《Interest Rate Risk Modeling》阅读笔记——第六章:用利率期货对冲

第六章:用利率期货对冲

思维导图

转换因子释疑

转换因子是特定假设下可交割券全价与标准券全价的比值,用该比值近似真实情况下可交割券全价与标准券全价的比例关系。

标准券有 6% 的票息率,每半年付息一次,发行日为期货合约到期月份的首个交割日。若发行日期限结构保持水平,利率为 6%,则标准券全价是 $100,可交割券(到期月份要向下取整到 0 月)全价是,

\[\frac{c \times 100}{2} \left(\frac{1}{0.03} - \frac{1}{0.03(1+0.03)^{2n}} \right) + \frac{100}{0.03(1+0.03)^{2n}} \]

转换因子等于,

\[CF_0 = \frac{c}{2} \left(\frac{1}{0.03} - \frac{1}{0.03(1+0.03)^{2n}} \right) + \frac{1}{0.03(1+0.03)^{2n}} \]

其他情况类似。

因此,国债期货的机制可以理解为:以标准券为基准,为所有可交割券估价。若期货合约价格 \(FP\),CTD 的转换因子是 \(CF\),那么交割时多头相当于用净价 \(FP \times CF\) 购买了 CTD。

利率期货久期向量的推导

Eurodollar 期货的久期向量

未来 \(s\) 年的 \(t\) 年期远期利率 \(f(s, s+t)\) 和瞬时远期利率 \(f(s)\) 之间存在关系:

\[f(s,s+t)t = \int_{s}^{s+t}f(x)dx\\ \Delta f(s,s+t)t = \int_{s}^{s+t}\Delta f(x)dx \]

(连续复利)期货利率 \(f^*\) 和远期利率 \(f\) 之间存在“凸性修正”关系:

\[f(s,s+t) = f^*(s,s+t) - \frac{1}{2} \sigma^2 s(s+t) \]

所以

\[\Delta f(s,s+t) = \Delta f^*(s,s+t) \]

已知:

\[CP = 1000000[1-(100-Q)/400]\\ q=100-Q\\ \]

那么

\[\Delta CP = -2500 \times \Delta q \]

如果(连续复利)期货利率由 \(f^*(s,s+90/365)\) 变为 \(f^{* \prime}(s,s+90/365)\)(记 \(\Delta f^{*} = f^{* \prime} - f^{*}\)),那么

\[\begin{aligned} \Delta q &=q^{\prime} - q\\ &= (e^{f^{* \prime}(s,s+90/365) \times(90/365)} - e^{f^{*}(s,s+90/365)\times(90/365)})\times 400\\ &=e^{f^{*}(s,s+90/365)\times(90/365)}(e^{\Delta f^{*}(s,s+90/365)\times(90/365)} - 1)\times 400\\ \end{aligned} \]

根据 \(e^x - 1 \approx x\)

\[\begin{aligned} \Delta CP &= -2500 \times \Delta q\\ &=-1000000\times e^{f^{*}(s,s+90/365)\times(90/365)}(e^{\Delta f^{*}(s,s+90/365)\times(90/365)} - 1)\\ &\approx -1000000\times e^{f^{*}(s,s+90/365)\times(90/365)} \Delta f^{*}(s,s+90/365)\times(90/365)\\ &=-1000000\times ((100-Q)/400+1)\Delta f^{*}(s,s+90/365)\times(90/365)\\ \end{aligned} \]

如果:

\[\Delta y(t) = \Delta A_0 + \Delta A_1 t + \Delta A_2 t^2 + \Delta A_3 t^3 + \cdots\\ \Delta f(t) = \Delta A_0 + 2\Delta A_1 t + 3\Delta A_2 t^2 + 4\Delta A_3 t^3 + \cdots \]

那么

\[\begin{aligned} &\Delta f^{*}(s,s+90/365)\times(90/365) \\ &= \Delta f(s,s+90/365)\times(90/365)\\ &=\int_{s}^{s+90/365} \Delta f(t)dt\\ &=\int_{s}^{s+90/365} \Delta A_0 + 2\Delta A_1 t + 3\Delta A_2 t^2 + 4\Delta A_3 t^3 + \cdots dx\\ &=\Delta A_0(90/365) + \Delta A_1\left[(s+90/365)^2-s^2 \right] + \Delta A_2\left[(s+90/365)^3-s^3 \right] + \Delta A_3\left[(s+90/365)^4-s^4 \right] + \cdots \end{aligned} \]

最终

\[\frac{\Delta CP}{CP} = -D^f(1)\times \Delta A_0 -D^f(2)\times \Delta A_1 -D^f(3)\times \Delta A_2 + \cdots\\ \begin{aligned} D^f(1) &= K(Q)\times(90/365)\\ D^f(2) &= K(Q)\times[(s+90/365)^2-s^2]\\ D^f(3) &= K(Q)\times[(s+90/365)^3-s^3]\\ \end{aligned} \\ K(Q)=\left(1+\frac{100-Q}{400} \right) / \left(1-\frac{100-Q}{400} \right)=\frac{500-Q}{300+Q} \]

国债期货的久期向量

记:

