基于因子Copula的债务抵押债券定价模型研究
|
摘要
债务抵押债券(CDO)是一种新兴的组合类信用衍生产品,基于抵押债务的信用,通过资产证券化技术,将债券、贷款等金融资产组建资产池,重新分割投资回报和风险,以满足不同投资者的需要。自1987年由Drexel Burnham Lamert创立以来,该产品因其独特的收益风险特征得到迅速发展,成为最具典型的结构化金融创新工具。据美国证券行业和金融市场协会(SIFMA)统计,CDO近十年的全球发行量累计已超过两万亿美元。特别地,自2007年源于美国住房抵押贷款的金融危机以来,CDO的定价模型和风险管理得到了理论界和实务界的广泛关注。作为一种组合类信用衍生产品,CDO定价的关键是如何由标的资产的违约率、回收率、提前偿付率及各个资产之间的相关性得到资产池的损失分布。因此,本文直面金融危机以来的宏观经济背景和信用风险态势,紧扣CDO市场发展和研究热点,借鉴国内外现有的相关研究成果,着眼于CDO资产池损失的“尖峰厚尾”特征和“动态相关”特性,兼顾CDO标的资产的随机回收率和提前偿付率,构建了基于因子Copula的定价模型并进行了数值模拟,相关研究成果简述如下：
(1)针对标准高斯因子Copula模型的“相关性微笑”现象,将资产价值的市场共同因子和个体异质因子分别用混合高斯分布和标准高斯分布刻画,建立了随机相关结构条件下基于混合高斯因子Copula的CDO定价模型,有效减弱了“相关性微笑”现象对CDO定价准确性的影响。
在贝努利随机相关结构和三状态随机相关结构条件下,给出了CDO资产池损失分布的具体表达式,进而基于CDO分券层损失面的预期损失与收益面的预期收益相等的无套利定价原理,得出了CDO分券层在贝努利随机相关结构下的合理信用价差。
(2)针对CDO资产池损失分布的“尖峰厚尾”特征,将资产价值的市场共同因子和个体异质因子用标准高斯和NIG的混合分布刻画,利用半解析法和傅里叶变换及其逆变换等建立了随机相关结构和局部相关结构条件下的混合NIG因子Copula定价模型,弥补了CDO资产池损失的“尖峰厚尾”特征和“动态相关”特性等风险因素刻画方面的不足。
采用随机相关和局部相关两种形式刻画相关结构。随机相关结构中重点讨论贝努利相关和三状态相关两种情形；局部相关结构中主要分析相关系数服从两点分布的形式。利用傅里叶变换及其逆变换,得出了CDO资产池损失分布的求解方法。
(3)针对回收率为常数的CDO定价模型“相关性微笑”、高级分券层定价失效的不足,通过将回收率设定为系统性风险这一市场共同因子的函数,构建了考虑回收率随机特征的CDO因子Copula定价模型,使之适宜于高分券层价差的估算。
综合考虑了回收率的下列三种特征：随机回收率、市场共同因子服从混合高斯分布及用贝努利相关、三状态相关刻画的随机相关结构。通过对资产池违约边界、随机回收率和CDO分券层定价的数值模拟,发现：混合高斯分布可以有效地用于刻画市场共同因子这一系统性风险因素的尾部特征；随机回收率可较有效地用于刻画回收率与市场共同因子及违约相关结构的市场特征。上述特点为CDO高分券层价差的较合理估算奠定了基础。
(4)针对标的资产为信贷资产等有提前偿付风险的CDO产品,通过高斯OU强度过程,结合Levy因子Copula结构,构建了考虑提前偿付风险因素的CDO因子Copula定价模型,实现了CDO定价中提前偿付因素的风险度量。
利用高斯OU过程来模拟提前偿付强度和违约强度之间的动态相关性,并以Shifted Gamma过程为例,将因子Copula结构由标准高斯因子Copula推广到了Levy因子Copula,以更好地拟合金融市场的“厚尾”特征和“跳扩散”现象。在此基础上,得到了CDO资产池提前偿付分布和违约损失分布的计算方法和CDO分券层价差的半解析解。
综上所述,本文从CDO的因子Copula定价模型入手,着力考察了影响CDO定价准确性的厚尾因子Copula结构、相关系数、违约回收率和提前偿付率等四个因素,藉此丰富完善了因子Copula定价模型。特别地,数值模拟表明,本文构建的因子Copula定价模型较现有模型在解决CDO定价中的“相关性微笑”、高分券层定价失效和提前偿付风险度量等方面有一定的优势,相应的研究成果有益于为金融机构CDO等结构化金融工具的产品设计与风险管理提供新思路。
As a new class of credit derivatives portfolio, Collateralized Debt Obligations (CDO) securitizes bonds, loans and other financial assets into an assets pool to re-split return and risk for diversified investors, based on mortgage debt credit. Since first issuance in 1987 by Drexel Burnham Lambert, CDO has got a tremendous growth due to its unique profit/risk characteristics and has become the most typical structured instrument of financial innovations. According to statistics of Securities Industry and Financial Markets Association (SIFMA), the total global CDO issuance in recent 10 years has more than 2 trillion U.S. dollars. In particular, since the 2007 financial crisis originated by the U.S. Subprime Mortgage, CDO pricing models and risk management have been widespread community concerned in theory and practice. As a credit derivatives portfolio, the key of CDO pricing is how to get the loss distribution of assets pool by default rates, recovery rates, prepayment rates of the underlying assets and correlations between different underlying assets. Therefore, facing the macroeconomic background and credit risk situation after the financial crisis, closely concentrating on CDO market development and research focus, referring to existing relevant research outcomes, focusing on the fat tail and dynamic correlation characteristics of the CDO assets pool loss, and accounting for the random recovery rates and prepayment rates of CDO underlying assets, this paper constructed several CDO pricing models based on factor Copula and calibrated the models by numerical simulations. The relevant research results are summarized below:
(1) Aiming at the "Correlation Smile" in the standard Gaussian Factor Copula model, this dissertation expresses the common market factor as the mixtures of Gaussian distributions and idiosyncratic factor as standard Gaussian distribution, and constructs CDO Pricing model based on Gaussian mixture factor Copula of the stochastic correlation structures. All of the above effectively amends CDO pricing's error caused by the "Correlation Smile".
Under the conditions of Bernoulli stochastic correlation structure and three-state stochastic correlation structure, this dissertation gives the specific expression of the cumulative default loss probability distribution for the entire assets pool, based on the no-arbitrage pricing theory where CDO tranches'expected loss is equal to its expected premium, then obtains CDO tranches'credit spread of Bernoulli stochastic correlation structure.
