动率期限结构， 比如， 日对数收益率的波动率与月对数收益率的波动率的关系。 以时间\(t\)为基础， 距离\(t\)时刻\(h\)期（比如\(h\)个交易日）的对数收益率为 \[\begin{aligned} r_{t,h} = \sum_{i=1}^h r_{t+i} \end{aligned}\] 于是 \[ E(r_{t,h} | F_t) = \sum_{i=1}^h E( r_{t+i} | F_t ) \] \(h\)期的条件方差，即波动率平方为 \[\begin{aligned} \text{Var}(r_{t,h} | F_t) = \sum_{i=1}^h \text{Var}(r_{t+i} | F_t) + \sum_{1 \leq i < j \leq h} \text{Cov}(r_{t+i}, r_{t+j} | F_t) \end{aligned}\] 实证分析和有效市场理论都认为协方差接近零，所以可假定 \[\begin{aligned} \text{Var}(r_{t,h} | F_t) = \sum_{i=1}^h \text{Var}(r_{t+i} | F_t) \end{aligned}\] 对于GARCH模型，这就是 \[\begin{aligned} \sigma_{t,h}^2 = \text{Var}(r_{t,h} | F_t) = \sum_{\ell=1}^h \sigma_t^2(\ell) \end{aligned}\] 其中\(\sigma_{t,h}^2\)表示以\(h\)期为单位的基于时刻\(t\)计算的条件方差， 即\(h\)期的对数收益率的波动率平方， \(\sigma_t^2(\ell)\)是基于时刻\(t\)的单期对数收益率的波动率的超前\(\ell\)步预测。

考虑GARCH(1,1)模型的超前\(\ell\)步预测问题。模型为： \[\begin{align} \sigma_t^2 =& \alpha_0 + \alpha_1 a_{t-1}^2 + \beta_1 \sigma_{t-1}^2 \tag{22.1} \end{align}\] 其中\(\alpha_0>0\), \(\alpha_1, \beta_1 \in [0, 1)\), \(\alpha_1 + \beta_1 < 1\)。 已证明 \[ \text{Var}(a_t) = \sigma^2 = \frac{\alpha_0}{1 - \alpha_1 - \beta_1} \] 可以将(22.1)改写成 \[\begin{align} (\sigma_t^2 - \sigma^2) = \alpha_1 (a_{t-1}^2 - \sigma^2) + \beta_1 (\sigma_{t-1}^2 - \sigma^2) \tag{22.2} \end{align}\] 这个式子可以看成是\(a_t^2\)的一步预测\(E(a_t^2|F_{t-1})\)与长期预测\(\sigma^2\)的偏离的模型。

波动率的基于\(F_t\)的超前一步预测为 \[ \sigma_t^2(1) = \alpha_0 + \alpha_1 a_{t}^2 + \beta_1 \sigma_{t}^2 \] 超前\(\ell\)步预测为 \[\begin{aligned} \sigma_t^2(\ell) = \alpha_0 + (\alpha_1 + \beta_1) \sigma_t^2(\ell - 1), \ \ell=2,3,\dots \end{aligned}\] 以\(\sigma^2 = \alpha_0/(1-\alpha_1 - \beta_1)\)代入上式可变成 \[\begin{aligned} \sigma_t^2(\ell) - \sigma^2 =& (\alpha_1 + \beta_1) [\sigma_t^2(\ell-1) - \sigma^2] \\ =& (\alpha_1 + \beta_1)^{\ell-1} [\sigma_t^2(1) - \sigma^2] \end{aligned}\]

当\(\alpha_1 + \beta_1