| According to the no-arbitrage pricing theory the present thesis first reveals the key for synthetic CDO pricing is the loss distributions of the portfolio of reference instruments over different time horizons. At the same time, this paper analyzes non-linear default correlation with Copula and, in accordance with the requirements of risk-neutral valuation principle and martingale approach, figures out risk-neutral probability of default of individual assets in the portfolio of reference from the CDS prices.Then, Analysis is focused on derivation of the market prices and loss distribution of tranches of CDO under the Gaussian Copula model and analyzing defects of Gaussian Copula CDO pricing model as to illustrates the need for its improvement. On the basis of predecessors, having considered particularity of equity tranche and senior tranche, the preaent paper derives the exact expressions of distribution of expected loss of CDO tranche and the underlying asset portfolio. At the same time, the artic comparatively analyzes differences between implied portfolio loss distribution and portfolio loss distribution of the single-factor Gaussian Copula model and explains the two major deficiencies, that is, implied Default Correlation smile and implied loss distribution within the single-factor Gaussian,which affects it pricing power. In addition, internal consistency of these two defects were explained.The most important part of this paper is the proposal for a new CDO pricing model——double symmetric normal inverse Gaussian copula with random factor loadings model for synthetic CDO pricing. Following a comparative analysis of the double NIG Copula and random factor loading, emphasis is placed on study of double symmetric normal inverse Gaussian copula with random factor loadings model and through rigorous mathematical derivation this paper figures out analytical solutions or semi-analytical types for parameters in model, distributions of the portfolio of reference instruments and the expected loss distribution and market prices of tranches of CDO. In order to avoid deficiencies in L. Annelis (2005,2007) that estimation of model parameter is very unstable, the calculation speed is extremely slow and the error of calculation is large mainly because any analytical solution for model parameters in L. Annelis (2005,2007) can not be obtained, this paper uses the symmetric NIG Copula instead of generalization NIG Copula.As interference effects of random factor loadings and complexity of NIG Copula, this paper can only firgure out semi-analytical solutions of the expected loss distribution of CDO tranche and even some parameters can only be expressed in a semi-analytical equation, the solutions to this problem is that for some parameters of the semi-analytical type questions, this article will be translated into non-linear optimization problem to solve.Another important part of this thesis is that the present thesis comparatively analyzes the differences in CDO pricing among double symmetric normal inverse Gaussian copula with random factor loadings model, one factor Gaussian Copula model, double NIG Copula model and random factor loading Gaussian Copula modle and concluds that double symmetric normal inverse Gaussian copula with random factor loadings model have the biggest fitting ability to market data and the minimum pricing error. Meanwhile, the paper also studys impacts of recovery rate, interest rates, credit quality of the underlying assets and default correlation on CDO prices.The results show that CDO tranche spreads change in inversely direction with the credit quality of the underlying asset pool, but senior tranche reflects more sensitively. equity tranche spreads change reversely with default correlation coefficient, while senior tranche presents in the same direction of change. CDO spreads is not sensitive to interest rates, which in turn also illustrates the assumption of constant interest rate in field of CDO pricing has some rationality. |