(2) Aiming at the fat tail characteristics of the CDO assets pool loss, this dissertation expresses the common market factor and idiosyncratic factor as the mixtures of standard Gaussian distributions and NIG distribution, constructs CDO Pricing model based on NIG mixture factor Copula of the stochastic correlation structure and local correlation structure by the semi-analytical method and the Fourier transform and its inverse transform, and improves the risk measure of the fat tail and dynamic correlation characteristics of the CDO assets pool loss.
We consider two correlation structures:the stochastic correlation structure and the local correlation structure, focus on Bernoulli stochastic correlation structure and three-state stochastic correlation structure for the stochastic correlation structure; analyze the two points distribution case for local correlation structure, and derive the CDO assets pool cumulative loss by Fourier transform and its inverse transform.
(3) Aiming at the "Correlation Smile" and senior tranches'pricing failure in the standard Gaussian Factor Copula model where the recovery rate is a deterministic constant, this dissertation expresses the recovery rate as a function of common market factor for systemic risk, and constructs CDO factor Copula Pricing model based on random recovery to suitable for senior tranches'spread pricing.
We consider the recovery with the following three features:random recovery rate, the common market factor as the mixtures of Gaussian distributions, Bernoulli and three-state stochastic correlation structure. After the numerical simulations for default barriers of assets pool, random recovery and CDO tranche pricing, it is shown that the model with Gaussian mixture factor copula can capture fat-tails, and the random recovery model can be used to describe the system risk and default correlations effectively. The features above lay the foundation for more reasonable estimate of CDO senior tranches spread.
(4) Aiming at CDO products based on loans and other underlying assets with prepayment risk, using the Gaussian OU intensities process and the Levy factor Copula structure, we construct a CDO factor Copula pricing model based on prepayment risk, and realize the risk measure of prepayment risk in CDO pricing models.
By capturing the negative correlation between the default and prepayment intensities via the Gaussian OU process, this dissertation introduces the Shifted Gamma process to extend the standard Gaussian factor Copula model to Levy factor Copula model in order to better fit the fat-fail and jump diffusion of financial markets. On this basis, it derives the calculating method for the prepayment distribution and default loss distribution of CDO assets pool, and then gets the semi-analytical solution of CDO tranches'spread.
In summary, focusing on CDO factor Copula pricing model, this paper studied the risk factors of CDO pricing accuracy by the fat-tail factor Copula, correlation coefficient, recovery rate and prepayment rates, enriched and improved the factor Copula pricing model. In particular, numerical simulations shows that our factor Copula pricing models better fit the "Correlation Smile", senior tranches'pricing failure and risk measure of prepayment risk in CDO pricing models compared with the existing models, the corresponding research results are useful to provide new ideas about products design and risk management of CDOs and other structured financial instruments for financial institutions.
(1)针对标准高斯因子Copula模型的“相关性微笑”现象,将资产价值的市场共同因子和个体异质因子分别用混合高斯分布和标准高斯分布刻画,建立了随机相关结构条件下基于混合高斯因子Copula的CDO定价模型,有效减弱了“相关性微笑”现象对CDO定价准确性的影响。
在贝努利随机相关结构和三状态随机相关结构条件下,给出了CDO资产池损失分布的具体表达式,进而基于CDO分券层损失面的预期损失与收益面的预期收益相等的无套利定价原理,得出了CDO分券层在贝努利随机相关结构下的合理信用价差。
(2)针对CDO资产池损失分布的“尖峰厚尾”特征,将资产价值的市场共同因子和个体异质因子用标准高斯和NIG的混合分布刻画,利用半解析法和傅里叶变换及其逆变换等建立了随机相关结构和局部相关结构条件下的混合NIG因子Copula定价模型,弥补了CDO资产池损失的“尖峰厚尾”特征和“动态相关”特性等风险因素刻画方面的不足。
采用随机相关和局部相关两种形式刻画相关结构。随机相关结构中重点讨论贝努利相关和三状态相关两种情形；局部相关结构中主要分析相关系数服从两点分布的形式。利用傅里叶变换及其逆变换,得出了CDO资产池损失分布的求解方法。
(3)针对回收率为常数的CDO定价模型“相关性微笑”、高级分券层定价失效的不足,通过将回收率设定为系统性风险这一市场共同因子的函数,构建了考虑回收率随机特征的CDO因子Copula定价模型,使之适宜于高分券层价差的估算。
综合考虑了回收率的下列三种特征：随机回收率、市场共同因子服从混合高斯分布及用贝努利相关、三状态相关刻画的随机相关结构。通过对资产池违约边界、随机回收率和CDO分券层定价的数值模拟,发现：混合高斯分布可以有效地用于刻画市场共同因子这一系统性风险因素的尾部特征；随机回收率可较有效地用于刻画回收率与市场共同因子及违约相关结构的市场特征。上述特点为CDO高分券层价差的较合理估算奠定了基础。
(4)针对标的资产为信贷资产等有提前偿付风险的CDO产品,通过高斯OU强度过程,结合Levy因子Copula结构,构建了考虑提前偿付风险因素的CDO因子Copula定价模型,实现了CDO定价中提前偿付因素的风险度量。
利用高斯OU过程来模拟提前偿付强度和违约强度之间的动态相关性,并以Shifted Gamma过程为例,将因子Copula结构由标准高斯因子Copula推广到了Levy因子Copula,以更好地拟合金融市场的“厚尾”特征和“跳扩散”现象。在此基础上,得到了CDO资产池提前偿付分布和违约损失分布的计算方法和CDO分券层价差的半解析解。
综上所述,本文从CDO的因子Copula定价模型入手,着力考察了影响CDO定价准确性的厚尾因子Copula结构、相关系数、违约回收率和提前偿付率等四个因素,藉此丰富完善了因子Copula定价模型。特别地,数值模拟表明,本文构建的因子Copula定价模型较现有模型在解决CDO定价中的“相关性微笑”、高分券层定价失效和提前偿付风险度量等方面有一定的优势,相应的研究成果有益于为金融机构CDO等结构化金融工具的产品设计与风险管理提供新思路。
As a new class of credit derivatives portfolio, Collateralized Debt Obligations (CDO) securitizes bonds, loans and other financial assets into an assets pool to re-split return and risk for diversified investors, based on mortgage debt credit. Since first issuance in 1987 by Drexel Burnham Lambert, CDO has got a tremendous growth due to its unique profit/risk characteristics and has become the most typical structured instrument of financial innovations. According to statistics of Securities Industry and Financial Markets Association (SIFMA), the total global CDO issuance in recent 10 years has more than 2 trillion U.S. dollars. In particular, since the 2007 financial crisis originated by the U.S. Subprime Mortgage, CDO pricing models and risk management have been widespread community concerned in theory and practice. As a credit derivatives portfolio, the key of CDO pricing is how to get the loss distribution of assets pool by default rates, recovery rates, prepayment rates of the underlying assets and correlations between different underlying assets. Therefore, facing the macroeconomic background and credit risk situation after the financial crisis, closely concentrating on CDO market development and research focus, referring to existing relevant research outcomes, focusing on the fat tail and dynamic correlation characteristics of the CDO assets pool loss, and accounting for the random recovery rates and prepayment rates of CDO underlying assets, this paper constructed several CDO pricing models based on factor Copula and calibrated the models by numerical simulations. The relevant research results are summarized below:
(1) Aiming at the "Correlation Smile" in the standard Gaussian Factor Copula model, this dissertation expresses the common market factor as the mixtures of Gaussian distributions and idiosyncratic factor as standard Gaussian distribution, and constructs CDO Pricing model based on Gaussian mixture factor Copula of the stochastic correlation structures. All of the above effectively amends CDO pricing's error caused by the "Correlation Smile".
