目录
第八章：基于 LIBOR 模型用互换和利率期权进行对冲
思维导图
推导浮息债在重置日（reset date）的价格
记首个重置日 \(T_0=0\) 观察到的即期期限结构是 \(Y(t)\)，对应零息债券的价格是，
\[ P(T_0,T_i) = e^{-Y(T_i)}，i=1,\dots,n \]
根据 LIBOR 远期利率的定义，
\[ \begin{aligned} 1 + \tau L(T_0,T_i,T_{i+1}) &= \frac{P(T_0,T_{i})}{P(T_0,T_{i+1})}\\ \tau L(T_0,T_i,T_{i+1}) &= \frac{P(T_0,T_{i}) - P(T_0,T_{i+1})}{P(T_0,T_{i+1})} \end{aligned} \]
面额是 \(F\) 的浮息债在 \(T_0\) 的预期现金流如下：
\[ \begin{aligned} T_1&: CF_1 = F \times \tau \times L(T_0, T_0, T_1)\\ T_2&: CF_2 = F \times \tau \times L(T_0, T_1, T_2)\\ \vdots \\ T_n&: CF_n = F \times \tau \times L(T_0, T_{n-1}, T_n) + F\\ \end{aligned} \]
这些现金流的贴现值是：
\[ \begin{aligned} P &= \sum_{i=1}^n CF_i \times P(T_0,T_i)\\ &=\sum_{i=1}^n F \times \tau \times L(T_0, T_{i-1}, T_i) \times P(T_0,T_i) + F\times P(T_0,T_n)\\ &=\sum_{i=1}^n F \times \frac{P(T_0,T_{i-1}) - P(T_0,T_{i})}{P(T_0,T_{i})} \times P(T_0,T_i) + F\times P(T_0,T_n)\\ &=F\times P(T_0,T_0)\\ &=F \end{aligned} \]