第十章：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
[egin{aligned}
PCD(i) &= -frac{1}{P} frac{partial P}{partial c^*_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
[egin{aligned}
PCC(i,j) &= -frac{1}{P} frac{partial^2 P}{partial c^*_i partial c^*_j}\
&=-sqrt{lambda_i}sqrt{lambda_j}frac{1}{P} frac{partial^2 P}{partial c_i partial c_j}\
end{aligned}
]
其中
[egin{aligned}
frac{partial^2 P}{partial c_i partial c_j}&=
frac{partialleft(frac{partial P}{partial c_i}
ight)}{partial c_j}\
&=frac{partialleft(sum_{l=1}^k frac{partial P}{partial y(t_l)} mu_{il}
ight)}{partial c_j}\
&=sum_{l=1}^k frac{partial^2 P}{partial y(t_l) partial c_j} mu_{il}\
end{aligned}
]
又有
[egin{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
[egin{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
[egin{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}
]