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，
[egin{aligned}
1 + au L(T_0,T_i,T_{i+1}) &= frac{P(T_0,T_{i})}{P(T_0,T_{i+1})}\
au 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：
[egin{aligned}
T_1&: CF_1 = F imes au imes L(T_0, T_0, T_1)\
T_2&: CF_2 = F imes au imes L(T_0, T_1, T_2)\
vdots \
T_n&: CF_n = F imes au imes L(T_0, T_{n-1}, T_n) + F\
end{aligned}
]
这些现金流的贴现值是：
[egin{aligned}
P &= sum_{i=1}^n CF_i imes P(T_0,T_i)\
&=sum_{i=1}^n F imes au imes L(T_0, T_{i-1}, T_i) imes P(T_0,T_i) + F imes P(T_0,T_n)\
&=sum_{i=1}^n F imes frac{P(T_0,T_{i-1}) - P(T_0,T_{i})}{P(T_0,T_{i})} imes P(T_0,T_i) + F imes P(T_0,T_n)\
&=F imes P(T_0,T_0)\
&=F
end{aligned}
]