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Author : Gohou Ferdinand DANON As popular vehicles for trading a portfolio of credit risks, we focus on a Synthetic Collateralized Debt Obligation swaps (Synthetic CDOs), in terms of pricing and risk analysis. Our purpose is not to create a new concept in these stylised facts of correlation products. Instead, we attempt to assess the key idea behind the standard credit derivatives pricing model in order to fully capture the essential of the risk of a synthetic CDO swaps. To this end, we provide a step by step description of the one factor Gaussian Copula model which is said to overcome computation costs inherent to the use of Monte Carlo simulation in the standard Gaussian copula model. This thesis also presents the double-t distribution suggested by Hull and White (2004) as an extension of the one factor Gaussian copula where they used a multi factor framework. For practical purpose, we use Microsoft Excel to calculate a synthetic CDO tranche price based on the computation of a homogenous portfolio of credit defaults under the one factor Gaussian copula model. We compared our empirical results in terms of prices relative to our homogenous assumptions with the market quotes. We recognized that even if the CDO pricing theoretical side in terms of relationship between the default correlation risk and tranches prices is satisfied, our model prices do not match the market quotes. The thesis then goes on to present a way to assess the demanding credit risk analysis in light of such appealing issue. We also introduce other problems that we would like to understand better such as the implied and base correlations. We highlight the intuition behind them in terms of pricing and risk analysis. Finally the recent trouble of Bears Stearns funds Discuss this paper
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by Abel Elizalde Some investors in the Collateralized Debt Obligations (CDOs) market have been publicly accused of not fully understanding the risks and dynamics of these products. They won Discuss this paper
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by Jean-Paul Laurent & Jon Gregory We consider a factor approach to the pricing of basket credit derivatives and synthetic CDO tranches. Our purpose is to deal in a convenient way with dependent defaults for a large number of names. We provide semi-explicit expressions of the stochastic intensities of default times and pricing formulae for basket default swaps and CDO tranches. Two cases are studied in detail: mean-variance mixture models and frailty models. We also compare prices under Gaussian and Clayton copulas
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Abstract: In this paper we investigate one factor models that extend the classical Gaussian copula model for pricing CDOs. The proposed models are very tractable and perform significantly better than the classical Gaussian copula model. Moreover, we introduce the concept of Levy base correlation. The obtained Levy base correlation curve is much flatter than the corresponding Gaussian one. This indicates that the models do fit the observed data much better. Additionally, flat base correlation curves are also much more reliable for pricing of bespoke tranches. Discuss this paper
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Anna Kalemanova Bernd Schmid Ralf Werner Abstract This paper presents an extension of the popular Large Homogeneous Portfolio (LHP) approach to the pricing of CDOs. LHP (which has already become a standard model in practice) assumes a flat default correlation structure over the reference credit portfolio and models default using a one factor Gaussian copula. However, this model fails to fit the prices of different CDO tranches simultaneously which leads to the well known implied correlation smile. Many researchers explain this phenomenon with the lack of tail dependence and propose to use a Student t copula. Incorporating the effect of tail dependence into the one factor portfolio credit model yields significant pricing improvement. However, the computation time increases dramatically as the Student t distribution is not stable under convolution. This makes it impossible to use the model for computationally intensive applications such as the determination of the optimal asset allocation in an investor Discuss this paper
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Lijuan Cao Zhang Jingqing Lim Kian Guan Zhonghui Zhao
Abstract This paper studies the pricing of collateral debt obligation (CDO) using Monte Carlo and analytic methods. Both methods are developed within the framework of the reduced form model. One-factor Gaussian Copula is used for treating default correlations amongst the collateral portfolio. Based on the two methods, the portfolio loss, the expected loss in each CDO tranche, tranche spread and the default delta sensitivity are analyzed with respect to different parameters such as maturity, default correlation, default intensity or hazard rate, and recovery rate. We provide a careful study of the effects of different parametric impact. Our results show that Monte Carlo method is slow and not robust in the calculation of default delta sensitivity. The analytic approach has comparative advantages for pricing CDO. We also employ empirical data to investigate the implied default correlation and base correlation of the CDO. The implication of extending the analytical approach to incorporating Levy processes is also discussed.
