﻿ 担保信用等级变换的利率互换衍生品定价
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 同济大学学报(自然科学版)  2018, Vol. 46 Issue (11): 1609-1614.  DOI: 10.11908/j.issn.0253-374x.2018.11.021 0

### 引用本文

LIANG Jin, ZOU Hongchun. Pricing of Interest Rate Swap Derivatives for Assuring Credit Rating Migration[J]. Journal of Tongji University (Natural Science), 2018, 46(11): 1609-1614. DOI: 10.11908/j.issn.0253-374x.2018.11.021

### 文章历史

Pricing of Interest Rate Swap Derivatives for Assuring Credit Rating Migration
LIANG Jin , ZOU Hongchun
School of Mathematical Sciences, Tongji University, Shanghai 200092, China
Abstract: Considering the valuation of a protected swap on credit rating migration, under the framework of structural methods, a pricing model was established for protecting the loss caused by the first credit rating migration, where the credit ratings depend on the interest rate and have two grades. In the pricing model, an independent variable of the model was defined by the value of low-grade zero-coupon, which is a new perspective. A partial differential equation(PDE) pricing model was derived by the hedging method. A semi-closed solution was obtained by dimensionality reduction technique. The numerical solution was calculated by the explicit finite difference method. Finally, the dependency of parameters of the model was discussed and the results show that there is a monotonically decreasing relationship between premium value and each parameter.
Key words: interest rate swap    credit rating migration    dimensionality reduction    finite difference method

1 模型的建立

1.1 基本假设

(1) 假设市场是完备的，不存在套利机会.

(2) 利率产品由一张面值为1、到期日为T的零息票债券构成.

(3) 担保的是固定利率支付方的信用等级，保费由利率信用等级保护买方(浮动利率支付方)在期初一次性支付给卖方(相应的担保浮动利率支付方也可以做类似考虑).

(4) 存在一个信用等级边界ra.当随机利率rra，利率处于高等级；当rra，利率处于低等级.初始时刻随机利率r0ra，即初始时刻利率处于高等级.

(5) 市场利率模型为Vasicek模型, 如下所示:

 ${\rm{d}}{r_t} = a\left( {\theta - {r_t}} \right){\rm{d}}t + \left( {{\sigma _1}{1_{\left\{ {{r_t} \ge {r_{\rm{a}}}} \right\}}} + {\sigma _2}{1_{\left\{ {{r_t} < {r_{\rm{a}}}} \right\}}}} \right){\rm{d}}{W_t}$

(6) 合约担保的是固定利率支付方信用等级的下降，并且只考虑担保信用等级的首次迁移.随机利率首次发生信用等级迁移的时刻定义为

 $\tau = \inf \left\{ {t\left| {{r_0} > {r_{\rm{a}}},{r_t} \le {r_{\rm{a}}}} \right.} \right\}$
1.2 现金流分析 1.2.1 零息票定价公式[12]

 ${\rm{d}}{r_t} = a\left( {\theta - {r_t}} \right){\rm{d}}t + \sigma {\rm{d}}{W_t}$

 $\left\{ \begin{array}{l} \frac{{\partial P}}{{\partial t}} + {{\cal L}_0}P = 0,\;\;\;r \in {\bf{R}},t \in \left[ {0,T} \right)\\ P\left( {r,T} \right) = 1,\;\;\;r \in {\bf{R}} \end{array} \right.$ (1)

 ${{\cal L}_0} = \frac{{{\sigma ^2}}}{2}\frac{{{\partial ^2}}}{{\partial {r^2}}} + a\left( {\theta - r} \right)\frac{\partial }{{\partial r}} - r$

 $P\left( {r,t;T} \right) = A\left( t \right){{\rm{e}}^{ - rB\left( t \right)}}$ (2)

 $\begin{array}{l} A\left( t \right) = \exp \left( {\frac{1}{{{a^2}}}\left( {B{{\left( t \right)}^2} - \left( {T - t} \right)} \right)\left( {{a^2}\theta - \frac{{{\sigma ^2}}}{2}} \right) - } \right.\\ \;\;\;\;\;\;\;\;\;\;\;\left. {\frac{{{\sigma ^2}}}{{4a}}B{{\left( t \right)}^2}} \right) \end{array}$
 $B\left( t \right) = \frac{1}{a}\left( {1 - {{\rm{e}}^{ - a\left( {T - t} \right)}}} \right)$
1.2.2 现金流损失分析

