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,
\[
P(T_0,T_i) = e^{-Y(T_i)T_i},i=1,\dots,n
\]
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:
\[
\begin{aligned}
P &= \sum_{i=1}^n CF_i \times P(T_0,T_i)\\
&=\sum_{i=1}^n F \times \tau \times L(T_0, T_{i-1}, T_i) \times P(T_0,T_i) + F\times P(T_0,T_n)\\
&=\sum_{i=1}^n F \times \frac{P(T_0,T_{i-1}) - P(T_0,T_{i})}{P(T_0,T_{i})} \times P(T_0,T_i) + F\times P(T_0,T_n)\\
&=F\times P(T_0,T_0)\\
&=F
\end{aligned}
\]