组合信用风险模型及应用研究
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摘要
组合信用风险研究在信用资产风险管理和多标的信用衍生品定价方面具有重要作用。2008年金融危机爆发后，金融监管部门对银行等金融行业机构施加更加严格的监管，对它们的组合风险管理能力提出更高的要求；另一方面，金融危机的发生也进一步暴露了单因子高斯模型用于以合成CDO为代表的多标的信用衍生品定价所存在的缺陷。面对当前全球经济不景气以及信用质量恶化的大坏境，如何提出更合理的组合信用风险模型管理信用风险、评估信用衍生品价值已成为学术界和实业界共同关注的热点课题。
组合信用风险模型研究的核心在于如何刻画各个信用主体之间的相关性，进而建立起各主体信用迁移之间或违约之间的相关性。本文将针对现有模型在组合信用风险管理或多标的信用衍生品定价方面的某些不足，建立一些与市场经验事实更吻合的新模型。本文具体的主要工作和贡献有以下几点：
第一，运用分组关联函数为度量债券组合信用风险建模。考虑到债券组合的债券发行公司来自不同行业，尽管这些公司都受系统风险影响，但每个行业都有各自行业具体的风险，因此为债券组合信用风险建模时需考虑行业之间的差异。本文运用分组分组关联函数将行业先分类，然后再综合。该模型能使同行业公司之间的债券信用风险相关性大于不同行业之间的相关性。实证分析表明，分组关联函数模型比不考虑行业差异的关联函数模型更能刻画组合的极端风险。
第二，运用极值理论建立一个用于研究债券组合信用风险的离散形式首次通过时间模型。以公司的年度或半年最大负日对数收益率作为状态变量，然后运用极值理论得到各状态变量的分布函数；最后考虑到违约的行业、地区聚集性，利用多元极值关联函数——分层Gumbel关联函数建立状态变量之间的联系。该模型可看成是以Merton违约为基础的CreditMetrics模型在Black-Cox违约情形中的对应物。实证分析表明，该模型比CreditMetrics模型和不分层Gumbel关联函数模型所得违约损失的高分位数尾部更厚，即就压力测试而言，该模型相对更为审慎。
第三，在结构模型框架内，运用经济时间（时变）布朗运动建立非连续、具有跳跃性的资产价值过程以及考虑公司资产价值为Hawkes跳跃扩散过程情形下的违约相关性。由于存在违约突发性，我们建立一个以独立经济时间为“基底”，个体经济时间为“基底”线性组合的多元违约相关性模型。该模型能刻画违约突发性，且能刻画从违约独立到违约完全相依的各种关系。另一方面，由于金融市场中的资产价格存在波动率聚集现象，即资产价格大的波动常被另一个大的波动跟随，利用能刻画跳跃聚集的Hawkes过程取代泊松过程建立服从跳跃扩散过程的资产价值。模型中的每个资产价值过程中有两个独立的Hawkes过程，其中一个刻画公共的跳跃，另一个刻画个体的跳跃。从而通过资产价值过程中的相关布朗运动和公共Hawkes过程建立起违约相关性。数值结果表明，资产价值过程的公共期望跳跃次数的增加并一定会增加违约相关性，但公共跳跃的聚集程度越高，则违约相关系数越大；此外，个体跳跃聚集程度越高，则违约相关系数越小。
第四，利用MGB2分布为违约时间建模并用之为合成CDO定价。与期权市场中观察到的隐含波动率微笑类似，合成CDO市场中观察到隐含相关系数微笑，这说明市场基准模型——单因子高斯模型用于合成CDO定价不够理想。由于隐含相关系数并不始终存在，从而以基相关系数作为判断模型优劣的标准更为合理。从基相关系数角度看，MGB2模型具有很强的弹性；当模型中的参数变化时，它能生成各种形状的基相关系数。此外，与双t分布一样，MGB2模型能很好地拟合市场价格隐含的基相关系数，但市场标准模型和Clayton模型无法实现该点。
The study of portfolio credit risk plays an important role in credit risk management andcredit derivative pricing. Since the2008financial crisis, the financial institutions have beenfacing a series of more strict requirements on risk management implemented by the financialregulatory authorities, which means that they have to improve their abilities of managing port-folio risk. On the other hand, the financial crisis has revealed the shortcomings of one-factorGaussian model for pricing the portfolio credit derivative products of which the synthetic CDOis a typical one. In the circumstance of global economic gloom and deteriorated credit quality, itcurrently is a very popular topic in academy and industry that how to build the more reasonableportfolio credit risk models for managing the credit risk and evaluating the credit derivatives.
The core of portfolio credit risk modeling is to correlate all the entities, and then to buildthe dependence between the credit migrations of the entities or between defaults. Taking intoaccountthedisadvantagesoftheexistingmodelsforportfoliocreditriskmanagementandmulti-name credit derivative pricing, this paper aims to propose some new models so that they can fitthe empirical facts or market data better. The specific work and contributions of this thesis areas follows.