Under the conditions of Bernoulli stochastic correlation structure and three-state stochastic correlation structure, this dissertation gives the specific expression of the cumulative default loss probability distribution for the entire assets pool, based on the no-arbitrage pricing theory where CDO tranches'expected loss is equal to its expected premium, then obtains CDO tranches'credit spread of Bernoulli stochastic correlation structure.
(2) Aiming at the fat tail characteristics of the CDO assets pool loss, this dissertation expresses the common market factor and idiosyncratic factor as the mixtures of standard Gaussian distributions and NIG distribution, constructs CDO Pricing model based on NIG mixture factor Copula of the stochastic correlation structure and local correlation structure by the semi-analytical method and the Fourier transform and its inverse transform, and improves the risk measure of the fat tail and dynamic correlation characteristics of the CDO assets pool loss.
We consider two correlation structures:the stochastic correlation structure and the local correlation structure, focus on Bernoulli stochastic correlation structure and three-state stochastic correlation structure for the stochastic correlation structure; analyze the two points distribution case for local correlation structure, and derive the CDO assets pool cumulative loss by Fourier transform and its inverse transform.
(3) Aiming at the "Correlation Smile" and senior tranches'pricing failure in the standard Gaussian Factor Copula model where the recovery rate is a deterministic constant, this dissertation expresses the recovery rate as a function of common market factor for systemic risk, and constructs CDO factor Copula Pricing model based on random recovery to suitable for senior tranches'spread pricing.
We consider the recovery with the following three features:random recovery rate, the common market factor as the mixtures of Gaussian distributions, Bernoulli and three-state stochastic correlation structure. After the numerical simulations for default barriers of assets pool, random recovery and CDO tranche pricing, it is shown that the model with Gaussian mixture factor copula can capture fat-tails, and the random recovery model can be used to describe the system risk and default correlations effectively. The features above lay the foundation for more reasonable estimate of CDO senior tranches spread.
(4) Aiming at CDO products based on loans and other underlying assets with prepayment risk, using the Gaussian OU intensities process and the Levy factor Copula structure, we construct a CDO factor Copula pricing model based on prepayment risk, and realize the risk measure of prepayment risk in CDO pricing models.
By capturing the negative correlation between the default and prepayment intensities via the Gaussian OU process, this dissertation introduces the Shifted Gamma process to extend the standard Gaussian factor Copula model to Levy factor Copula model in order to better fit the fat-fail and jump diffusion of financial markets. On this basis, it derives the calculating method for the prepayment distribution and default loss distribution of CDO assets pool, and then gets the semi-analytical solution of CDO tranches'spread.
In summary, focusing on CDO factor Copula pricing model, this paper studied the risk factors of CDO pricing accuracy by the fat-tail factor Copula, correlation coefficient, recovery rate and prepayment rates, enriched and improved the factor Copula pricing model. In particular, numerical simulations shows that our factor Copula pricing models better fit the "Correlation Smile", senior tranches'pricing failure and risk measure of prepayment risk in CDO pricing models compared with the existing models, the corresponding research results are useful to provide new ideas about products design and risk management of CDOs and other structured financial instruments for financial institutions.
引文
[1]Cifuentes A, Connor G 0. The Binomial Expension Method Applied to CBO/CLO Analysis[R]. Moody's Investors Service, Newyork. Moody's Special Report, Dec 13th 1996.
[2]McGinty L, Beinstein E, Ahluwalia R et al. Introducing Base Correlations[R]. JP Morgan, Newyork. JP Morgan,2004.
[3]Li D X. On Default Correlation:a Copula Approach [J]. Journal of Fixed Income,2000, 9(4):43-54.
[4]Schonbucher P J, Schubert D. Copula-Dependent Defaults in Intensity Models[EB/OL]. December 2001, http://ssrn. com/abstract=301968,2007.
[5]Mashal R, Naldi M, Zeevi A. On the Dependence of Equity and Asset Returns[J]. Risk, 2003,16(10):83-87.
[6]Andersen L, Sidenius J, Basu S. All your Hedges in One Basket [J]. Risk,2003,16(11): 67-72.
[7]Demarta S, McNeil A J. The t Copula and Related Copulas[J]. International Statistical Review,2005,73(1):111-129.
[8]Embrechts P, Lindskog F, McNeil A. Modelling Dependence with Copulas and Applications to Risk Management [A]. Handbook of Heavy Tailed Distributions in Finance, Amsterdam, North-Holland Publishing Company:Rachev S, Elsevier,2003:329-384.
[9]Frey R, McNeil A. Dependent Defaults in Models of Portfolio Credit Risk[J]. Journal of Risk,2003,6(1):59-92.
[10]0'Kane D, Schloegl L. A Note on the Large Homogeneous Portfolio Approximation with the Student-t Copula[J]. Finance and Stochastics,2005,9(4):577-584.
[11]Lindskog F, McNeil A. Common Poisson Shock Models:Applications to Insurance and Credit Risk Modelling[J], ASTIN Bulletin,2003,33(2):209-238.
[12]Mashal R, Zeevi A. Inferring the Dependence Structure of Financial Assets:Empirical Evidence and Implications[EB/OL].2003, http://www. faculty. idc. ac. il/roy/,2007.