 $\begin{array}{l} {\rm{d}}{P_{2t}} = \left( {\frac{{\partial {P_2}}}{{\partial t}} + \frac{{\sigma _2^2}}{2}\frac{{{\partial ^2}{P_2}}}{{\partial r_2^2}} + a\left( {\theta - {r_{2t}}} \right)\frac{{\partial {P_2}}}{{\partial {r_2}}}} \right){\rm{d}}t + \\ \;\;\;\;\;\;\;\;\;{\sigma _2}\frac{{\partial {P_2}}}{{\partial {r_2}}}{\rm{d}}{W_t} \end{array}$

 $\frac{{\partial {P_2}}}{{\partial {r_2}}} = - B\left( t \right){P_2}$

 $\frac{{{\rm{d}}{P_{2t}}}}{{{P_{2t}}}} = {r_{2t}}{\rm{d}}t - B\left( t \right){\rm{d}}{W_t}$ (3)

 $V\left( {t,{r_1},{P_2}} \right) = E\left( {{{\rm{e}}^{ - \int_t^\tau {{r_{1s}}{\rm{d}}s} }}{{\left( {{P_{1\tau }} - {P_{2\tau }}} \right)}^ + }{1_{\left\{ {\tau < T} \right\}}}\left| {{{\cal F}_t}} \right.} \right)$
1.3 偏微分方程的推导

 ${\mathit{\Pi }_t} = {V_t} - {\Delta _{1t}}{P_{1t}} - {\Delta _{2t}}{P_{2t}}$

 ${\rm{d}}{\mathit{\Pi }_t} = {\rm{d}}{V_t} - {\Delta _{1t}}{\rm{d}}{P_{1t}} - {\Delta _{2t}}{\rm{d}}{P_{2t}} = {r_{1t}}{\mathit{\Pi }_t}{\rm{d}}t$ (4)

 $\frac{{\partial V}}{{\partial t}} + {{\cal L}_1}V + a\left( {\theta - {r_1}} \right)\frac{{\partial V}}{{\partial {r_1}}} + {r_1}{P_2}\frac{{\partial V}}{{\partial {P_2}}} - {r_1}V = 0$ (5)

 ${{\cal L}_1} = \frac{{\sigma _1^2}}{2}\frac{{{\partial ^2}}}{{\partial r_1^2}} + \frac{{\sigma _2^2}}{2}B{\left( t \right)^2}P_2^2\frac{{{\partial ^2}}}{{\partial P_2^2}} - {\sigma _1}{\sigma _2}B\left( t \right){P_2}\frac{{{\partial ^2}}}{{\partial {r_1}\partial {P_2}}}$

 ${\rm{d}}{{\tilde r}_1} = {\rm{d}}{r_1} = a\left( {\theta - {r_{\rm{a}}} - {{\tilde r}_{{\rm{1}}t}}} \right){\rm{d}}t + {\sigma _1}{\rm{d}}{W_t}$ (6)

 ${\mathit{\Sigma }_1}:\left\{ {{{\tilde r}_1} > 0,{P_2} > 0,0 \le t < T} \right\}$

 $\left\{ \begin{array}{l} \frac{{\partial V}}{{\partial t}} + {{\tilde {\cal L}}_1}V + a\left( {\theta - {r_{\rm{a}}} - {{\tilde r}_1}} \right)\frac{{\partial V}}{{\partial {{\tilde r}_1}}} + \\ \;\;\;\;\left( {{{\tilde r}_1} - {r_{\rm{a}}}} \right){P_2}\frac{{\partial V}}{{\partial {P_2}}} - \left( {{{\tilde r}_1} + {r_{\rm{a}}}} \right)V = 0\\ V\left( {t,0,{P_2}} \right) = f\left( t \right),\;\;\;\;{P_2} > 0,0 \le t < T\\ V\left( {T,{{\tilde r}_1},{P_2}} \right) = 0,\;\;\;\;\;{{\tilde r}_1} > 0,{P_2} > 0 \end{array} \right.$ (7)
 ${{\tilde {\cal L}}_1} = \frac{{\sigma _1^2}}{2}\frac{{{\partial ^2}}}{{\partial \tilde r_1^2}} + \frac{{\sigma _2^2}}{2}B{\left( t \right)^2}P_2^2\frac{{{\partial ^2}}}{{\partial P_2^2}} - {\sigma _1}{\sigma _2}B\left( t \right){P_2}\frac{{{\partial ^2}}}{{\partial {{\tilde r}_1}\partial {P_2}}}$
 $f\left( t \right) = {\left( {{P_{1t,{r_{\rm{a}}}}} - {P_{2t,{r_{\rm{a}}}}}} \right)^ + }$