First, we apply grouped t-copula to study the default risk of the bond portfolio. Due to thefact that the bond issuers are from different industry sectors and each industry sector bears itsown specific risk, though they are subject to systematic risks, the difference between industrysectors should be taken into account when modeling the credit risk of a bond portfolio. Inthe thesis, we apply grouped t-copula to study the credit risk from sector to sector, and thencombine them together. This model means that the default dependence between firms from thesame industry sectors is stronger than that from different ones. Besides, the empirical analysisshows that the grouped t-copula model can characterize the extremal tail risk of loss distributionbetter than the usual t-copula.
Second, extreme value theroy is used to build a discrete-form first passage time credit riskmodel for studying the default risk of the bond portfolio. The annual or semi-annual maximumnegative daily return is regarded as the state variable in the model, and the distributions of themare derived by extreme value theory. Because it is observed in reality that defaults cluster in the same industry sector or in the same region, hierarchical Gumbel copula which is an extremevalue copula is used to link the marginal distributions of state varibles. The proposed modelcan be regarded as the counterpart of CreditMertrics under the framework of Black-Cox defaultmodel. The empirical analysis indicates that the high-quantile tail of the loss distribution result-ing from the proposed model is heavier than those from CreditMetrics and the usual Gumbelcopula models, namley, the proposed model is relatively more conservative in terms of stresstesting.
Third, under the framework of structural default model, default correlation is studied intwo cases in which asset processes are constrcuted by applying business time (time-changed)Brownian motions and Hawkes jump-diffusion processes, respectively. Due to the unexpect-edness of some defaults, we build a model in which a set of business times are taken as thebasis vector and the business time for each film is assumed to be the linear combination of thosebusiness times. The proposed model is able to characterize extensive default correlation fromindependence to perfect dependence. On the other hand, because of the existence of volatilityclustering in financial markets, that is, large changes tend to be followed by large changes ofeither sign, the asset value is modelled by Hawkes jump-diffusion process rather than Poissonjump-diffusion process, and they are correlated by common Hawkes process as well as correlat-ed Brownian motions. The numerical examples illustrate that the default correlation coefficientwon’t necessarily increase as the expected common jump times increase; however, the moreclustering of the jumps in the common Hawkes process are, the larger the default correlation is.What is more, the more clustering of the jumps in idiosyncratic Hawkes process, the less thedefault correlation is.
Forth, MGB2distribution is applied to model the default times and evaluate the syntheticCDO. Similar to the implied volatility smile arising in option market, implied correlation smileis observed in synthetic CDO market. That means that the standard model of pricing syntheticCDO which is one-factor Gaussian model is not good enough. Since implied correlation doesnot always exist, it is more reasonable to take base correlation as the tool to judge a model.From the perspective of base correlation, MGB2model is so flexible that it can generates manypatterns of base correlation with varying parameters. Furthermore, MGB2model can matchthe market implied base correlation as well as double t model does, which Gaussian model and Clayton model cannot make.
组合信用风险模型研究的核心在于如何刻画各个信用主体之间的相关性，进而建立起各主体信用迁移之间或违约之间的相关性。本文将针对现有模型在组合信用风险管理或多标的信用衍生品定价方面的某些不足，建立一些与市场经验事实更吻合的新模型。本文具体的主要工作和贡献有以下几点：
第一，运用分组关联函数为度量债券组合信用风险建模。考虑到债券组合的债券发行公司来自不同行业，尽管这些公司都受系统风险影响，但每个行业都有各自行业具体的风险，因此为债券组合信用风险建模时需考虑行业之间的差异。本文运用分组分组关联函数将行业先分类，然后再综合。该模型能使同行业公司之间的债券信用风险相关性大于不同行业之间的相关性。实证分析表明，分组关联函数模型比不考虑行业差异的关联函数模型更能刻画组合的极端风险。
第二，运用极值理论建立一个用于研究债券组合信用风险的离散形式首次通过时间模型。以公司的年度或半年最大负日对数收益率作为状态变量，然后运用极值理论得到各状态变量的分布函数；最后考虑到违约的行业、地区聚集性，利用多元极值关联函数——分层Gumbel关联函数建立状态变量之间的联系。该模型可看成是以Merton违约为基础的CreditMetrics模型在Black-Cox违约情形中的对应物。实证分析表明，该模型比CreditMetrics模型和不分层Gumbel关联函数模型所得违约损失的高分位数尾部更厚，即就压力测试而言，该模型相对更为审慎。
第三，在结构模型框架内，运用经济时间（时变）布朗运动建立非连续、具有跳跃性的资产价值过程以及考虑公司资产价值为Hawkes跳跃扩散过程情形下的违约相关性。由于存在违约突发性，我们建立一个以独立经济时间为“基底”，个体经济时间为“基底”线性组合的多元违约相关性模型。该模型能刻画违约突发性，且能刻画从违约独立到违约完全相依的各种关系。另一方面，由于金融市场中的资产价格存在波动率聚集现象，即资产价格大的波动常被另一个大的波动跟随，利用能刻画跳跃聚集的Hawkes过程取代泊松过程建立服从跳跃扩散过程的资产价值。模型中的每个资产价值过程中有两个独立的Hawkes过程，其中一个刻画公共的跳跃，另一个刻画个体的跳跃。从而通过资产价值过程中的相关布朗运动和公共Hawkes过程建立起违约相关性。数值结果表明，资产价值过程的公共期望跳跃次数的增加并一定会增加违约相关性，但公共跳跃的聚集程度越高，则违约相关系数越大；此外，个体跳跃聚集程度越高，则违约相关系数越小。
第四，利用MGB2分布为违约时间建模并用之为合成CDO定价。与期权市场中观察到的隐含波动率微笑类似，合成CDO市场中观察到隐含相关系数微笑，这说明市场基准模型——单因子高斯模型用于合成CDO定价不够理想。由于隐含相关系数并不始终存在，从而以基相关系数作为判断模型优劣的标准更为合理。从基相关系数角度看，MGB2模型具有很强的弹性；当模型中的参数变化时，它能生成各种形状的基相关系数。此外，与双t分布一样，MGB2模型能很好地拟合市场价格隐含的基相关系数，但市场标准模型和Clayton模型无法实现该点。