[13]Greenberg A, Mashal R, Naldi M et al. Tuning Correlation and Tail Risk to the Market Prices of Liquid Tranches[R]. Lehman Brothers:Newyork, Lehman Brothers, Quantitative Research Quarterly,2004.
[14]Schonbucher P J. Taken to the Limit:Simple and Not-So-Simple Loan Loss Distributions[EB/OL]. August 2002, http://ssrn. com/abstract=378640,2007.
[15]Rogge E, Schonbucher P J. Modelling Dynamic Portfolio Credit Risk [EB/OL]. April 2002, http://www. schonbucher. de/papers,2007.
[16]Friend A, Rogge E. Correlation at First Sight [J], Economic Notes,2005,34(2):155-183.
[17]Laurent J P, Gregory J. Basket Default Swaps, CDOs and Factor Copulas[J]. Journal of Risk,2005 7(4):103-122.
[18]Gregory J, Laurent J P. I Will Survive[J]. Risk,2003,16(6):103-107.
[19]Madan D B, Konikov M, Marinescu M. Credit and basket default swaps[J]. Journal of Credit. Risk,2006,2(1):67-87.
[20]Duffie D, Singleton K J. Simulating correlated defaults[EB/OL]. May 1999, http://www. stanford, edu/duffie/working. htm,2007.
[21]Wong D. Copula from the Limit of a Multivariate Binary Model[EB/OL]. December 2000, http://www. defaultrisk. com/pp_corr_25. htm,2007.
[22]Giesecke K. A Simple Exponential Model for Dependent Defaults[J]. Journal of Fixed Income,2003,13(3):74-83.
[23]Elouerkhaoui Y. Credit Risk:Correlation with a Difference[EB/OL].2003, http://ralyx. inria. fr/,2007.
[24]Hull J, White A. Valuation of a CDO and an nth to Default CDS without Monte Carlo Simulation[J]. Journal of Derivatives,2004,12(2):8-23.
[25]Burtschell X, Gregory J and Laurent J P. Beyond the Gaussian copula:Stochastic and Local Correlation[J] Journal of Credit Risk,2007,3(1):31-62.
[26]Andersen L, Sidenius J. Extensions to the Gaussian Copula:Random Recovery and Random Factor Loadings [J]. Journal of Credit Risk,2005, 1(1):29-70.
[27]Gregory J, Laurent J P. In the Core of Correlation [J]. Risk,2004,17(10):87-91.
[28]Gordy M, Jones D. Random Tranches[J]. RISK,2003,16(3):78-83.
[29]Davis M, Lo V. Infectious Defaults [J]. Quantitative Finance,2001,1(4):382-387.
[30]Jarrow R, Yu F. Counterparty Risk and the Pricing of Defaultable Securities [J]. Journal of Finance,2001,56(5):1765-1799.
[31]Kalemanova A, Schmid B, Werner R. The Normal Inverse Gaussian Distribution for Synthetic CDO Pricing[J]. Journal of Derivatives,2007,14(3):80-93.
[32]Thomas M. Pricing CDOs with Correlated Variance Gamma Distributions[EB/OL]. February 2006, http://www. defaultrisk. com/pp_crdrv103. htm,2007.
[33]Henrik B. Comparative Analysis of Hyperbolic Copulas Induced by a One Factor Levy Model[EB/OL]. August,2006, http://www. defaultrisk. com/pp_CDO_17. htm,2007.
[34]Prange D, Scherer W. Correlation Smile Matching with Alpha-Stable Distributions and Fitted Archimedan Copula Models[EB/OL]. March,2006, http://www. defaultrisk. com/pp_corr_46. htm,2007.
[35]Burtschell X, Gregory, Laurent J P.. Beyond the Gaussian Copula:Stochastic and Local Correlation[EB/OL]. January 2007, http://www. defaultrisk. com/pp_corr_76. htm,2007.
[36]Turc J, Very P, Benhamou D. Pricing CDOs with A Smile[R]. Societe Generale:Paris, Societe Generale Credit Research, February 2005.
[37]Pykhtin M, Dev A. Coarse-Grained CDOs [J]. Risk,2003,16(1):113-116.
[38]Debuysscher A, Szego M. The Fourier Transform Method-Technical Document[R]. Moody's Investors Service, Newyork, Moody's,2003.
[39]Albanese C, Chen O, Dalessandro A et al. Dynamic Credit Correlation Modeling [EB/OL]. May,2006, http://www. defaultrisk. com/pp_corr_75. htm,2007.
[40]Baxter M. Levy Process Dynamic Modeling of Single Name Credits and CDO Tranches [EB/OL]. April,2006, http://www. defaultrisk. com/pp_crdrvll0. htm,2007.
[41]Hull J, White A. Valuing Credit Derivatives Using an Implied Copula Approach[J]. Journal of Derivatives,2006,14(2):8-28.
[42]Totouom D, Armstrong M. Dynamic Copula Processes:A new way of modelling CDO tranches[EB/OL]. November 2005, http://www. defaultrisk. com/pp_crdrv_99. htm,2007.
[43]Chapovsky A, Rennie A, Tavares P A C. Stochastic Intensity Modelling for Structured Credit Exotics[EB/OL]. October,2006, http://www. defaultrisk. com/pp_crdrvl36. htm,2007.
[44]Graziano G, Rogers L. A Dynamic Approach to the Modeling of Correlation Credit Derivatives using Markov Chains[EB/OL]. November,2006, http://www. defaultrisk.com/pp_crdrv_88. htm,2007
[45]Hurd T R, Kuznetsov A. Fast CDO Computations in the Affine Markov Chain Model [EB/OL]. November,2006, http://www. defaultrisk. com/pp_crdrv_65. htm,2007.
[46]Joshi M S, Stacey A M. Intensity Gamma:A New Approach to Pricing Portfolio Credit Derivatives [J], Risk,2006,16(6):78-83.
[47]J. Sidenius, V. Piterbrag, L. Andersen. A New Framework for Dynamic Credit Portfolio. Loss Modeling[EB/OL]. June,2006, http://www. defaultrisk. com/pp_model_83. htm,2007.
[48]Bennani N. The Forward Loss Model:A Dynamic Term Structure Approach for the Pricing of Portfolio Credit Derivatives[EB/OL]. November,2005, http://www. defaultrisk. com/pp_crdrv_95. htm,2007
[49]Walker M B. CDO Valuation:Term Structure, Tranche Structure, and Loss Distributions[EB/OL]. January,2007, http://www. defaultrisk. com/pp_crdrv109. htm,2007.
