CDO Pricing Model Based On Factor Copula

Collateralized debt obligation (CDO) is a security based on one or more categories and decentralized assets as a pledge.These assets include corporate bonds, emerging mar-ket bonds, assets pledged bonds, mortgage bonds, real estate investment trust, bank debt and so on. As an innovative financial derivative,CDO combines the advantage of asset securitization and credit derivatives and is trusted by investors in general.Financial institu-tions make use of CDO to reach the purpose of transferring risk or taking spread, and es-pecially use this product to design different credit risk bonds series to meet investors with a variety of risk preferences.In order to price credit derivatives portfolio, we must consider factors as default probability of all reference assets pool, default correlation between refer-ence assets and recovery rates after default.Among the factors, default correlation is very important, because it determines the default probability that all reference assets in CDO credit assets pool default at the same time. Thus, we may obtain the potential loss distri-bution of assets pool.There’s no consensus in mainstream financial community for CDO pricing. The general models include BET model, factor copula model, affine intensity model.However,there are shortcomings and deficiencies in these existing models.Thus,in this paper,we combine the existing domestic and international research results,in-depth s-tudy the CDO pricing mechanism, apply the frontier methods of financial engineering to innovate, construct reasonable pricing models and promote the development of pricing theory and practice. Based on the fat tail and dynamic correlation characteristics of the CDO assets pool loss, and introducing random recovery rate,this article constructed the fat-tail mixed copula model for CDO pricing with the structure of stochastic correlation, local correlation and random recovery rate. Besides,numerical analysis was carried out to compare the advantages and disadvantages of the models.The relevant research results are summarized as follows:1. The lack of fat-tail characteristic and the existence of "correlation smile" in the standard Gaussian Copula model leads to the result that the model couldn’t match the mar-ket quotation well.This inspired us to assume that the systematic factor and idiosyncratic factors subject to the fat-tail mixed G-VG distribution.G-VG distribution has good fat-tail and convolution properties,which makes it suitable for CDO pricing.Besides,this article extends constant correlation to dynamic correlation, including stochastic correlation and local correlation. Thus we establish the mixed G-VG copula model with2-state,3-state stochastic correlation and local correlation structure to reflect the fat tail and dynamic cor-relation characteristics of the CDO reference assets pool loss. As for calculation,we apply Fast Fourier Transform and the inverse transform to give the calculation method of CDO reference assets pool loss,carry out numerical simulation and on this basis obtain the fairy credit spread of every CDO tranche.2. Due to the low efficiency of Fast Fourier Transform and its inverse transform, we consider the CDO pricing problem under the large homogeneous portfolio(LHP) approx-imation. We still assume the systematic factor and idiosyncratic factors subjecting to the mixed G-VG distribution and construct the mixed G-VG copula model with2-state,3-state stochastic correlation structure and local correlation structure. The analytical expressions of conditional default probability, cumulative loss distribution function and the expect-ed tranche loss are given.According to the CDO pricing formula,we may obtain the fairy credit spread of CDO tranches.The empirical analysis is carried out in Gaussian Copula model> mixed G-VG Copula model and Copula models with stochastic correlation, and the results are compared. The results show that the mixed G-VG Copula model with2-state correlation fits the market quotation best.Because this model combines fat tail copula and stochastic correlation,thus it has more free parameters and may bring more flexibility in-to the correlation structure.As the computational efficiency is considered,the mixed G-VG copula model is stable under convolution to some extent, so it may simplify and accelerate computation.This will facilitate practitioners to use the model for CDO pricing.3. Aiming at the "correlation smile" and senior tranche, super senior tranche’s pricing failure in the standard Gaussian Copula model,where the recovery is a determined constant,this article introduces random recovery rate into CDO pricing model.The involved random recovery rate model is Amraoui and Hitier model.We still consider the mixed G-VG Copula model with2-state and3-state stochastic correlation structure.We generalized stand Gaussian Copula model in3aspects and carry out numerical analysis for CDO pric-ing.The results show that random recovery rate can reflect better the impact of systematic factor and idiosyncratic factors on recovery rate.Thus the new model may better simulate the loss distribution of the reference assets pool and the fairy credit spread of the CDO tranches.In summary,this article selected credit derivatives as our research field and focused on the pricing problem of structured financial products-CDO.Aiming at the defects in ex-isting CDO models,this article improved CDO pricing models from different perspectives, including the introduction of mixed fat tail distribution、dynamic correlation and random recovery rate and constructed new CDO pricing models.We carried out numerical simula-tion and compared these models.The results shows that compared with the old model,our new models fit the market quotation better and improve "correlation smile" problem.We sincerely hope that corresponding research results contribute to provide new ideas for the-orists and practitioners about CDO products design and risk management.