如何推導出變異數的風險中性期望的近似值？
在論文Bollerslev、Tauchen 和 Zhou (2009 RFS)上，作者們談到了等式 (15)：
相應的模型隱含風險中性條件期望 $$E^Q_t(\sigma^2_{r,t+1})=E_t(\sigma^2_{r,t+1}M_{t+1})E_t( M_{t+1})^{-1}$$ 不容易以封閉形式計算。
然而，可以計算以下接近對數線性近似： $$E^Q_t(\sigma^2_{r,t+1}) \approx \log[e^{-r_{f,t}} E_t[ e^{m_{t+1}+\sigma^2_{r,t+1}}]] -\frac{1}{2}Var_t(\sigma_{r,t+1}^2) = E_t( \sigma^2_{r,t+1})+(\theta - 1)\kappa_1 [A_\sigma + A_q \kappa_1^2(A_\sigma^2 + A_q^2 \varphi_q^2)\varphi_q^2 ]q_t$$
我完全理解如何從第一個平等到第二個平等。但是最後一個相等，我不知道它來自哪裡。
首先，我想這些條款： $\log[e^{-r_{f,t}} E_t[e^{m_{t+1}}]]$ 取消。但是他如何擺脫$E_t[e^{\sigma^2_{r,t+1}}]$？
我們首先列出假設。\begin{align*} g_{t+1} &= \mu_g + \sigma_{g, t} z_{g, t+1}, \tag{1}\ \sigma_{g, t+1}^ 2 &= a_{\sigma} + \rho_{\sigma} \sigma_{g, t}^2 + \sqrt{q_t} z_{\sigma, t+1}, \tag{2} \ q_{t +1} &= a_{q} + \rho_q q_t + \varphi_q \sqrt{q_t} z_{q, t+1}。\tag{3} \end{align*} 此外， \begin{align*} r_{t+1} &= -\ln \delta +\psi^{-1} \mu_g - \frac{(1-\ gamma)^2}{2\theta} \sigma_{g, t}^2 + (\kappa_1 \rho_q-1)A_q q_t \ & \quad +\sigma_{g, t}z_{g, t+1 } +\kappa_1\sqrt{q_t} (A_{\sigma}z_{\sigma, t+1} + A_q \varphi_q z_{q, t+1})。\tag{10} %\sigma_{r, t}^2 &= \sigma_{g, t}^2 + \kappa_1^2(A_{\sigma}^2 + A_q^2 \varphi_q^2)q_t, \tag{12} \end{align*} 從 (2) 和 (3), \begin{align*} \sigma_{r, t+1}^2 &= \sigma_{g, t+1}^2 + \kappa_1^2(A_{\sigma}^2 + A_q^2 \varphi_q^2)q_{t+1}, \tag{13}\ & =a_{\sigma} + \rho_{\sigma} \sigma_{g, t}^2 + \sqrt{q_t} z_{\sigma, t+1} \ &\quad + \kappa_1^2(A_{ \sigma}^2 + A_q^2 \varphi_q^2)(a_{q} + \rho_q q_t + \varphi_q \sqrt{q_t} z_{q, t+1})。\end{align*} 從 (1) 和 (10)，\begin{align*} m_{t+1} &= \theta \ln \delta - \theta \psi^{-1}g_{t+1 }+(\theta-1)r_{t+1} \tag{4}\ &=\theta \ln \delta - \theta \psi^{-1}(\mu_g + \sigma_{g, t} z_{g, t+1})\ &\quad +(\theta-1)\bigg[-\ln \delta +\psi^{-1} \mu_g - \frac{(1-\gamma)^ 2}{2\theta} \sigma_{g, t}^2 + (\kappa_1 \rho_q-1)A_q q_t\ &\quad +\sigma_{g, t}z_{g, t+1} +\ kappa_1\sqrt{q_t} (A_{\sigma}z_{\sigma, t+1} + A_q \varphi_q z_{q, t+1})\bigg]。