[50]Schonbucher P J. Portfolio Losses and the Term Structure of Loss Transition Rates: A new Methodology for The Pricing of Portfolio Credit Derivatives [EB/OL]. June 2006, http://www. defaultrisk. com/pp_model_74. htm,2007.
[51]Errais E, Goldberg L R, Giesecke K. Pricing Credit from the Top Down with Affine Point Processes[EB/OL]. September,2007, http://www. defaultrisk. com/pp_CDO_16. htm,2007.
[52]Longsyaff F A, Rajan A. An Empirical Analysis of the Pricing of Collateralized Debt Obligations[J]. Journal of Finance,2008,63(2):529-563.
[53]Brigo D, Pallavicini A, Torresetti R. Calibration of CDO Tranches with The Dynamical Generalized-Poisson Loss Model[J]. Risk,2007,16(5):70-75.
[54]梁钧,黄智猛,黄培清.资产支持证券设计[J].预测,2000,1：60-62.
[55]朱世武.基于Copula函数度量违约相关性[J].统计研究,2005,4：61-64.
[56]冯谦,杨朝军.担保债权凭证定价——Copula函数的非参数估计与应用[J].运筹与管理,2006,15(5)：104-107.
[57]胡新华.含交易对手风险的公司债券和信用衍生品的定价问题研究[D].上海：上海交通大学理学院数学系,2007.
[58]刘冬.抵押债务证券(CDO)的信用风险分析[D].成都：西南财经大学金融研究中心,2007.
[59]袁子甲,李仲飞.基于因子模型的CDO定价研究：止态单因子模型的三种推广[EB/OL]. 03,2007, http://www. ccfr. org. cn/cicf2007/download. php?paper=20070119163654. PDF,2007.
[60]张维,邱勇,郝刚.基于多银行贷款池的合成CDO设计[J].现代财经,2007,27(9)：6-13.
[61]穆放,宋曼洁,陈治津.债务抵押债券定价模型探讨及实施研究[J].经济纵横,2007,10：10-13.
[62]Li D X. On Default Correlation:A Copula Approach. Journal of Fixed Income,2000, 9(3):43-54.
[63]Laurent J P, Gregory J. Basket default swaps, CDOs and factor copulas. Working paper, http://www. defaultrisk. com/pp_crdrv_25. htm,2003.
[64]Hull J, White A. Valuation of a CDO and an n-th to Default CDS without Monte Carlo Simulation. Journal of Derivatives.2004,12(2):8-23.
[65]Kalemanova A, Schmid B, Werner R. The Normal Inverse Gaussian Distribution for Synthetic CDO Pricing. Working paper, http://www. defaultrisk. com/pp_crdrv_91. htm, 2005.
[66]Giesecke K. A Simple Exponential Model for Dependent Defaults, Journal of Fixed Income, 2003,13(3):74-83.
[67]Ferrarese C. A Comparative Analysis of Correlation Skew Modeling Techniques for CDO Index Tranches. Working paper, http://www. defaultrisk. com/pp_crdrv_91. htm,2006.
[68]Burtschell X, Gregory J, and Laurent J P. Beyond the Gaussian Copula:Stochastic and Local Correlation,. Working paper, http://laurent.jeanpaul.free.fr/ stochastic_local_correlation. pdf,2005.
[69]Xu G, Extending Gaussian Copula with Jumps to Match Correlation Smile. Working paper, http://www. defaultrisk. com/pp_corr100. htm,2006.
[70]Schonbucher P J. Credit Derivatives Pricing Models:Models, Pricing, and Implementation[M]. Wiley,2003.
[71]冯谦,杨朝军.担保债权凭证定价——Copula函数的非参数估计与应用[J].运筹与管理,2006,15(5)：104-107.
[72]Li D X. On Default Correlation:a Copula Approach[J]. Journal of Fixed Income,2000, 9(3):43-54.
[73]Hull J, White A. Valuation of a CDO and an nth to Default CDS without Monte Carlo Simulation[J], Journal of Derivatives,2004,12(2):8-23.
[74]Andersen L, Sidenius J and Basu S. All Your Hedges in One Basket [J], Risk,2003,16(11): 67-72.
[75]Gregory J, Laurent J P. In the Core of Correlation [J], Risk,2004,17(10):87-91.
[76]Andersen L, Sidenius J. Extensions to the Gaussian Copula:Random Recovery and Random Factor Loadings[J], Journal of Credit Risk,2005,1(1):29-70.
[77]Ferrarese C. A Comparative Analysis of Correlation Skew Modeling Techniques for CDO Index Tranches(workingpaper), http://www. defaultrisk. com/pp\_CDO\_02. htm,2006.
[78]Cherubini U, Luciano E, Vecchiato V. Copula Methodsin Finance[M]. New York:Wiley Finance,2004.
[79]Nolan J P. Numerical Calculation of Stable Densities and Distribution Functions[J], Commun. Statist.-Stochastic Models,1997,13(4):759-774.
[80]Kalemanova A, Schmid B, Werner R. The Normal Inverse Gaussian Distribution for Synthetic CDO[J], Journal of Derivatives,2007,14(3):80-93.
[81]Laurent J P, Gregory J. Basket Default Swaps, CDOs and Factor Copulas[J], Journal of Risk,2005,7(4):103-122.
[82]Barndorff N. Process of Normal Inverse Gaussian type [J], Finance and Stochastic,1997, 2(1):41-68.
[83]Bracewell R N. The Fourier Transform & Its Applications[M]. New York:McGraw-Hill Companies Inc.,2002.
[84]Ech-Chatbi C. CDS and CDO Pricing with Stochastic Recovery[R]. Working Paper,2008.
[85]Andersen L, Sidenius J. Extensions to the Gaussian Copula:Random Recovery and Random Factor Loadings[J].2004,1(1):29-70.
[86]Krekel M. Pricing Distressed CDOs with Base Correlation and Stochastic Recovery[R]. Working Paper,2008.
[87]Amraoui S, Hitier S. Optimal Stochastic Recovery for Base Correlation[R]. Working Paper, 2008.
[88]Elizalde A. Credit risk Models IV:Understanding and Pricing CDOs[R]. Working Paper, 2006.
[89]Kalemanova A, Schmid B, Werner R. The Normal Inverse Gaussian Distribution for Synthetic CDO Pricing[J]. Journal of Derivatives,2007,14(3):80-93.
[90]Vasicek O. Probability of Loss on Loan Portfolio[R]. Working Paper,1987.
[91]Vasicek O. Limiting Loan Loss Distribution[R]. Working Paper,1991.
