《Interest Rate Risk Modeling》第十章：主成分模型与 VaR 分析

主成分模型与 VaR 分析

思维导图

一些想法

• NS 家族模型的参数有经济意义，同时参数变化的行为类似主成分，考虑基于 NS 模型参数的风险度量。
• 尝试用（多元）GARCH 滤波利率变化，对残差应用 PCA。

推导 PCD、PCC 和 KRD、KRC 的关系

利用主成分系数矩阵的正交性。

PCD 和 KRD

\[\begin{aligned} PCD(i) &= -\frac{1}{P} \frac{\partial P}{\partial c^{\ast}_i}\\&= -\sqrt{\lambda_i} \frac{1}{P} \frac{\partial P}{\partial c_i}\\ &=-\sqrt{\lambda_i} \frac{1}{P} \frac{\partial P}{\partial c_i} \sum_{j=1}^k \mu_{ij}^2\\ &=-\sqrt{\lambda_i} \frac{1}{P} \sum_{j=1}^k \frac{\partial P}{\partial c_i} \mu_{ij}^2\\ &=-\sqrt{\lambda_i} \frac{1}{P} \sum_{j=1}^k \frac{\partial P}{\partial c_i} \frac{\partial c_i}{\partial y(t_j)} \mu_{ij}\\ &=- \sqrt{\lambda_i} \frac{1}{P} \sum_{j=1}^k \frac{\partial P}{\partial y(t_j)} \mu_{ij}\\ &=\sqrt{\lambda_i}\sum_{j=1}^k KRD(j) \mu_{ij}\\ &=\sum_{j=1}^k KRD(j) l_{ji} \end{aligned}\]

PCC 和 KRC

\[\begin{aligned} PCC(i,j) &= -\frac{1}{P} \frac{\partial^2 P}{\partial c^{\ast}_i \partial c^{\ast}_j}\\ &=-\sqrt{\lambda_i}\sqrt{\lambda_j}\frac{1}{P} \frac{\partial^2 P}{\partial c_i \partial c_j}\\ \end{aligned}\]

其中

\[\begin{aligned} \frac{\partial^2 P}{\partial c_i \partial c_j}&= \frac{\partial\left(\frac{\partial P}{\partial c_i}\right)}{\partial c_j}\\ &=\frac{\partial\left(\sum_{l=1}^k \frac{\partial P}{\partial y(t_l)} \mu_{il}\right)}{\partial c_j}\\ &=\sum_{l=1}^k \frac{\partial^2 P}{\partial y(t_l) \partial c_j} \mu_{il}\\ \end{aligned}\]

又有

\[\begin{aligned} \frac{\partial^2 P}{\partial y(t_l) \partial c_j}&= \frac{\partial^2 P}{\partial y(t_l) \partial c_j} \sum_{n=1}^k \mu_{jn}^2\\ &=\sum_{n=1}^k \frac{\partial^2 P}{\partial y(t_l) \partial c_j} \mu_{jn}^2\\ &=\sum_{n=1}^k \frac{\partial^2 P}{\partial y(t_l) \partial c_j} \frac{\partial c_j}{\partial y(t_n)} \mu_{jn}\\ &=\sum_{n=1}^k \frac{\partial^2 P}{\partial y(t_l) \partial y(t_n)} \mu_{jn}\\ \end{aligned}\]

所以

\[\begin{aligned} \frac{\partial^2 P}{\partial c_i \partial c_j}&= \sum_{l=1}^k \sum_{n=1}^k \frac{\partial^2 P}{\partial y(t_l) \partial y(t_n)} \mu_{jn} \mu_{il} \end{aligned}\]

最终

\[\begin{aligned} PCC(i,j) &= -\sqrt{\lambda_i}\sqrt{\lambda_j}\frac{1}{P} \sum_{l=1}^k \sum_{n=1}^k \frac{\partial^2 P}{\partial y(t_l) \partial y(t_n)} \mu_{jn} \mu_{il}\\ &=\sum_{l=1}^k \sum_{n=1}^k KRC(l,n) l_{nj}l_{li} \end{aligned}\]