The study of portfolio credit risk plays an important role in credit risk management andcredit derivative pricing. Since the2008financial crisis, the financial institutions have beenfacing a series of more strict requirements on risk management implemented by the financialregulatory authorities, which means that they have to improve their abilities of managing port-folio risk. On the other hand, the financial crisis has revealed the shortcomings of one-factorGaussian model for pricing the portfolio credit derivative products of which the synthetic CDOis a typical one. In the circumstance of global economic gloom and deteriorated credit quality, itcurrently is a very popular topic in academy and industry that how to build the more reasonableportfolio credit risk models for managing the credit risk and evaluating the credit derivatives.
The core of portfolio credit risk modeling is to correlate all the entities, and then to buildthe dependence between the credit migrations of the entities or between defaults. Taking intoaccountthedisadvantagesoftheexistingmodelsforportfoliocreditriskmanagementandmulti-name credit derivative pricing, this paper aims to propose some new models so that they can fitthe empirical facts or market data better. The specific work and contributions of this thesis areas follows.
First, we apply grouped t-copula to study the default risk of the bond portfolio. Due to thefact that the bond issuers are from different industry sectors and each industry sector bears itsown specific risk, though they are subject to systematic risks, the difference between industrysectors should be taken into account when modeling the credit risk of a bond portfolio. Inthe thesis, we apply grouped t-copula to study the credit risk from sector to sector, and thencombine them together. This model means that the default dependence between firms from thesame industry sectors is stronger than that from different ones. Besides, the empirical analysisshows that the grouped t-copula model can characterize the extremal tail risk of loss distributionbetter than the usual t-copula.
Second, extreme value theroy is used to build a discrete-form first passage time credit riskmodel for studying the default risk of the bond portfolio. The annual or semi-annual maximumnegative daily return is regarded as the state variable in the model, and the distributions of themare derived by extreme value theory. Because it is observed in reality that defaults cluster in the same industry sector or in the same region, hierarchical Gumbel copula which is an extremevalue copula is used to link the marginal distributions of state varibles. The proposed modelcan be regarded as the counterpart of CreditMertrics under the framework of Black-Cox defaultmodel. The empirical analysis indicates that the high-quantile tail of the loss distribution result-ing from the proposed model is heavier than those from CreditMetrics and the usual Gumbelcopula models, namley, the proposed model is relatively more conservative in terms of stresstesting.
Third, under the framework of structural default model, default correlation is studied intwo cases in which asset processes are constrcuted by applying business time (time-changed)Brownian motions and Hawkes jump-diffusion processes, respectively. Due to the unexpect-edness of some defaults, we build a model in which a set of business times are taken as thebasis vector and the business time for each film is assumed to be the linear combination of thosebusiness times. The proposed model is able to characterize extensive default correlation fromindependence to perfect dependence. On the other hand, because of the existence of volatilityclustering in financial markets, that is, large changes tend to be followed by large changes ofeither sign, the asset value is modelled by Hawkes jump-diffusion process rather than Poissonjump-diffusion process, and they are correlated by common Hawkes process as well as correlat-ed Brownian motions. The numerical examples illustrate that the default correlation coefficientwon’t necessarily increase as the expected common jump times increase; however, the moreclustering of the jumps in the common Hawkes process are, the larger the default correlation is.What is more, the more clustering of the jumps in idiosyncratic Hawkes process, the less thedefault correlation is.
Forth, MGB2distribution is applied to model the default times and evaluate the syntheticCDO. Similar to the implied volatility smile arising in option market, implied correlation smileis observed in synthetic CDO market. That means that the standard model of pricing syntheticCDO which is one-factor Gaussian model is not good enough. Since implied correlation doesnot always exist, it is more reasonable to take base correlation as the tool to judge a model.From the perspective of base correlation, MGB2model is so flexible that it can generates manypatterns of base correlation with varying parameters. Furthermore, MGB2model can matchthe market implied base correlation as well as double t model does, which Gaussian model and Clayton model cannot make.
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