[92]Li D. On Default Correlation:a Copula Function Approach[J]. Journal of Fixed Income, 2000,9(4):43-54.
[93]Burtschell X, Gregory J, Laurent J P. Beyond the Gaussian Copula:Stochastic and Local Correlation [J]. Journal of Credit Risk,2007,3(1):31-62.
[94]Xu G. Extending Gaussian Copula with Jumps to Match Correlation Smile[R]. Working Paper, 2006.
[95]杨瑞成,秦学志,陈田.随机相关结构下单因子混合高斯模型CDO定价问题[J].大连理工大学学报,2009,49(4)：587-592.
[96]Shek H, Uematsu S, Wei Zh. Valuation of Loan CDS and CDX[R]. Working Paper,2007.
[97]Dobranszky P, Schoutens W. Generic Levy One-factor Models for the Joint Modelling of Prepayment and Default:Modelling LCDX[R]. Working Paper,2008.
[98]Wei Zh. Valuation of Loan CDS Under Intensity Based Model[R]. Working Paper,2007.
[99]Dobranszky P. Joint Modelling of CDS and LCDS Spreads with Correlated Default and Prepayment Intensities and with Stochastic Recovery Rate[R]. Working Paper,2008.
[100]Duffie D, Pan J, Singleton K. Transform Analysis and Asset Pricing for Affine Jump-diffusions[J]. Econometrica,2000,68(6):1343-1376.
[101]Prabhu N. U. Stochastic Storage Processes[M]. New York:Springer,1998.
[102]Barndorff-Nielsen O. E., Mikosch T, Resnich S. Levy process:Theory and applications[M]. Boston:Birkhauser,2001.
[103]Kyprianou A. E. Introductory Lectures on Fluctuations of Levy Processes with Applications[M]. Berlin:Springer,2006.
[104]Berton J. Levy Process[M]. Cambridge:Cambridge University Press,1996.
[105]Schoutens W. Levy Processes in Finance-Pricing Financial Derivatives[M]. Chichester: Wiley,2003.
[106]Albrecher H, Ladoucette S, Schoutens W. A Generic One Factor Levy Model for Pricing Synthetic CDOs[M]. In:Advances in Mathematical Fianance. Boston:Birkhauser,2007: 259-277.
[107]Garcia J, Goossens S, Masol V et al. Levy Base Correlation [J]. Wilmott Journal.2009, 1(2):95-100.
[2]McGinty L, Beinstein E, Ahluwalia R et al. Introducing Base Correlations[R]. JP Morgan, Newyork. JP Morgan,2004.
[3]Li D X. On Default Correlation:a Copula Approach [J]. Journal of Fixed Income,2000, 9(4):43-54.
[4]Schonbucher P J, Schubert D. Copula-Dependent Defaults in Intensity Models[EB/OL]. December 2001, http://ssrn. com/abstract=301968,2007.
[5]Mashal R, Naldi M, Zeevi A. On the Dependence of Equity and Asset Returns[J]. Risk, 2003,16(10):83-87.
[6]Andersen L, Sidenius J, Basu S. All your Hedges in One Basket [J]. Risk,2003,16(11): 67-72.
[7]Demarta S, McNeil A J. The t Copula and Related Copulas[J]. International Statistical Review,2005,73(1):111-129.
[8]Embrechts P, Lindskog F, McNeil A. Modelling Dependence with Copulas and Applications to Risk Management [A]. Handbook of Heavy Tailed Distributions in Finance, Amsterdam, North-Holland Publishing Company:Rachev S, Elsevier,2003:329-384.
[9]Frey R, McNeil A. Dependent Defaults in Models of Portfolio Credit Risk[J]. Journal of Risk,2003,6(1):59-92.
[10]0'Kane D, Schloegl L. A Note on the Large Homogeneous Portfolio Approximation with the Student-t Copula[J]. Finance and Stochastics,2005,9(4):577-584.
[11]Lindskog F, McNeil A. Common Poisson Shock Models:Applications to Insurance and Credit Risk Modelling[J], ASTIN Bulletin,2003,33(2):209-238.
[12]Mashal R, Zeevi A. Inferring the Dependence Structure of Financial Assets:Empirical Evidence and Implications[EB/OL].2003, http://www. faculty. idc. ac. il/roy/,2007.
[13]Greenberg A, Mashal R, Naldi M et al. Tuning Correlation and Tail Risk to the Market Prices of Liquid Tranches[R]. Lehman Brothers:Newyork, Lehman Brothers, Quantitative Research Quarterly,2004.
[14]Schonbucher P J. Taken to the Limit:Simple and Not-So-Simple Loan Loss Distributions[EB/OL]. August 2002, http://ssrn. com/abstract=378640,2007.
[15]Rogge E, Schonbucher P J. Modelling Dynamic Portfolio Credit Risk [EB/OL]. April 2002, http://www. schonbucher. de/papers,2007.
[16]Friend A, Rogge E. Correlation at First Sight [J], Economic Notes,2005,34(2):155-183.
[17]Laurent J P, Gregory J. Basket Default Swaps, CDOs and Factor Copulas[J]. Journal of Risk,2005 7(4):103-122.
[18]Gregory J, Laurent J P. I Will Survive[J]. Risk,2003,16(6):103-107.
[19]Madan D B, Konikov M, Marinescu M. Credit and basket default swaps[J]. Journal of Credit. Risk,2006,2(1):67-87.
[20]Duffie D, Singleton K J. Simulating correlated defaults[EB/OL]. May 1999, http://www. stanford, edu/duffie/working. htm,2007.
[21]Wong D. Copula from the Limit of a Multivariate Binary Model[EB/OL]. December 2000, http://www. defaultrisk. com/pp_corr_25. htm,2007.
[22]Giesecke K. A Simple Exponential Model for Dependent Defaults[J]. Journal of Fixed Income,2003,13(3):74-83.
[23]Elouerkhaoui Y. Credit Risk:Correlation with a Difference[EB/OL].2003, http://ralyx. inria. fr/,2007.
[24]Hull J, White A. Valuation of a CDO and an nth to Default CDS without Monte Carlo Simulation[J]. Journal of Derivatives,2004,12(2):8-23.
[25]Burtschell X, Gregory J and Laurent J P. Beyond the Gaussian copula:Stochastic and Local Correlation[J] Journal of Credit Risk,2007,3(1):31-62.
[26]Andersen L, Sidenius J. Extensions to the Gaussian Copula:Random Recovery and Random Factor Loadings [J]. Journal of Credit Risk,2005, 1(1):29-70.
[27]Gregory J, Laurent J P. In the Core of Correlation [J]. Risk,2004,17(10):87-91.