\end{align*} 因此，\begin{align*} %E_t(m_{t+1}) &= \theta \ln \delta - \theta \psi^{-1}\mu_g + (\theta-1 )\bigg[-\ln \delta +\psi^{-1} \mu_g - \frac{(1-\gamma)^2}{2\theta} \sigma_{g, t}^2 + (\kappa_1 \rho_q-1)A_q q_t\bigg],\ %{\rm Cov}t(m{t+1}, r_{t+1}) &= -\gamma \sigma_ {g, t}^2 + (\theta -1) \kappa_1^2 q_t\big(A_{\sigma}^2 + A_q^2 \varphi_q^2\big),\tag{11}\ {\ rm Cov}t(m{t+1}, \sigma_{r, t+1}^2) &=(\theta -1)\kappa_1 \Big[A_{\sigma}+A_q\kappa_1^2\big (A_{\sigma}^2 + A_q^2 \varphi_q^2\big) \varphi_q^2 \Big]q_t 。\end{align*} 此外，根據 $m_{t+1}$ 和 $\sigma_{r, t+1}^2$ 的條件正態性，\begin{align*} E_t^Q\left(\sigma_ {r, t+1}^2\right) &=E_t\left(\sigma_{r, t+1}^2M_{t+1}\right)/E_t(M_{t+1})\ & \近似 \ln\left(e^{-r_{f, t}} E_t\left(e^{m_{t+1}+\sigma_{r, t+1}^2} \right) \right) - \frac{1}{2} {\rm Var}t\left(\sigma{r, t+1}^2\right) \tag{}\ &=\ln\left(e^{- r_{f, t}} e^{E_t(m_{t+1}) + \frac{1}{2}{\rm Var}t(m{t+1})+E_t(\sigma_{r, t+1}^2)+ \frac{1}{2} {\rm Var}t\left(\sigma{r, t+1}^2\right) + {\rm Cov}t(m{t+1}, \sigma_{r, t+1}^2)} \right) \ &\quad- \frac{1 }{2} {\rm Var}t\left(\sigma{r, t+1}^2\right)\ &=\ln\left(e^{-r_{f, t}} E_t\left (e^{m_{t+1}}\right) e^{E_t(\sigma_{r, t+1}^2)+ \frac{1}{2} {\rm Var}t\left(\ sigma{r, t+1}^2\right) + {\rm Cov}t(m{t+1}, \sigma_{r, t+1}^2)} \right) - \frac{1} {2} {\rm Var}t\left(\sigma{r, t+1}^2\right)\ &=E_t\left(\sigma_{r, t+1}^2\right) + { \rm Cov}t(m{t+1}, \sigma_{r, t+1}^2) \ &=E_t\left(\sigma_{r, t+1}^2\right) + (\ theta-1)\kappa_1\Big[A_{\sigma} + A_q \kappa_1^2 \big(A_{\sigma}^2 + A_q^2 \varphi_q^2\big)\varphi_q^2 \Big]q_t。\結束{對齊} t+1}^2)+ \frac{1}{2} {\rm Var}t\left(\sigma{r, t+1}^2\right) + {\rm Cov}t(m{t +1}, \sigma_{r, t+1}^2)} \right) - \frac{1}{2} {\rm Var}t\left(\sigma{r, t+1}^2\右)\ &=E_t\left(\sigma_{r, t+1}^2\right) + {\rm Cov}t(m{t+1}, \sigma_{r, t+1}^2 ) \ &=E_t\left(\sigma_{r, t+1}^2\right) + (\theta-1)\kappa_1\Big[A_{\sigma} + A_q \kappa_1^2 \big(A_ {\sigma}^2 + A_q^2 \varphi_q^2\big)\varphi_q^2 \Big]q_t。