信用衍生品定价的传染模型
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摘要
信用衍生品是20世纪90年代出现的一种从其他资产(如贷款、债券或其他金融资产)衍生而来的金融产品,它将信用风险从市场风险中分离出来,并使信用风险变得可以交易和管理,从根本上改变了传统的信用风险管理模式。上个世纪的亚洲金融危机和2007年的美国次贷危机说明传染风险对信用衍生品转移、重组和转换信用风险尤为重要,因此,我们需要通过优化模型不断地对信用衍生品深入研究,从而使理论成果可以更好地指导实践。本文主要研究带有交易对手风险和利率风险的债券、信用违约互换(Credit Default Swap,简称CDS)和异质组合CDO的定价问题。
本文按两条线来研究债券和CDS定价的传染模型,一条线是公司的违约与无违约期限结构是否有关,在不同的利率模型下考虑有交易对手风险的债券和CDS的定价问题;另一条线是在原生―丛生和环形违约两种框架下将模型拓展。前两章对信用衍生品的背景和概念作了介绍,总结了信用资产间关于相关性的三种约化模型和描述传染风险的三种方法,也就是计算有传染风险的资产的违约时间联合分布的方法。第三章详细介绍了短期利率和远期利率两种模型,并着重讨论了三种短期利率模型,推导出一些重要的结果。第四章和第五章按照上面两条线对债券和CDS进行定价,得到了其价格的显式表示,并做了数值模拟。第六章研究异质参考组合的CDO的定价,同时也给出了算例。第七章是对本文研究的总结及一些展望。本文在已有成果的基础上,基于违约强度过程对信用衍生品的定价进行研究,主要得到了如下新的结果:
第一、将Jarrow和Yu (2001)的模型进行推广。Jarrow和Yu引入了交易对手风险的概念,与以往的强度模型不同,解决了传统的结构化方法和约化方法所忽略的问题。他们提出了原生-丛生模型和环形违约模型,并在原生-丛生框架下研究了债券和CDS的定价,其中的无风险利率服从Vasicek模型。我们将模型推广,一方面,若公司违约和短期利率无关,短期利率服从其它仿射过程(如CIR模型和Hull-White模型),分别在原生-丛生和环形违约框架下给出了债券和两方违约的信用违约互换(CDS)的显式价格。另一方面,若公司违约和宏观因子利率相关,同样在上述两种框架下考虑利率服从不同仿射扩散过程,分别给出了债券和CDS的显式价格。
第二、将Bai,Hu和Ye(2007)的模型进行推广。Bai,Hu和Ye首次提出衰减模型,考虑交易对手对违约影响的衰减效应。本文在原生-丛生和环形违约框架下分别研究可违约债券和CDS的定价,模型中短期利率满足仿射扩散过程。Bai,Hu和Ye (2007,2008)讨论了利率为常数和服从Vasicek模型的信用衍生品定价的衰减模型。本文考虑短期利率服从其它仿射模型(如CIR模型和Hull-White模型)的情形,利用Park(2008)的方法得到了有关随机利率的几个重要的随机微分方程,并利用它们研究公司违约和宏观因子利率相关时,带有衰减效应的债券和CDS的定价。其中在环形违约框架下,为了得到其显式价格,我们利用测度变换的方法计算公司的条件生存概率。
第三、本文将上述两种模型进一步推广。前面的模型中短期利率都服从扩散过程,为了使模型更全面、更真实地反映实际市场,我们进一步考虑短期利率的跳扩散(Jump-diffusion)模型。同时,也考虑了交易对手风险对债券和CDS定价的影响。公司的违约与短期利率是有关的,在原生-丛生和环形违约框架下,本文分别得到了与利率相关的几个结果,进而推导出债券和CDS的显式价格,还对债券价格和CDS的价格做了数值模拟。
第四、研究了带传染风险的异质组合的担保债务凭证(Collateralized Debt Obligations,简称CDO)的定价。CDO定价的关键是对参考资产组合违约损失的估计,已有的大多数违约损失模型都假设资产组合是同质的,如部分条件独立模型和Copula模型。受单资产信用衍生品风险模型的启发,本文将Bai,Hu和Ye (2007)的传染模型应用于CDO定价中,给出CDO各层的解析价格。CDO参考组合由两类无息债券组成,两类债券的发行公司间存在某种传染风险。另外,Zheng和Jiang(2009)提出一种违约相关的因子传染模型,模型的参考资产池中包含几类相互条件独立的资产组合,同一类中风险资产的违约具有传染性,作者运用总风险函数构造法计算出了违约时间的联合分布,并在同一类组合中资产违约概率和传染率都相等的前提下(同质假设)给出篮子CDS价格的解析表达式。从而启发我们将这种模型应用于CDO定价中,给出了与前面不同的异质组合的CDO价格。最后还给出了CDO的算例,数值模拟出各层的价格。
Credit derivatives were the financial assets derived from the loan, the bond or other assetsin 1990s. They separated the credit risk from market risk and made the credit risk be tradeableand manageable. The traditional management mode was changed. The financial crisis in thelast century and the occurrence of American subprime mortgage crisis further showed that thecontagious risk was important for transferring and reorganizing the credit risk. Therefore, weneed the fair valuation of credit derivatives and make the theory more helpful. The study on theirpricing is fundamental. This thesis focuses on the pricing of bonds and credit default swaps withcounterparty risk and the pricing of Collateralized Debt Obligations with heterogeneous portfolio.
