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 同济大学学报(自然科学版)  2018, Vol. 46 Issue (12): 1754-1760.  DOI: 10.11908/j.issn.0253-374x.2018.12.019 0

### 引用本文

ZHAO Dan, XU Chenglong. Conditional Monte Carlo Hybrid Acceleration Method Under Stochastic Interest Rate Model and Its Applications[J]. Journal of Tongji University (Natural Science), 2018, 46(12): 1754-1760. DOI: 10.11908/j.issn.0253-374x.2018.12.019

### 文章历史

1. 同济大学 数学科学学院，上海 200092;
2. 上海财经大学 数学学院，上海 200433

Conditional Monte Carlo Hybrid Acceleration Method Under Stochastic Interest Rate Model and Its Applications
ZHAO Dan 1, XU Chenglong 2
1. School of Mathematical Sciences, Tongji University, Shanghai 200092, China;
2. School of Mathematical Sciences, Shanghai University of Finance and Economics, Shanghai 200433, China
Abstract: This paper mainly studies acceleration methods of the Monte Carlo simulation method for the pricing of European call options under the assumption of the CIR(Cox-Ingersoll-Ross) stochastic interest rate model. A new control variable based on the combination of the conditional expectation formula and the control variable technique is presented. The theoretical analysis and numerical results show that this method, with a new control variable, can improve the computation efficiency. Then it is applied to the computation of Greeks. The numerical results illustrate that the new method is more accurate and stable than the classical Monte Carlo method. It can also be applied to basket options, Asian option, and other high-dimension cases.
Key words: stochastic interest rate    conditional Monte Carlo simulation method    variance reduction    Greeks

1 随机利率模型下期权定价

1.1 随机利率下资产价格模型

 $\frac{{{\rm{d}}{S_t}}}{{{S_t}}} = {r_t}{\rm{d}}t + {\sigma _s}{\rm{d}}{W_t}$ (1)

 ${\rm{d}}{r_t} = \alpha \left( {\theta - {r_t}} \right){\rm{d}}t + {\sigma _r}\sqrt {{r_t}} {\rm{d}}{Z_t}$ (2)

 ${\rm{d}}{W_t} = \rho {\rm{d}}{Z_t} + \sqrt {1 - {\rho ^2}} {\rm{d}}{{\tilde Z}_t}$ (3)

 $\begin{array}{l} {S_T} = {S_t}\exp \left[ {\int_t^T {{r_s}{\rm{d}}s} - \frac{1}{2}\sigma _s^2\left( {T - t} \right) + \rho {\sigma _s}\left( {{Z_T} - {Z_t}} \right) + } \right.\\ \;\;\;\;\;\;\;\left. {{\sigma _s}\sqrt {1 - {\rho ^2}} \left( {{{\tilde Z}_T} - {{\tilde Z}_t}} \right)} \right] \end{array}$ (4)

 $\begin{array}{l} {S_T} = {S_t}\xi \left( {t,T} \right)\exp \left[ {\left( {\bar r - \frac{1}{2}\sigma _s^2\left( {1 - {\rho ^2}} \right)} \right)\left( {T - t} \right) + } \right.\\ \;\;\;\;\;\;\;\left. {{\sigma _s}\sqrt {1 - {\rho ^2}} \left( {{{\tilde Z}_T} - {{\tilde Z}_t}} \right)} \right] \end{array}$ (5)

 $\bar r\left( {t,T} \right) = \frac{1}{{T - t}}\int_t^T {{r_s}{\rm{d}}s}$ (6)
 $\xi \left( {t,T} \right) = \exp \left[ { - \frac{1}{2}{\rho ^2}\sigma _s^2\left( {T - t} \right) + \rho {\sigma _s}\left( {{Z_T} - {Z_t}} \right)} \right]$ (7)
1.2 期权定价的模拟方法 1.2.1 标准蒙特卡罗模拟方法

 ${V_t} = {E^Q}\left[ {{{\rm{e}}^{ - \bar r\left( {T - t} \right)}} \cdot {{\left( {{S_T} - K} \right)}^ + }} \right]$ (8)