\結束{對齊*} t+1}^2)+ \frac{1}{2} {\rm Var}t\left(\sigma{r, t+1}^2\right) + {\rm Cov}t(m{t +1}, \sigma_{r, t+1}^2)} \right) - \frac{1}{2} {\rm Var}t\left(\sigma{r, t+1}^2\右)\ &=E_t\left(\sigma_{r, t+1}^2\right) + {\rm Cov}t(m{t+1}, \sigma_{r, t+1}^2 ) \ &=E_t\left(\sigma_{r, t+1}^2\right) + (\theta-1)\kappa_1\Big[A_{\sigma} + A_q \kappa_1^2 \big(A_ {\sigma}^2 + A_q^2 \varphi_q^2\big)\varphi_q^2 \Big]q_t。\結束{對齊*}
對數線性近似 (*) 的解釋。
關於對數線性近似 ()，由於論文沒有提供解釋，我們在下面提供了一種可能的解釋。具體來說，請注意 \begin{align} e^{\sigma_{r, t+1}^2} &\approx 1+ \sigma_{r, t+1}^2 + \frac{1}{2} \left(\sigma_{r, t+1}^2\right)^2\ &\約 1+ \sigma_{r, t+1}^2 + \frac{1}{2} \Big[\大(\sigma_{r, t+1}^2\big)^2 - \left(E_t\big(\sigma_{r, t+1}^2\big)\right)^2\Big]\ &\約 1+ \sigma_{r, t+1}^2 + \frac{1}{2}{\rm Var}t \big(\sigma{r, t+1}^2\big)。\end{align*} 然後，\begin{align*} \ln\left(e^{-r_{f, t}} E_t\left(e^{m_{t+1}+\sigma_{r, t +1}^2} \right) \right) &\近似 \ln\left(e^{-r_{f, t}} E_t\left(e^{m_{t+1}}\left(1+ \sigma_{r, t+1}^2 + \frac{1}{2}{\rm Var}t \big(\sigma{r, t+1}^2\big) \right) \right) \右)\ &\約 \ln \left(1 +e^{-r_{f, t}} E_t\left(\sigma_{r, t+1}^2M_{t+1}\right) + \frac{1}{2}{\rm Var}t \big(\sigma{r, t+1}^2\big) \right)\ \ &\近似 e^{-r_{f, t}} E_t\left(\sigma_{r, t+1}^2M_{t+1}\right) + \frac{1}{2}{\rm Var}t \big(\sigma{r, t+1}^2\big)\ &= E_t\left(\sigma_{r, t+1}^2M_{t+1}\right)/E_t( M_{t+1}) + \frac{1}{2}{\rm Var}t \big(\sigma{r, t+1}^2\big)。\end{align*} 即 \begin{align*} E_t\left(\sigma_{r, t+1}^2M_{t+1}\right)/E_t(M_{t+1}) &\約 \ln\left(e^{-r_{f, t}} E_t\left(e^{m_{t+1}+\sigma_{r, t+1}^2} \right) \right) - \frac{1}{2}{\rm Var}t \big(\sigma{r, t+1}^2\big)。\結束{對齊*} t+1}^2\大）。\end{align*} 即 \begin{align*} E_t\left(\sigma_{r, t+1}^2M_{t+1}\right)/E_t(M_{t+1}) &\約 \ln\left(e^{-r_{f, t}} E_t\left(e^{m_{t+1}+\sigma_{r, t+1}^2} \right) \right) - \frac{1}{2}{\rm Var}t \big(\sigma{r, t+1}^2\big)。\結束{對齊*} t+1}^2\大）。\end{align*} 即 \begin{align*} E_t\left(\sigma_{r, t+1}^2M_{t+1}\right)/E_t(M_{t+1}) &\約 \ln\left(e^{-r_{f, t}} E_t\left(e^{m_{t+1}+\sigma_{r, t+1}^2} \right) \right) - \frac{1}{2}{\rm Var}t \big(\sigma{r, t+1}^2\big)。\結束{對齊*}