We study the pricing of bonds and credit default swaps from two aspects. One is that thedefaults of the assets depend on the short interest rate or not and the other is that we considerthe pricing respectively in the primary-secondary framework and looping default framework. Thefirst two chapters introduce the setting and notions of credit derivatives, the three correlated modeland three methods of computing the joint distribution of default times in the contagious model.Chapter 3 introduces the short interest rate and the forward interest rate in details. We take moreimportance on three types of the short interest rate: Vasicek model, CIR model and Hull-Whitemodel which are used in the following chapters. Chapter 4 and Chapter 5 discuss the pricing ofbonds and CDS from the two aspects above and give their explicit prices. Chapter 6 consider thepricing of CDO with heterogeneous portfolio and give an example. In this paper, we obtain somenew results by the intensity-based approach of default event as following: In this thesis, the mainresults are obtained with .
Firstly, we extend the models in Jarrow and Yu (2001). They proposed the counterparty riskwhich was different from the traditional intensity-based model and solved the problems ignoredby the traditionally structural model and reduced-form model. They introduced primary-secondarymodel and priced bonds and CDS with Vasicek interest rate. We extend their models and considerother interest rate following diffusion processes. On the one hand, we price bonds and CDS whenthe defaults of firms are independent of the default-free term structure. On the other hand, wederive the prices of bonds and CDS when the defaults of firms depend on the default-free termstructure.
Secondly, we extend the models in Bai, Hu and Ye (2007). They firstly introduced the conta- gious model with attenuation effect. We price bonds and CDS respectively in primary-secondarymodel and looping default model. Bai, Hu and Ye (2007,2008) only considered the interest rate asa constant and the Vasicek diffusion process. We discuss the other diffusion processes. By usingthe methods in Park (2008), we obtain some important differential equations on the interest rateand price bonds and CDS with a hyperbolic attenuation effect.
Thirdly, we further generalize the above models and is extended by adding a hyperbolic at-tenuation effect. We consider the interest rate satisfying jump-diffusion process and obtain somenew results. Moreover, We obtain one explicit solution for bonds and CDS in the similar modelabove. Meanwhile, we make use of Matlab and Monte-Carlos simulation to get the price curves oftheir.
