22 波动率模型的应用

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[egin{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[egin{aligned} ext{Var}(r_{t,h} | F_t)= sum_{i=1}^h ext{Var}(r_{t+i} | F_t)+ sum_{1 leq i < j leq h} ext{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[egin{aligned} ext{Var}(r_{t,h} | F_t)= sum_{i=1}^h ext{Var}(r_{t+i} | F_t)end{aligned}]对于GARCH模型，这就是[egin{aligned}sigma_{t,h}^2 = ext{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：[egin{align}sigma_t^2 =& alpha_0 + alpha_1 a_{t-1}^2 + eta_1 sigma_{t-1}^2 ag{22.1}end{align}]其中(alpha_0>0),(alpha_1, eta_1 in [0, 1)),(alpha_1 + eta_1 < 1)。已证明[ ext{Var}(a_t) = sigma^2= frac{alpha_0}{1 - alpha_1 - eta_1}]可以将(22.1)改写成[egin{align}(sigma_t^2 - sigma^2)= alpha_1 (a_{t-1}^2 - sigma^2) + eta_1 (sigma_{t-1}^2 - sigma^2) ag{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 + eta_1 sigma_{t}^2]超前(ell)步预测为[egin{aligned}sigma_t^2(ell)= alpha_0 + (alpha_1 + eta_1) sigma_t^2(ell - 1), ell=2,3,dotsend{aligned}]以(sigma^2 = alpha_0/(1-alpha_1 - eta_1))代入上式可变成[egin{aligned}sigma_t^2(ell) - sigma^2=& (alpha_1 + eta_1) [sigma_t^2(ell-1) - sigma^2] \=& (alpha_1 + eta_1)^{ell-1} [sigma_t^2(1) - sigma^2]end{aligned}]

当(alpha_1 + eta_1