 ${V_0} = {E^Q}\left[ {{{\rm{e}}^{ - \bar rT}}{{\left( {{S_T} - K} \right)}^ + }} \right]$ (9)

 ${r_{k + 1}} = {r_k} + \alpha \left( {\theta - {r_k}} \right) \cdot \Delta t + {\sigma _r} \cdot \sqrt {{r_k} \cdot \Delta t} \cdot {\varepsilon _k}$
 $\begin{array}{l} {S_{k + 1}} = {S_k}\exp \left\{ {{r_k} \cdot \Delta t - \frac{1}{2}\sigma _s^2 \cdot \Delta t + \rho {\sigma _s} \cdot \sqrt {\Delta t} \cdot {\varepsilon _k} + } \right.\\ \;\;\;\;\;\;\;\;\;\left. {{\sigma _s}\sqrt {\left( {1 - {\rho ^2}} \right) \cdot \Delta t} \cdot {{\tilde \varepsilon }_k}} \right\} \end{array}$

$\bar r\left( {0,T} \right) = \frac{{{{\tilde r}_T}}}{T}$，由式(6)，${{{\tilde r}_T}}$可由下式计算得到：

 ${{\tilde r}_0} = 0,{{\tilde r}_{k + 1}} = {{\tilde r}_k} + {r_k} \cdot \Delta t$ (10)

 ${{\bar V}_0} = \frac{1}{M}\sum\limits_{i = 1}^M {\left( {{{\rm{e}}^{ - \bar rT}}{{\left( {S_T^{\left( i \right)} - K} \right)}^ + }} \right)}$

 $P\left( {\left| {{{\bar V}_0} - {V_0}} \right| < \frac{{\sigma {\lambda _\alpha }}}{{\sqrt M }}} \right) = 1 - \alpha$

 $\sigma _M^2 = \frac{1}{{M - 1}}\sum\limits_{i = 1}^M {{{\left( {\bar V_0^{\left( i \right)} - {{\bar V}_0}} \right)}^2}}$

1.2.2 条件蒙特卡罗方法

 $E\left[ Y \right] = E\left[ {E\left[ {Y\left| X \right.} \right]} \right]$ (11)

 $\begin{array}{l} {\rm{Var}}\left( Y \right) = {\rm{Var}}\left( {E\left[ {Y\left| X \right.} \right]} \right) + \\ \;\;\;\;\;\;\;E\left[ {{\rm{Var}}\left( {Y\left| X \right.} \right)} \right] > {\rm{Var}}\left( {E\left[ {Y\left| X \right.} \right]} \right) \end{array}$

 $\begin{array}{*{20}{c}} {V\left( {S,t} \right) = {V_t} = E\left[ {E\left[ {{{\rm{e}}^{\int_t^T {{r_s}{\rm{d}}s} }} \cdot {{\left( {{S_T} - K} \right)}^ + }\left| {{{\tilde Z}_{\rm{t}}}} \right.} \right]} \right] = }\\ {E\left[ {{V_{{\rm{BS}}}}\left( {t,T,{S_t}\xi \left( {t,T} \right);\bar r\left( {t,T} \right),{\sigma _s}\sqrt {1 - {\rho ^2}} } \right)} \right]} \end{array}$ (12)

 ${V_0} = E\left[ {{V_{{\rm{BS}}}}\left( {0,T,{S_0}\xi \left( {0,T} \right);\bar r\left( {0,T} \right),{\sigma _s}\sqrt {1 - {\rho ^2}} } \right)} \right]$ (13)

 $\begin{array}{*{20}{c}} {{{\bar \xi }_0} = 0,{{\bar \xi }_{k + 1}} = {{\bar \xi }_k} - \frac{{{\rho ^2}\sigma _s^2}}{2}\Delta t + \rho {\sigma _s}\sqrt {\Delta t} \cdot {\varepsilon _k},}\\ {k = 0,1, \cdots ,N - 1} \end{array}$