At last, we study the contagious model of pricing CDO with heterogeneous portfolio. Thekey of pricing CDO is to valuate the credit risk of the reference assets. Until now, many results arebased on the homogeneous assumption, such as the conditionally independent model and Copulamodel. In this thesis, we apply the hyperbolic attenuation model of two parties in Bai, Hu andYe (2007) to price CDO and obtain the analytic form of CDO tranche’s price. In addition, weconsider the contagious model in Zheng and Jiang(2009). They only discussed the pricing of CDSwith many assets. We apply a comparatively simple model to the reference portfolio and obtainthe analytic price of CDO. Moreover, we give the example and the pricing curve of each tranche.
本文按两条线来研究债券和CDS定价的传染模型,一条线是公司的违约与无违约期限结构是否有关,在不同的利率模型下考虑有交易对手风险的债券和CDS的定价问题;另一条线是在原生―丛生和环形违约两种框架下将模型拓展。前两章对信用衍生品的背景和概念作了介绍,总结了信用资产间关于相关性的三种约化模型和描述传染风险的三种方法,也就是计算有传染风险的资产的违约时间联合分布的方法。第三章详细介绍了短期利率和远期利率两种模型,并着重讨论了三种短期利率模型,推导出一些重要的结果。第四章和第五章按照上面两条线对债券和CDS进行定价,得到了其价格的显式表示,并做了数值模拟。第六章研究异质参考组合的CDO的定价,同时也给出了算例。第七章是对本文研究的总结及一些展望。本文在已有成果的基础上,基于违约强度过程对信用衍生品的定价进行研究,主要得到了如下新的结果:
第一、将Jarrow和Yu (2001)的模型进行推广。Jarrow和Yu引入了交易对手风险的概念,与以往的强度模型不同,解决了传统的结构化方法和约化方法所忽略的问题。他们提出了原生-丛生模型和环形违约模型,并在原生-丛生框架下研究了债券和CDS的定价,其中的无风险利率服从Vasicek模型。我们将模型推广,一方面,若公司违约和短期利率无关,短期利率服从其它仿射过程(如CIR模型和Hull-White模型),分别在原生-丛生和环形违约框架下给出了债券和两方违约的信用违约互换(CDS)的显式价格。另一方面,若公司违约和宏观因子利率相关,同样在上述两种框架下考虑利率服从不同仿射扩散过程,分别给出了债券和CDS的显式价格。
第二、将Bai,Hu和Ye(2007)的模型进行推广。Bai,Hu和Ye首次提出衰减模型,考虑交易对手对违约影响的衰减效应。本文在原生-丛生和环形违约框架下分别研究可违约债券和CDS的定价,模型中短期利率满足仿射扩散过程。Bai,Hu和Ye (2007,2008)讨论了利率为常数和服从Vasicek模型的信用衍生品定价的衰减模型。本文考虑短期利率服从其它仿射模型(如CIR模型和Hull-White模型)的情形,利用Park(2008)的方法得到了有关随机利率的几个重要的随机微分方程,并利用它们研究公司违约和宏观因子利率相关时,带有衰减效应的债券和CDS的定价。其中在环形违约框架下,为了得到其显式价格,我们利用测度变换的方法计算公司的条件生存概率。
第三、本文将上述两种模型进一步推广。前面的模型中短期利率都服从扩散过程,为了使模型更全面、更真实地反映实际市场,我们进一步考虑短期利率的跳扩散(Jump-diffusion)模型。同时,也考虑了交易对手风险对债券和CDS定价的影响。公司的违约与短期利率是有关的,在原生-丛生和环形违约框架下,本文分别得到了与利率相关的几个结果,进而推导出债券和CDS的显式价格,还对债券价格和CDS的价格做了数值模拟。
第四、研究了带传染风险的异质组合的担保债务凭证(Collateralized Debt Obligations,简称CDO)的定价。CDO定价的关键是对参考资产组合违约损失的估计,已有的大多数违约损失模型都假设资产组合是同质的,如部分条件独立模型和Copula模型。受单资产信用衍生品风险模型的启发,本文将Bai,Hu和Ye (2007)的传染模型应用于CDO定价中,给出CDO各层的解析价格。CDO参考组合由两类无息债券组成,两类债券的发行公司间存在某种传染风险。另外,Zheng和Jiang(2009)提出一种违约相关的因子传染模型,模型的参考资产池中包含几类相互条件独立的资产组合,同一类中风险资产的违约具有传染性,作者运用总风险函数构造法计算出了违约时间的联合分布,并在同一类组合中资产违约概率和传染率都相等的前提下(同质假设)给出篮子CDS价格的解析表达式。从而启发我们将这种模型应用于CDO定价中,给出了与前面不同的异质组合的CDO价格。最后还给出了CDO的算例,数值模拟出各层的价格。
Credit derivatives were the financial assets derived from the loan, the bond or other assetsin 1990s. They separated the credit risk from market risk and made the credit risk be tradeableand manageable. The traditional management mode was changed. The financial crisis in thelast century and the occurrence of American subprime mortgage crisis further showed that thecontagious risk was important for transferring and reorganizing the credit risk. Therefore, weneed the fair valuation of credit derivatives and make the theory more helpful. The study on theirpricing is fundamental. This thesis focuses on the pricing of bonds and credit default swaps withcounterparty risk and the pricing of Collateralized Debt Obligations with heterogeneous portfolio.