[28]Gordy M, Jones D. Random Tranches[J]. RISK,2003,16(3):78-83.
[29]Davis M, Lo V. Infectious Defaults [J]. Quantitative Finance,2001,1(4):382-387.
[30]Jarrow R, Yu F. Counterparty Risk and the Pricing of Defaultable Securities [J]. Journal of Finance,2001,56(5):1765-1799.
[31]Kalemanova A, Schmid B, Werner R. The Normal Inverse Gaussian Distribution for Synthetic CDO Pricing[J]. Journal of Derivatives,2007,14(3):80-93.
[32]Thomas M. Pricing CDOs with Correlated Variance Gamma Distributions[EB/OL]. February 2006, http://www. defaultrisk. com/pp_crdrv103. htm,2007.
[33]Henrik B. Comparative Analysis of Hyperbolic Copulas Induced by a One Factor Levy Model[EB/OL]. August,2006, http://www. defaultrisk. com/pp_CDO_17. htm,2007.
[34]Prange D, Scherer W. Correlation Smile Matching with Alpha-Stable Distributions and Fitted Archimedan Copula Models[EB/OL]. March,2006, http://www. defaultrisk. com/pp_corr_46. htm,2007.
[35]Burtschell X, Gregory, Laurent J P.. Beyond the Gaussian Copula:Stochastic and Local Correlation[EB/OL]. January 2007, http://www. defaultrisk. com/pp_corr_76. htm,2007.
[36]Turc J, Very P, Benhamou D. Pricing CDOs with A Smile[R]. Societe Generale:Paris, Societe Generale Credit Research, February 2005.
[37]Pykhtin M, Dev A. Coarse-Grained CDOs [J]. Risk,2003,16(1):113-116.
[38]Debuysscher A, Szego M. The Fourier Transform Method-Technical Document[R]. Moody's Investors Service, Newyork, Moody's,2003.
[39]Albanese C, Chen O, Dalessandro A et al. Dynamic Credit Correlation Modeling [EB/OL]. May,2006, http://www. defaultrisk. com/pp_corr_75. htm,2007.
[40]Baxter M. Levy Process Dynamic Modeling of Single Name Credits and CDO Tranches [EB/OL]. April,2006, http://www. defaultrisk. com/pp_crdrvll0. htm,2007.
[41]Hull J, White A. Valuing Credit Derivatives Using an Implied Copula Approach[J]. Journal of Derivatives,2006,14(2):8-28.
[42]Totouom D, Armstrong M. Dynamic Copula Processes:A new way of modelling CDO tranches[EB/OL]. November 2005, http://www. defaultrisk. com/pp_crdrv_99. htm,2007.
[43]Chapovsky A, Rennie A, Tavares P A C. Stochastic Intensity Modelling for Structured Credit Exotics[EB/OL]. October,2006, http://www. defaultrisk. com/pp_crdrvl36. htm,2007.
[44]Graziano G, Rogers L. A Dynamic Approach to the Modeling of Correlation Credit Derivatives using Markov Chains[EB/OL]. November,2006, http://www. defaultrisk.com/pp_crdrv_88. htm,2007
[45]Hurd T R, Kuznetsov A. Fast CDO Computations in the Affine Markov Chain Model [EB/OL]. November,2006, http://www. defaultrisk. com/pp_crdrv_65. htm,2007.
[46]Joshi M S, Stacey A M. Intensity Gamma:A New Approach to Pricing Portfolio Credit Derivatives [J], Risk,2006,16(6):78-83.
[47]J. Sidenius, V. Piterbrag, L. Andersen. A New Framework for Dynamic Credit Portfolio. Loss Modeling[EB/OL]. June,2006, http://www. defaultrisk. com/pp_model_83. htm,2007.
[48]Bennani N. The Forward Loss Model:A Dynamic Term Structure Approach for the Pricing of Portfolio Credit Derivatives[EB/OL]. November,2005, http://www. defaultrisk. com/pp_crdrv_95. htm,2007
[49]Walker M B. CDO Valuation:Term Structure, Tranche Structure, and Loss Distributions[EB/OL]. January,2007, http://www. defaultrisk. com/pp_crdrv109. htm,2007.
[50]Schonbucher P J. Portfolio Losses and the Term Structure of Loss Transition Rates: A new Methodology for The Pricing of Portfolio Credit Derivatives [EB/OL]. June 2006, http://www. defaultrisk. com/pp_model_74. htm,2007.
[51]Errais E, Goldberg L R, Giesecke K. Pricing Credit from the Top Down with Affine Point Processes[EB/OL]. September,2007, http://www. defaultrisk. com/pp_CDO_16. htm,2007.
[52]Longsyaff F A, Rajan A. An Empirical Analysis of the Pricing of Collateralized Debt Obligations[J]. Journal of Finance,2008,63(2):529-563.
[53]Brigo D, Pallavicini A, Torresetti R. Calibration of CDO Tranches with The Dynamical Generalized-Poisson Loss Model[J]. Risk,2007,16(5):70-75.
[54]梁钧,黄智猛,黄培清.资产支持证券设计[J].预测,2000,1：60-62.
[55]朱世武.基于Copula函数度量违约相关性[J].统计研究,2005,4：61-64.
[56]冯谦,杨朝军.担保债权凭证定价——Copula函数的非参数估计与应用[J].运筹与管理,2006,15(5)：104-107.
[57]胡新华.含交易对手风险的公司债券和信用衍生品的定价问题研究[D].上海：上海交通大学理学院数学系,2007.
[58]刘冬.抵押债务证券(CDO)的信用风险分析[D].成都：西南财经大学金融研究中心,2007.
[59]袁子甲,李仲飞.基于因子模型的CDO定价研究：止态单因子模型的三种推广[EB/OL]. 03,2007, http://www. ccfr. org. cn/cicf2007/download. php?paper=20070119163654. PDF,2007.
[60]张维,邱勇,郝刚.基于多银行贷款池的合成CDO设计[J].现代财经,2007,27(9)：6-13.
[61]穆放,宋曼洁,陈治津.债务抵押债券定价模型探讨及实施研究[J].经济纵横,2007,10：10-13.
[62]Li D X. On Default Correlation:A Copula Approach. Journal of Fixed Income,2000, 9(3):43-54.
[63]Laurent J P, Gregory J. Basket default swaps, CDOs and factor copulas. Working paper, http://www. defaultrisk. com/pp_crdrv_25. htm,2003.
[64]Hull J, White A. Valuation of a CDO and an n-th to Default CDS without Monte Carlo Simulation. Journal of Derivatives.2004,12(2):8-23.