 ${P_{1t,{r_{\rm{a}}}}} = {A_1}\left( t \right){{\rm{e}}^{ - {r_{\rm{a}}}B\left( t \right)}},\;\;\;\;{P_{2t,{r_{\rm{a}}}}} = {A_2}\left( t \right){{\rm{e}}^{ - {r_{\rm{a}}}B\left( t \right)}}$

2 模型求解与数值分析 2.1 偏微分方程求解

 $\left\{ \begin{array}{l} \frac{{\partial \tilde V}}{{\partial t}} + \frac{1}{2}{\left( {{\sigma _2} - {\sigma _1}} \right)^2}B{\left( t \right)^2}{y^2}\frac{{{\partial ^2}\tilde V}}{{\partial {y^2}}} + \\ \;\;\;\;{r_{\rm{a}}}y\frac{{\partial \tilde V}}{{\partial y}} - {r_{\rm{a}}}\tilde V = 0\\ \tilde V\left( {t,{y_0}} \right) = g\left( t \right),\;\;\;\;\;0 \le t < T\\ \tilde V\left( {T,y} \right) = 0,\;\;\;\;\;y > {y_0} \end{array} \right.$ (8)

 ${y_0} = \frac{{{P_{2t,{r_{\rm{a}}}}}}}{{{{\tilde P}_{1t,0}}}} = \frac{{{P_{2t,{r_{\rm{a}}}}}}}{{{P_{1t,{r_{\rm{a}}}}}}}$
 $g\left( t \right) = \frac{{f\left( t \right)}}{{{{\tilde P}_{1t,0}}}} = \frac{{f\left( t \right)}}{{{P_{1t,0}}}} = {\left( {1 - \frac{{{P_{2t,{r_{\rm{a}}}}}}}{{{P_{1t,{r_{\rm{a}}}}}}}} \right)^ + } = {\left( {1 - {y_0}} \right)^ + }$

 $s = T - \int_0^t {\left( {{\sigma _2} - {\sigma _1}} \right)B\left( \omega \right){\rm{d}}\omega } ,\;\;\;x = \ln \left( {y{{\rm{e}}^{{r_{\rm{a}}}\left( {T - t} \right)}}} \right),$
 $u = \tilde V{{\rm{e}}^{\frac{1}{8}s - \frac{1}{2}x}}{{\rm{e}}^{{r_{\rm{a}}}\left( {T - t} \right)}}$

 $\left\{ \begin{array}{l} \frac{{\partial u}}{{\partial s}} - \frac{1}{2}\frac{{{\partial ^2}u}}{{\partial {x^2}}} = 0\\ u\left( {s,{x_0}} \right) = h\left( s \right),\;\;\;\;0 \le s < T\\ u\left( {0,x} \right) = 0,\;\;\;\;x \in {\bf{R}} \end{array} \right.$ (9)

 ${x_0} = \ln \left( {{y_0}{{\rm{e}}^{{r_{\rm{a}}}\left( {T - t} \right)}}} \right),\;\;\;\;h\left( s \right) = g\left( {T - s} \right){{\rm{e}}^{\frac{1}{8}s + \frac{1}{2}x}}$

 $z = x - {x_0},\;\;\;\;\;\varphi \left( {s,z} \right) = u\left( {s,z} \right) - h\left( s \right)$

 $\begin{array}{*{20}{c}} {\varphi \left( {s,z} \right) = - \int_{ - \infty }^{ + \infty } {\mathit{\Gamma }\left( {z - \xi ,s} \right)h\left( 0 \right){\rm{d}}\xi } - }\\ {\int_0^s {h'\left( \omega \right){\rm{d}}\omega } \int_{ - \infty }^{ + \infty } {\mathit{\Gamma }\left( {z - \xi ,s - \omega } \right){\rm{d}}\xi } } \end{array}$