We study the pricing of bonds and credit default swaps from two aspects. One is that thedefaults of the assets depend on the short interest rate or not and the other is that we considerthe pricing respectively in the primary-secondary framework and looping default framework. Thefirst two chapters introduce the setting and notions of credit derivatives, the three correlated modeland three methods of computing the joint distribution of default times in the contagious model.Chapter 3 introduces the short interest rate and the forward interest rate in details. We take moreimportance on three types of the short interest rate: Vasicek model, CIR model and Hull-Whitemodel which are used in the following chapters. Chapter 4 and Chapter 5 discuss the pricing ofbonds and CDS from the two aspects above and give their explicit prices. Chapter 6 consider thepricing of CDO with heterogeneous portfolio and give an example. In this paper, we obtain somenew results by the intensity-based approach of default event as following: In this thesis, the mainresults are obtained with .
Firstly, we extend the models in Jarrow and Yu (2001). They proposed the counterparty riskwhich was different from the traditional intensity-based model and solved the problems ignoredby the traditionally structural model and reduced-form model. They introduced primary-secondarymodel and priced bonds and CDS with Vasicek interest rate. We extend their models and considerother interest rate following diffusion processes. On the one hand, we price bonds and CDS whenthe defaults of firms are independent of the default-free term structure. On the other hand, wederive the prices of bonds and CDS when the defaults of firms depend on the default-free termstructure.
Secondly, we extend the models in Bai, Hu and Ye (2007). They firstly introduced the conta- gious model with attenuation effect. We price bonds and CDS respectively in primary-secondarymodel and looping default model. Bai, Hu and Ye (2007,2008) only considered the interest rate asa constant and the Vasicek diffusion process. We discuss the other diffusion processes. By usingthe methods in Park (2008), we obtain some important differential equations on the interest rateand price bonds and CDS with a hyperbolic attenuation effect.
Thirdly, we further generalize the above models and is extended by adding a hyperbolic at-tenuation effect. We consider the interest rate satisfying jump-diffusion process and obtain somenew results. Moreover, We obtain one explicit solution for bonds and CDS in the similar modelabove. Meanwhile, we make use of Matlab and Monte-Carlos simulation to get the price curves oftheir.
At last, we study the contagious model of pricing CDO with heterogeneous portfolio. Thekey of pricing CDO is to valuate the credit risk of the reference assets. Until now, many results arebased on the homogeneous assumption, such as the conditionally independent model and Copulamodel. In this thesis, we apply the hyperbolic attenuation model of two parties in Bai, Hu andYe (2007) to price CDO and obtain the analytic form of CDO tranche’s price. In addition, weconsider the contagious model in Zheng and Jiang(2009). They only discussed the pricing of CDSwith many assets. We apply a comparatively simple model to the reference portfolio and obtainthe analytic price of CDO. Moreover, we give the example and the pricing curve of each tranche.
引文
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