[65]Kalemanova A, Schmid B, Werner R. The Normal Inverse Gaussian Distribution for Synthetic CDO Pricing. Working paper, http://www. defaultrisk. com/pp_crdrv_91. htm, 2005.
[66]Giesecke K. A Simple Exponential Model for Dependent Defaults, Journal of Fixed Income, 2003,13(3):74-83.
[67]Ferrarese C. A Comparative Analysis of Correlation Skew Modeling Techniques for CDO Index Tranches. Working paper, http://www. defaultrisk. com/pp_crdrv_91. htm,2006.
[68]Burtschell X, Gregory J, and Laurent J P. Beyond the Gaussian Copula:Stochastic and Local Correlation,. Working paper, http://laurent.jeanpaul.free.fr/ stochastic_local_correlation. pdf,2005.
[69]Xu G, Extending Gaussian Copula with Jumps to Match Correlation Smile. Working paper, http://www. defaultrisk. com/pp_corr100. htm,2006.
[70]Schonbucher P J. Credit Derivatives Pricing Models:Models, Pricing, and Implementation[M]. Wiley,2003.
[71]冯谦,杨朝军.担保债权凭证定价——Copula函数的非参数估计与应用[J].运筹与管理,2006,15(5)：104-107.
[72]Li D X. On Default Correlation:a Copula Approach[J]. Journal of Fixed Income,2000, 9(3):43-54.
[73]Hull J, White A. Valuation of a CDO and an nth to Default CDS without Monte Carlo Simulation[J], Journal of Derivatives,2004,12(2):8-23.
[74]Andersen L, Sidenius J and Basu S. All Your Hedges in One Basket [J], Risk,2003,16(11): 67-72.
[75]Gregory J, Laurent J P. In the Core of Correlation [J], Risk,2004,17(10):87-91.
[76]Andersen L, Sidenius J. Extensions to the Gaussian Copula:Random Recovery and Random Factor Loadings[J], Journal of Credit Risk,2005,1(1):29-70.
[77]Ferrarese C. A Comparative Analysis of Correlation Skew Modeling Techniques for CDO Index Tranches(workingpaper), http://www. defaultrisk. com/pp\_CDO\_02. htm,2006.
[78]Cherubini U, Luciano E, Vecchiato V. Copula Methodsin Finance[M]. New York:Wiley Finance,2004.
[79]Nolan J P. Numerical Calculation of Stable Densities and Distribution Functions[J], Commun. Statist.-Stochastic Models,1997,13(4):759-774.
[80]Kalemanova A, Schmid B, Werner R. The Normal Inverse Gaussian Distribution for Synthetic CDO[J], Journal of Derivatives,2007,14(3):80-93.
[81]Laurent J P, Gregory J. Basket Default Swaps, CDOs and Factor Copulas[J], Journal of Risk,2005,7(4):103-122.
[82]Barndorff N. Process of Normal Inverse Gaussian type [J], Finance and Stochastic,1997, 2(1):41-68.
[83]Bracewell R N. The Fourier Transform & Its Applications[M]. New York:McGraw-Hill Companies Inc.,2002.
[84]Ech-Chatbi C. CDS and CDO Pricing with Stochastic Recovery[R]. Working Paper,2008.
[85]Andersen L, Sidenius J. Extensions to the Gaussian Copula:Random Recovery and Random Factor Loadings[J].2004,1(1):29-70.
[86]Krekel M. Pricing Distressed CDOs with Base Correlation and Stochastic Recovery[R]. Working Paper,2008.
[87]Amraoui S, Hitier S. Optimal Stochastic Recovery for Base Correlation[R]. Working Paper, 2008.
[88]Elizalde A. Credit risk Models IV:Understanding and Pricing CDOs[R]. Working Paper, 2006.
[89]Kalemanova A, Schmid B, Werner R. The Normal Inverse Gaussian Distribution for Synthetic CDO Pricing[J]. Journal of Derivatives,2007,14(3):80-93.
[90]Vasicek O. Probability of Loss on Loan Portfolio[R]. Working Paper,1987.
[91]Vasicek O. Limiting Loan Loss Distribution[R]. Working Paper,1991.
[92]Li D. On Default Correlation:a Copula Function Approach[J]. Journal of Fixed Income, 2000,9(4):43-54.
[93]Burtschell X, Gregory J, Laurent J P. Beyond the Gaussian Copula:Stochastic and Local Correlation [J]. Journal of Credit Risk,2007,3(1):31-62.
[94]Xu G. Extending Gaussian Copula with Jumps to Match Correlation Smile[R]. Working Paper, 2006.
[95]杨瑞成,秦学志,陈田.随机相关结构下单因子混合高斯模型CDO定价问题[J].大连理工大学学报,2009,49(4)：587-592.
[96]Shek H, Uematsu S, Wei Zh. Valuation of Loan CDS and CDX[R]. Working Paper,2007.
[97]Dobranszky P, Schoutens W. Generic Levy One-factor Models for the Joint Modelling of Prepayment and Default:Modelling LCDX[R]. Working Paper,2008.
[98]Wei Zh. Valuation of Loan CDS Under Intensity Based Model[R]. Working Paper,2007.
[99]Dobranszky P. Joint Modelling of CDS and LCDS Spreads with Correlated Default and Prepayment Intensities and with Stochastic Recovery Rate[R]. Working Paper,2008.
[100]Duffie D, Pan J, Singleton K. Transform Analysis and Asset Pricing for Affine Jump-diffusions[J]. Econometrica,2000,68(6):1343-1376.
[101]Prabhu N. U. Stochastic Storage Processes[M]. New York:Springer,1998.
[102]Barndorff-Nielsen O. E., Mikosch T, Resnich S. Levy process:Theory and applications[M]. Boston:Birkhauser,2001.
[103]Kyprianou A. E. Introductory Lectures on Fluctuations of Levy Processes with Applications[M]. Berlin:Springer,2006.
[104]Berton J. Levy Process[M]. Cambridge:Cambridge University Press,1996.
[105]Schoutens W. Levy Processes in Finance-Pricing Financial Derivatives[M]. Chichester: Wiley,2003.
[106]Albrecher H, Ladoucette S, Schoutens W. A Generic One Factor Levy Model for Pricing Synthetic CDOs[M]. In:Advances in Mathematical Fianance. Boston:Birkhauser,2007: 259-277.
[107]Garcia J, Goossens S, Masol V et al. Levy Base Correlation [J]. Wilmott Journal.2009, 1(2):95-100.