快速随机波动率变换下的信用风险定价研究
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- 英文论文题名：The Study on the Pricing of Credit Risk under the Fast Stochastic Volatility
- 论文作者：周艳丽 ; 刘诗璨 ; 葛翔宇
- 英文论文作者：ZHOU Yan-li ; LIU Shi-can ; GE Xiang-yu ; School of Finance ; Zhongnan University of Economics and Law ; School of Statistics and Mathematics ; Zhongnan University of Economics and Law
- 年：2016
- 作者机构：中南财经政法大学金融学院;中南财经政法大学统计与数学学院;
- 论文关键词：随机波动率 ; 快速时间维度 ; 扰动理论 ; 算法逼近 ; 信用风险
- 英文论文关键词：stochastic volatility ; fast time scale ; perturbation theory ; algorithm approximation ; credit risk
- 会议召开时间：2016-11-12
- 会议录名称：第十八届中国管理科学学术年会论文集
- 英文会议录名称：Proceedings of the 18th Chinese Annual Conference on Management Science
- 语种：中文
- 分类号：F224;F830.9
- 学会代码：ZHYJ
- 会议名称：第十八届中国管理科学学术年会
- 会议地点：中国陕西西安
- 主办单位：中国优选法统筹法与经济数学研究会、西安交通大学、中国科学院科技战略咨询研究院、《中国管理科学》编辑部
- 学会名称：中国优选法统筹法与经济数学研究会
- 页数：10
- 文件大小：595k
- 原文格式：O
- 会议级别：全国
摘要
本文在快速随机波动率变换的情况下研究了单个信用衍生品定价的风险规避问题。本文首先在Merton最优化的问题中添加随机波动率,将其应用到信用风险的研究中。假设波动率过程是一种不同时间尺度的扰动过程,对其进行近似估计。由于价值过程可看成是一个不完全市场围绕着完全市场常数波动率的一种扰动,故这种近似方法是可行的。其次,当波动率变换是快速均值回归过程时,这种扰动过程是一种关于Hamilton-Jacobi-Bellman偏微分方程的奇异扰动过程。当波动率变换非常缓慢时,这种扰动过程是一般的扰动过程。两种扰动过程叠加在一起便是多元时间维度随机波动率过程。再者,本文提出的算法与线性期权定价问题具有较高的一致性,因而也发现了风险容忍度方程的一些新性质。本文最后还在快速时间维度的框架下研究了波动率对于信用风险的投资优化问题。由于新建模型是一个非线性PDE,故不能直接给出其解析解,于是本文引进一种扰动逼近的方法,该方法的关键中间步骤是把问题分成可解和扰动两个部分。通过扰动逼近的方法,可得一个近似的解析解。
In this paper,we study the risk aversion on valuing the single-name credit derivatives by introducing the fast —scale stochastic volatility correction.Firstly,the paper studies the Merton portfolio problem in the presence of stochastic volatility,and applies it to the credit risk problem.We assume the volatility process is a perturbation process with various time scale,and estimate it approximately.This approach is feasible,because we treat the value function as a perturbation problem in the incomplete market,which is moving around the constant volatility of the complete market.Secondly,if the volatility is fast mean—reverting,it is a singular perturbation defined by a nonlinear Hamilton—Jacobi-Bellman PDE;If it is slowly varying,it is a regular perturbation problem.In this paper,we add the two processes together,which is a multiscale stochastic volatility problem.Thirdly,the algorithm shares remarkable similarities with the linear option pricing problem,but it gives new properties to the risk tolerance function.Finely,this paper is to study how fast — scale volatility affect the portfolio optimizing problem with single name credit risk under the frame of the fast scale stochastic volatility.Due to our model is a non—linear PDE,analytical solution cannot be obtained directly.In order to get the closed form solution,we introduce a method called perturbation approximation.A critical feature of the technique is that we break the problem into two solvable and perturbation parts.By using the perturbation approximation method,the approximated analytical solution can be obtained.
In this paper,we study the risk aversion on valuing the single-name credit derivatives by introducing the fast —scale stochastic volatility correction.Firstly,the paper studies the Merton portfolio problem in the presence of stochastic volatility,and applies it to the credit risk problem.We assume the volatility process is a perturbation process with various time scale,and estimate it approximately.This approach is feasible,because we treat the value function as a perturbation problem in the incomplete market,which is moving around the constant volatility of the complete market.Secondly,if the volatility is fast mean—reverting,it is a singular perturbation defined by a nonlinear Hamilton—Jacobi-Bellman PDE;If it is slowly varying,it is a regular perturbation problem.In this paper,we add the two processes together,which is a multiscale stochastic volatility problem.Thirdly,the algorithm shares remarkable similarities with the linear option pricing problem,but it gives new properties to the risk tolerance function.Finely,this paper is to study how fast — scale volatility affect the portfolio optimizing problem with single name credit risk under the frame of the fast scale stochastic volatility.Due to our model is a non—linear PDE,analytical solution cannot be obtained directly.In order to get the closed form solution,we introduce a method called perturbation approximation.A critical feature of the technique is that we break the problem into two solvable and perturbation parts.By using the perturbation approximation method,the approximated analytical solution can be obtained.
引文
[1]Black F and Scholes M.The pricing of options and corporate liabilities.[J]The Journal of Political Economy,1973,81(3):637-654.