  • \(T\) = CTD 的剩余期限
  • \(C\) = CTD 的票息现金流(非年化)
  • \(F\) = CTD 的面额
  • \(CF\) = CTD 的转换因子
  • \(CP\) = CTD 的全价
  • \(\tau\) = 期货到期日与期货到期后债券首个付息日之间的距离
  • \(s\) = 期货到期日
  • \(n\) = 截止到期货到期日发生的付息次数
  • \(y(t)\):瞬时即期期限结构
  • 默认付息两次(美式规则)

那么,国债期货的价格是:

\[\begin{aligned} FP &=\frac{1}{CF} (CP - AI)\\ &= \frac{1}{CF} \left( \sum_{t=0}^{2(T-s-\tau)}\frac{C}{e^{(s +\tau + t\times 0.5)\times y(s +\tau + t\times 0.5)}} + \frac{F}{e^{T\times y(T)}} \right)e^{s \times y(s)} - \frac{C}{CF}\times \frac{0.5-\tau}{0.5} \end{aligned} \]

如果 \(y\) 变化到 \(y^{\prime}\)(记 \(\Delta y = y^{\prime}-y\)),那么

\[\begin{aligned} \Delta FP &= FP^{\prime} - FP\\ &= \frac{1}{CF} \left( \sum_{t=0}^{2(T-s-\tau)}\frac{C}{e^{(s +\tau + t\times 0.5)\times y^{\prime}(s +\tau + t\times 0.5)}} + \frac{F}{e^{T\times y^{\prime}(T)}} \right)e^{s \times y^{\prime}(s)} \\ &\ \ \ \ -\frac{1}{CF} \left( \sum_{t=0}^{2(T-s-\tau)}\frac{C}{e^{(s +\tau + t\times 0.5)\times y(s +\tau + t\times 0.5)}} + \frac{F}{e^{T\times y(T)}} \right)e^{s \times y(s)}\\ &=\frac{1}{CF}\left( \sum_{t=0}^{2(T-s-\tau)}\frac{C}{e^{(s +\tau + t\times 0.5)\times y(s +\tau + t\times 0.5)}}\left(e^{s\times \Delta y(s) - (s +\tau + t\times 0.5)\times\Delta y(s +\tau + t\times 0.5)} -1\right) + \frac{F}{e^{T\times y(T)}}\left(e^{s\times \Delta y(s) - T\times\Delta y(T)} -1\right) \right)e^{s \times y(s)} \end{aligned} \]

根据 \(e^x - 1 \approx x\)

\[\begin{aligned} \Delta FP &\approx \\ &\frac{1}{CF} \sum_{t=0}^{2(T-s-\tau)}\frac{C e^{s \times y(s)}}{e^{(s +\tau + t\times 0.5)\times y(s +\tau + t\times 0.5)}} [{s\times \Delta y(s) - (s +\tau + t\times 0.5)\times\Delta y(s +\tau + t\times 0.5)} ] \\ &+ \frac{F e^{s \times y(s)}}{{T\times y(T)}}\left[{s\times \Delta y(s) - T\times\Delta y(T)} \right] \end{aligned} \]

如果:

\[\Delta y(t) = \Delta A_0 + \Delta A_1 t + \Delta A_2 t^2 + \Delta A_3 t^3 + \cdots \]

那么

\[\begin{aligned} &\Delta FP \approx \\ &\frac{1}{CF} \sum_{t=0}^{2(T-s-\tau)}\frac{C e^{s \times y(s)}}{e^{(s +\tau + t\times 0.5)\times y(s +\tau + t\times 0.5)}} \times\\ &\left\{{s\times (\Delta A_0 + \Delta A_1 s + \Delta A_2 s^2 + \Delta A_3 s^3 +\cdots) - (s +\tau + t\times 0.5)\times \left[\Delta A_0 + \Delta A_1 (s +\tau + t\times 0.5) + \Delta A_2 (s +\tau + t\times 0.5)^2 + \Delta A_3 (s +\tau + t\times 0.5)^3 +\cdots \right]} \right\} \\ &+\frac{F e^{s \times y(s)}}{{T\times y(T)}}\left[s\times (\Delta A_0 + \Delta A_1 s + \Delta A_2 s^2 + \Delta A_3 s^3+\cdots) - T\times (\Delta A_0 + \Delta A_1 T + \Delta A_2 T^2 + \Delta A_3 T^3+\cdots) \right] \end{aligned} \]

最终

\[\begin{aligned} \frac{\Delta FP}{FP} &\approx -D(1)\times \Delta A_0 -D(2)\times \Delta A_1 -D(3)\times \Delta A_2 - \cdots - D(M)\times \Delta A_{M-1} -\cdots\\ D(m)&= \frac{e^{s \times y(s)}}{CF \times FP} \left( \sum_{t=0}^{2(T-s-\tau)}\frac{C\left((s+\tau+t\times 0.5)^m - s^m \right)}{e^{(s +\tau + t\times 0.5)\times y(s +\tau + t\times 0.5)}} + \frac{F(T^m - s^m)}{e^{T\times y(T)}} \right)\\ m&=1,2,3,\dots,M \end{aligned} \]

CME 上的 Eurodollar 教程

Introduction to Eurodollars

posted @ 2020-01-01 22:06  xuruilong100  阅读(...)  评论(...编辑  收藏