Research On CDO Pricing Models Based On Factor Copula

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.