1.2.3 基于条件期望的蒙特卡罗控制变量法

 $V\left( \mathit{\boldsymbol{b}} \right) = {V_{{\rm{BS}}}} - {\mathit{\boldsymbol{b}}^{\rm{T}}}\left( {\mathit{\boldsymbol{X}} - E\left[ \mathit{\boldsymbol{X}} \right]} \right)$

 $\begin{array}{*{20}{c}} {E\left[ {\bar r} \right] = \frac{1}{{T - t}}\int_t^T {E\left[ {{r_s}} \right]{\rm{d}}s} = \theta + \frac{{\left( {\theta - {r_0}} \right)}}{{\alpha \left( {T - t} \right)}}\left( {{{\rm{e}}^{ - \alpha T}} - {{\rm{e}}^{ - \alpha t}}} \right),}\\ {E\left[ \xi \right] = 1} \end{array}$

 $\left( {\begin{array}{*{20}{c}} {{\mathit{\boldsymbol{ \boldsymbol{\varSigma} }}_X}}&{{\mathit{\boldsymbol{ \boldsymbol{\varSigma} }}_{X{V_{{\rm{BS}}}}}}}\\ {\mathit{\boldsymbol{ \boldsymbol{\varSigma} }}_{X{V_{{\rm{BS}}}}}^{\rm{T}}}&{\sigma _{{V_{{\rm{BS}}}}}^2} \end{array}} \right)$

M次模拟情形下，二元控制变量下的估计值为

 $\begin{array}{*{20}{c}} {{{\bar V}_M}\left( \mathit{\boldsymbol{b}} \right) = {{\bar V}_{M,BS}} - {\mathit{\boldsymbol{b}}^{\rm{T}}}\left( {\mathit{\boldsymbol{\bar X}} - E\left[ \mathit{\boldsymbol{X}} \right]} \right) = }\\ {\frac{1}{M}\sum\limits_{i = 1}^M {\left( {{V_{i,{\rm{BS}}}} - {b_1}\left( {{{\bar r}^{\left( i \right)}} - E\left[ {\bar r} \right]} \right) - {b_2}\left( {{\xi ^{\left( i \right)}} - E\left[ \xi \right]} \right)} \right)} } \end{array}$ (14)

 $\begin{array}{l} {\rm{Var}}\left( {{{\bar V}_M}\left( \mathit{\boldsymbol{b}} \right)} \right) = {\rm{Var}}\left( {{{\bar V}_{M,\rm{BS}}} - {\mathit{\boldsymbol{b}}^{\rm{T}}}\left( {{{\mathit{\boldsymbol{\bar X}}}_M} - E\left[ \mathit{\boldsymbol{X}} \right]} \right)} \right) = \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\sigma _{{V_{{\rm{BS}}}}}^2 - 2{\mathit{\boldsymbol{b}}^{\rm{T}}}{\mathit{\boldsymbol{ \boldsymbol{\varSigma} }}_{X{V_{{\rm{BS}}}}}} + {\mathit{\boldsymbol{b}}^{\rm{T}}}{\mathit{\boldsymbol{ \boldsymbol{\varSigma} }}_X}\mathit{\boldsymbol{b}} \end{array}$

 $\mathit{\boldsymbol{b}} = \mathit{\boldsymbol{ \boldsymbol{\varSigma} }}_X^{ - 1}{\mathit{\boldsymbol{ \boldsymbol{\varSigma} }}_{X{V_{{\rm{BS}}}}}}$ (15)

 ${\rm{Var}}\left( {{{\bar V}_M}\left( \mathit{\boldsymbol{b}} \right)} \right) = \left( {1 - R_{X{V_{{\rm{BS}}}}}^2} \right)\sigma _{{V_{{\rm{BS}}}}}^2 < \sigma _{{V_{{\rm{BS}}}}}^2$

2 数值结果

 ${R_{{\rm{CMCC}}}} = \frac{{{\rm{st}}{{\rm{d}}_{{\rm{MC}}}}}}{{{\rm{st}}{{\rm{d}}_{{\rm{CMCC}}}}}}$