[2]Sircar R and Zariphopoulou T.Utility valuation of credit derivatives:Single and two-name cases[M].Advances in Mathematical Finance.Birkhauser Boston,2007,279-301.
[3]Sircar R and Thaleia Z.Utility valuation of multi-name credit derivatives and application to CDOs[J].Quantitative Finance,2010,10(2):195-208.
[4]Merton R C.Optimum consumption and portfolio rules in a continuous-time model[J].Journal of economic theory,1971,3(4):373-413.
[5]Heston S L.A closed-form solution for options with stochastic volatility with applications to bond and currency options[J].Review of financial studies,1993,6(2):327-343.
[6]Fouque J P,Papanicolaou G and Sircar K R,et al.Asymptotics of a two-scale stochastic volatility model[J].Comparative Biochemistry&physiology part B.Comparative Biochemistry,2000,53(2):187-190.
[7]Fouque J P,Sircar R and Zariphopoulou T,et al.Portfolio optimization and stochastic volatility asymptotics[J].Ssm Electronic Journal,2013,2(1):22-42.
[8]Fouque J P,Sircar R and Sφlna K.Multiname and multiscale default modeling[J].Multiscale Modeling&Simulation,2009,7(4):1956-1978.
[9]罗长青,朱慧明,欧阳资生等.跳跃—扩散条件下信用风险相关性度量的变结构Copula模型[J],中国管理科学,2014,22(3):1-12.
[10]吴建华,王新军,张颖.内生性回收率与信用风险度量研究[J].中国管理科学,2016,24(1):1-10.
[11]周艳丽,吴洋,葛翔宇等.一类高新技术企业专利权价值的实物期权评估方法[J].中国管理科学,2016,24(6):19-28.
[12]马俊海,张强.随机波动率LIBOR市场利率动态模型的理论估计与蒙特卡罗模拟[J].数量经济技术经济研究,2012,(1):118-134.
[13]韩立岩,郑承利.中国市政债券信用风险与发债规模研究[J].金融研究,2003,(2):85-94.
[14]曹道胜,何明升.商业银行信用风险模型的比较及其借鉴[J].金融研究,2006,(10):90-97.
[15]Papageorgiou E and Sircar R.Multiscaleintensity models for single name credit derivatives[J].Applied Mathematical Finance,2008,15(1):73-105.
[16]Bremaud P.Point processes and queues:martingale dynamics[M].New York:Springer-Verlag,1981.
[17]Fouque J P.Papanicolaou G,Sircar R,et al.Multiscale stochastic volatility for equity,interest rate,and credit derivatives[M].London:Cambridge University Press,2011.
[18]Duffie D andZariphopoulou T.Optimal investment with undiversifiable income risk[J].Mathematical Finance,1993,3(2):135-148.
[2]Sircar R and Zariphopoulou T.Utility valuation of credit derivatives:Single and two-name cases[M].Advances in Mathematical Finance.Birkhauser Boston,2007,279-301.
[3]Sircar R and Thaleia Z.Utility valuation of multi-name credit derivatives and application to CDOs[J].Quantitative Finance,2010,10(2):195-208.
[4]Merton R C.Optimum consumption and portfolio rules in a continuous-time model[J].Journal of economic theory,1971,3(4):373-413.
[5]Heston S L.A closed-form solution for options with stochastic volatility with applications to bond and currency options[J].Review of financial studies,1993,6(2):327-343.
[6]Fouque J P,Papanicolaou G and Sircar K R,et al.Asymptotics of a two-scale stochastic volatility model[J].Comparative Biochemistry&physiology part B.Comparative Biochemistry,2000,53(2):187-190.
[7]Fouque J P,Sircar R and Zariphopoulou T,et al.Portfolio optimization and stochastic volatility asymptotics[J].Ssm Electronic Journal,2013,2(1):22-42.
[8]Fouque J P,Sircar R and Sφlna K.Multiname and multiscale default modeling[J].Multiscale Modeling&Simulation,2009,7(4):1956-1978.
[9]罗长青,朱慧明,欧阳资生等.跳跃—扩散条件下信用风险相关性度量的变结构Copula模型[J],中国管理科学,2014,22(3):1-12.
[10]吴建华,王新军,张颖.内生性回收率与信用风险度量研究[J].中国管理科学,2016,24(1):1-10.
[11]周艳丽,吴洋,葛翔宇等.一类高新技术企业专利权价值的实物期权评估方法[J].中国管理科学,2016,24(6):19-28.
[12]马俊海,张强.随机波动率LIBOR市场利率动态模型的理论估计与蒙特卡罗模拟[J].数量经济技术经济研究,2012,(1):118-134.
[13]韩立岩,郑承利.中国市政债券信用风险与发债规模研究[J].金融研究,2003,(2):85-94.
[14]曹道胜,何明升.商业银行信用风险模型的比较及其借鉴[J].金融研究,2006,(10):90-97.
[15]Papageorgiou E and Sircar R.Multiscaleintensity models for single name credit derivatives[J].Applied Mathematical Finance,2008,15(1):73-105.
[16]Bremaud P.Point processes and queues:martingale dynamics[M].New York:Springer-Verlag,1981.
[17]Fouque J P.Papanicolaou G,Sircar R,et al.Multiscale stochastic volatility for equity,interest rate,and credit derivatives[M].London:Cambridge University Press,2011.
[18]Duffie D andZariphopoulou T.Optimal investment with undiversifiable income risk[J].Mathematical Finance,1993,3(2):135-148.
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