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Abstract Leif Andersen, Jakob Sidenius and Susanta Basu present new techniques for single-tranche CDO sensitivity and hedge ratio calculations. Using factorisation of the copula correlation matrix, discretisation of the conditional loss distribution followed by a recursion-based probability calculation, and derivation of analytical formulas for deltas, they demonstrate a significant improvement in computational speeds
Abstract This paper develops numerical approximations for pricing collateralized debt obligations (CDOs) and other portfolio credit derivatives in the multifactor Normal Copula model. A key aspect of pricing portfolio credit derivatives is capturing dependence between the defaults of the elements of the portfolio. But, compared with an independent-obligor model, pricing in a model with correlated defaults is more challenging. Our approach strikes a balance by reducing the problem of pricing in a model with correlated defaults to calculations involving only independent defaults. We develop approximations based on power series expansions in a parameter that scales the underlying correlations. These expansions express a CDO tranche price in a multifactor model as a series of prices in independent-obligor models, which are easy to compute. The approach builds on a classical approximation for multivariate Gaussian probabilities; we introduce an alternative representation that greatly reduces the number of terms required to evaluate the coefficients in the expansion. We also apply this method to the underlying problem of computing joint probabilities of multivariate normal random variables for which the correlation matrix has a factor structure.
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Nice explanation of "probability bucketing" fundamentals
Michael S. Gibson
Abstract: Synthetic collateralized debt obligations, or synthetic CDOs, are popular vehicles for trading the credit risk of a portfolio of assets. Following a brief summary of the devel- opment of the synthetic CDO market, I draw on recent innovations in modeling to present a pricing model for CDO tranches that does not require Monte Carlo simulation. I use the model to analyze the risk characteristics of the tranches of synthetic CDOs. The analysis shows that although the more junior CDO tranches { equity and mezzanine tranches { typ- ically contain a small fraction of the notional amount of the CDO's reference portfolio, theybear a majority of the credit risk. One implication is that credit risk disclosures relying on notional amounts are especially inadequate for rms that invest in CDOs. I show how the equity and mezzanine tranches can be viewed as leveraged exposures to the underlying credit risk of the CDO's reference portfolio. Even though mezzanine tranches are typically rated investment-grade, the leverage they possess implies their risk (and expected return) can be many times that of an investment-grade corporate bond. Discuss this paper
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by John Hull of the University of Toronto, and Alan White of the University of Toronto
Abstract: In this paper we develop two fast procedures for valuing tranches of collateralized debt obligations and nth to default swaps. The procedures are based on a factor copula model of times to default and are alternatives to using fast Fourier transforms. One involves calculating the probability distribution of the number of defaults by a certain time using a recurrence relationship; the other involves using a Discuss this paper
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Abstract: This paper presents and numerically implements a methodology to price credit derivative products referencing a portfolio of underlying assets. We develop a copula based framework to model the default dependency among obligors and offer algorithms for pricing Basket Default Swaps and Collateralized Debt Obligations. A risk neutral methodology by which [we] calibrate copula parameters to market quotes is taken into account, and different methods of calibration are illustrated and implemented. We numerically calculate the sensitivity of prices to the main state variables affecting their values, namely recovery rates, default correlation and credit quality of the underlying portfolio. By assuming two alternative specifications (Gaussian and Student Discuss this paper
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Fabio Mibielli Peixoto Abstract Monte Carlo simulation and a semi-analytical method are used to value a basket default swap and an homogeneous Collateralized Debt Obligation (CDO). The semianalytical technique is based on the one factor copula model proposed by J.P. Laurent and J. Gregory [1]. We study the properties of a CDO with Monte Carlo and compare the spread calculation with the one obtained by the factor model. Discuss this paper
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