《Interest Rate Risk Modeling》阅读笔记——第九章:关键利率久期和 VaR 分析

第九章:关键利率久期和 VaR 分析

思维导图

一些想法

  • 在解关键方程的时候施加 \(L^1\) 约束也许可以得到“稀疏解”,进而减少交易成本。
  • 借鉴样条插值拟合期限结构时选择 knot 的方法选择关键期限。

有关现金流映射技术的推导

已知,

\[\Delta y(t) = \begin{cases} \Delta y(t_{first}) & t \le t_{first}\\ \Delta y(t_{last}) & t \ge t_{last}\\ \alpha \Delta y(t_{left}) + (1-\alpha) \Delta y(t_{right})& \text{ else} \end{cases} \]

\[\alpha = \frac{t_{right}-t}{t_{right} - t_{left}} \]

\[t_{left} < t < t_{right} \]

求解 \(CF_{left}\)\(CF_{right}\)\(CF_0\) 使得:

\[\begin{aligned} P &= \frac{CF_t}{e^{y(t)t}} \\ &= \frac{CF_{left}}{e^{y(t_{left})t_{left}}} + \frac{CF_{right}}{e^{y(t_{right})t_{right}}} + CF_0 \end{aligned} \tag{1} \]

要求关键利率久期不变,那么:

\[\begin{aligned} \frac{1}{P} \frac{\partial P}{\partial y(t_{left})} &=\frac{1}{P} \frac{\partial P}{\partial y(t)} \frac{\partial y(t)}{\partial y(t_{left})}\\ &\approx\frac{1}{P} \frac{\partial P}{\partial y(t)} \frac{\Delta y(t)}{\Delta y(t_{left})}\\ &\approx-\frac{1}{P} \frac{CF_t\times t}{e^{y(t)t}} \alpha\\ &=-t\alpha \\ \frac{1}{P} \frac{\partial P}{\partial y(t_{left})} &=\frac{1}{P} \frac{\partial \left(\frac{CF_{left}}{e^{y(t_{left})t_{left}}} + \frac{CF_{right}}{e^{y(t_{right})t_{right}}} + CF_0 \right) }{\partial y(t_{left})}\\ &=-\frac{1}{P} \frac{CF_{left}\times t_{left}}{e^{y(t_{left})t_{left}}} \end{aligned} \]

解出

\[CF_{left} = \frac{t \alpha P e^{y(t_{left})t_{left}}}{t_{left}} \tag{2} \]

同理解出

\[CF_{right} = \frac{t (1-\alpha) P e^{y(t_{right})t_{right}}}{t_{right}} \tag{3} \]

(2)和(3)代入(1)解出

\[CF_0 = P \times \frac{(t-t_{left})(t-t_{right})}{t_{left} \times t_{right}} \]

posted @ 2020-01-30 22:51  xuruilong100  阅读(...)  评论(...编辑  收藏