 $s = \frac{{{\rm{std}}_{{\rm{MC}}}^2 \cdot {t_{{\rm{MC}}}}}}{{{\rm{std}}_{{\rm{CMCC}}}^2 \cdot {t_{{\rm{CMCC}}}}}}$

 图 1 期权价格与模拟次数关系图 Fig.1 Option price change for different simulation times

 图 2 标准误差与模拟次数关系图 Fig.2 Standard deviation change for different simulation times

 $s = \frac{{{\rm{std}}_{{\rm{MC}}}^2 \cdot {t_{{\rm{MC}}}}}}{{{\rm{std}}_{{\rm{CMCC}}}^2 \cdot {t_{{\rm{CMCC}}}}}} = {R^2}\frac{{{t_{{\rm{MC}}}}}}{{{t_{{\rm{CMCC}}}}}} = 347.726$

 ${\left( {A - K} \right)^ + } = \left( {A - K} \right)\left| {_{G \ge K}} \right. + \left( {A - K} \right)\left| {_{G < K < A}} \right.$

3 Greeks计算 3.1 Greeks介绍

 ${V_t} = E\left[ {{S_t}\xi N\left( {{{\hat d}_1}} \right) - K{{\rm{e}}^{ - \bar r\left( {T - t} \right)}} \cdot N\left( {{{\hat d}_2}} \right)} \right]$ (16)

 $\begin{array}{*{20}{c}} {{{\hat d}_1} = \frac{{\ln \frac{{{S_t}\xi }}{K} + \left( {\bar r + \frac{{\sigma _s^2}}{2}\left( {1 - {\rho ^2}} \right)} \right)\left( {T - t} \right)}}{{{\sigma _s}\sqrt {\left( {1 - {\rho ^2}} \right)\left( {T - t} \right)} }},}\\ {{{\hat d}_2} = {{\hat d}_1} - {\sigma _s}\sqrt {\left( {1 - {\rho ^2}} \right)\left( {T - t} \right)} } \end{array}$

N(x)表示标准正态分布的分布函数.

 $\Delta = \frac{{\partial V}}{{\partial S}} = E\left[ {\xi \cdot N\left( {{{\hat d}_1}} \right)} \right]$
 $\mathit{\Gamma } = \frac{{\partial \Delta }}{{\partial S}} = \frac{{{\partial ^2}V}}{{\partial {S^2}}} = E\left[ {\frac{{\xi \cdot N'\left( {{{\hat d}_1}} \right)}}{{S{\sigma _s}\sqrt {\left( {1 - {\rho ^2}} \right)\left( {T - t} \right)} }}} \right]$
 $\begin{array}{*{20}{c}} {\mathit{\Theta } = \frac{{\partial V}}{{\partial t}} = E\left[ { - \frac{{S\xi N'\left( {{{\hat d}_1}} \right) \cdot {\sigma _s}\sqrt {1 - {\rho ^2}} }}{{2\sqrt {T - t} }} - } \right.}\\ {\left. {K{r_t} \cdot {{\rm{e}}^{ - \bar r\left( {T - t} \right)}} \cdot N\left( {{{\hat d}_2}} \right)} \right]} \end{array}$
 $\begin{array}{l} \upsilon = \frac{{\partial V}}{{\partial {\sigma _s}}} = E\left[ {S\xi N'\left( {{{\hat d}_1}} \right) \cdot \sqrt {\left( {1 - {\rho ^2}} \right)\left( {T - t} \right)} - } \right.\\ \left. {S\xi N\left( {{{\hat d}_1}} \right) \cdot \left( { - {\rho ^2}{\sigma _s}\left( {T - t} \right) + \rho \left( {{Z_T} - {Z_t}} \right)} \right)} \right] \end{array}$ (17)
3.2 数值模拟

 图 3 不同模拟方法下标的资产初始价格与Δ值的标准误差关系图 Fig.3 Comparison of standard deviations of Δ by using different methods

 图 4 不同模拟方法下标的资产初始价格与Γ值的标准误差关系图 Fig.4 Comparison of standard deviations of Γ by using different methods
4 结论

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