 $\mathit{\Gamma }\left( {z - \xi ,s - \omega } \right) = \left\{ \begin{array}{l} \frac{1}{{\sqrt {2{\rm{ \mathsf{ π} }}\left( {s - \omega } \right)} }}{{\rm{e}}^{ - \frac{{{{\left( {z - \xi } \right)}^2}}}{{2\left( {s - \omega } \right)}}}},\;\;\;\;\;s > \omega \\ 0,\;\;\;\;s \le \omega \end{array} \right.$

 $V = {{\tilde P}_1}{{\rm{e}}^{ - {r_{\rm{a}}}\left( {T - t} \right)}}{{\rm{e}}^{\frac{1}{8}s - \frac{1}{2}x}}\left( {\varphi \left( {s,x} \right) + h\left( s \right)} \right)$ (10)
2.2 数值结果

 $\left( {\eta ,{{\tilde r}_1},{P_2}} \right) = \left[ {0,T} \right] \times \left[ {0,{R_1}} \right] \times \left[ {0,{R_2}} \right]$

 $\begin{array}{l} {V_{n + 1,i,j}} = {\pi _1}{V_{n,i,j}} + {\pi _2}{V_{n,i + 1,j}} + {\pi _3}{V_{n,i - 1,j}} + \\ \;\;\;\;\;\;\;\;\;\;\;\;{\pi _4}{V_{n,i,j + 1}} + {\pi _5}{V_{n,i,j - 1}} + {\pi _6}\left( {{V_{n,i + 1,j + 1}} - } \right.\\ \;\;\;\;\;\;\;\;\;\;\;\;\left. {{V_{n,i - 1,j + 1}} - {V_{n,i + 1,j - 1}} + {V_{n,i - 1,j - 1}}} \right) \end{array}$

 $\left\{ \begin{array}{l} {\pi _1} = 1 - \left( {{r_{\rm{a}}} + ih} \right){\tau _1} - \frac{{\sigma _1^2{\tau _1}}}{{{h^2}}}\\ {\pi _2} = \frac{{\sigma _1^2{\tau _1}}}{{2{h^2}}} + a\left( {\theta - {r_{\rm{a}}} - ih} \right)\frac{{{\tau _1}}}{{2h}}\\ {\pi _3} = \frac{{\sigma _1^2{\tau _1}}}{{2{h^2}}} - a\left( {\theta - {r_{\rm{a}}} - ih} \right)\frac{{{\tau _1}}}{{2h}}\\ {\pi _4} = \frac{{\sigma _2^2B{{\left( {n{\tau _1}} \right)}^2}{j^2}{\tau _1}}}{2} + \left( {{r_{\rm{a}}} + ih} \right)j\frac{{{\tau _1}}}{2}\\ {\pi _5} = \frac{{\sigma _2^2B{{\left( {n{\tau _1}} \right)}^2}{j^2}{\tau _1}}}{2} - \left( {{r_{\rm{a}}} + ih} \right)j\frac{{{\tau _1}}}{2}\\ {\pi _6} = \frac{{{\sigma _1}{\sigma _2}B\left( {n{\tau _1}} \right)j{\tau _1}}}{{4h}} \end{array} \right.$

 $\begin{array}{*{20}{c}} {{V_{n,0,j}} = f\left( {n{\tau _1}} \right),\;\;\;\;{V_{0, - \frac{{{R_1}}}{h},j}} = 0,\;\;\;\;{V_{0,\frac{{{R_1}}}{h},j}} = 0,}\\ {{V_{0,i,j}} = 0,\;\;\;\;{V_{n,i, - \frac{{{R_2}}}{h}}} = {P_{1n{\tau _1},{r_{\rm{a}}}}},\;\;\;{V_{n,i,\frac{{{R_2}}}{h}}} = 0} \end{array}$

 图 1 保费预期支付额与随机利率和时间的关系 Fig.1 Variation of expected premiums with random interest rates and time

 图 2 保费预期支付额和时间的关系 Fig.2 Variation of expected premiums with time

 图 3 回归速度对保费预期支付额的影响 Fig.3 Effect of regression speed on expected premiums

 图 4 回归均值对保费预期支付额的影响 Fig.4 Effect of regression means on expected premiums

 图 5 信用等级边界对保费预期支付额的影响 Fig.5 Effect of credit rating border on expected premiums
3 结语

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