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226
Extensions to the Gaussian copula: random recovery and random factor loadings
 JOURNAL OF CREDIT RISK
, 2004
"... This paper presents two new models of portfolio default loss that extend the standard Gaussian copula model yet preserve tractability and computational efficiency. In one extension, we randomize recovery rates, explicitly allowing for the empirically wellestablished effect of inverse correlation be ..."
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Cited by 77 (0 self)
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This paper presents two new models of portfolio default loss that extend the standard Gaussian copula model yet preserve tractability and computational efficiency. In one extension, we randomize recovery rates, explicitly allowing for the empirically wellestablished effect of inverse correlation between recovery rates and default frequencies. In another extension, we build into the model random systematic factor loadings, effectively allowing default correlations to be higher in bear markets than in bull markets. In both extensions, special cases of the models are shown to be as tractable as the Gaussian copula model and to allow efficient calibration to market credit spreads. We demonstrate that the models – even in their simplest versions – can generate highly significant pricing effects such as fat tails and a correlation “skew ” in synthetic CDO tranche prices. When properly calibrated, the skew effect of random recovery is quite minor, but the extension with random factor loadings can produce correlation skews similar to the steep skews observed in the market. We briefly discuss two alternative skew models, one based on the MarshallOlkin copula, the other on a spreaddependent correlation specification for the Gaussian copula.
A General Approach to Integrated Risk Management with Skewed, Fattailed Risks
, 2005
"... Integrated risk management in a financial institution requires an approach for aggregating risk types (market, credit, and operational) whose distributional shapes vary considerably. In this paper, we construct the joint risk distribution for a typical large, internationally active bank using the me ..."
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Cited by 58 (3 self)
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Integrated risk management in a financial institution requires an approach for aggregating risk types (market, credit, and operational) whose distributional shapes vary considerably. In this paper, we construct the joint risk distribution for a typical large, internationally active bank using the method of copulas. This technique allows us to incorporate realistic marginal distributions, both conditional and unconditional, that capture some of the essential empirical features of these risks such as skewness and fattails while allowing for a rich dependence structure. We explore the impact of business mix and interrisk correlations on total risk, whether measured by valueatrisk or expected shortfall. We find that given a risk type, total risk is more sensitive to differences in business mix or risk weights than to differences in interrisk correlations. There is a complex relationship between volatility and fattails in determining the total risk: depending on the setting, they either offset or reinforce each other. The choice of copula (normal versus Studentt), which determines the level of tail dependence, has a more modest effect on risk. We then compare the copulabased method with several conventional approaches to computing risk.
Beyond Correlation: Extreme Comovements Between Financial Assets
, 2002
"... This paper inv estigates the potential for extreme comov ements between financial assets by directly testing the underlying dependence structure. In particular, a tdependence structure, deriv ed from the Student t distribution, is used as a proxy to test for this extremal behav#a(0 Tests in three ..."
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Cited by 53 (5 self)
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This paper inv estigates the potential for extreme comov ements between financial assets by directly testing the underlying dependence structure. In particular, a tdependence structure, deriv ed from the Student t distribution, is used as a proxy to test for this extremal behav#a(0 Tests in three di#erent markets (equities, currencies, and commodities) indicate that extreme comov ements are statistically significant. Moreov er, the "correlationbased" Gaussian dependence structure, underlying the multiv ariate Normal distribution, is rejected with negligible error probability when tested against the tdependencealternativ e. The economic significance of these results is illustratedv ia three examples: comov ements across the G5 equity markets; portfoliov alueatrisk calculations; and, pricing creditderiv ativ es. JEL Classification: C12, C15, C52, G11. Keywords: asset returns, extreme comov ements, copulas, dependence modeling, hypothesis testing, pseudolikelihood, portfolio models, risk management. # The authorsw ould like to thankAndrew Ang, Mark Broadie, Loran Chollete, and Paul Glasserman for their helpful comments on an earlier version of this manuscript. Both authors arewS; the Columbia Graduate School of Business, email: {rm586,assaf.zeevi}@columbia.edu, current version available at www.columbia.edu\# rm586 1 Introducti7 Specification and identification of dependencies between financial assets is a key ingredient in almost all financial applications: portfolio management, risk assessment, pricing, and hedging, to name but a few. The seminal work of Markowitz (1959) and the early introduction of the Gaussian modeling paradigm, in particular dynamic Brownianbased models, hav e both contributed greatly to making the concept of co rrelatio almost synony...
The normal inverse gaussian distribution for synthetic CDO pricing
 Journal of derivatives
"... 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 defaults using a one factor Gauss ..."
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Cited by 49 (0 self)
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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 defaults 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 skew. Many researchers explain this phenomenon with the lack of tail dependence in the Gaussian copula and propose to use a Student tdistribution. 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 tdistribution 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’s portfolio over different asset classes including CDOs. We present a modification of the LHP model replacing the Student tdistribution with the Normal inverse Gaussian (NIG) distribution. We compare the properties of our new model with those of the Gaussian and the double tcopulas. The employment of the NIG distribution not only speeds up
Tail dependence functions and vine copulas
 Journal of Multivariate Analysis
"... Tail dependence and conditional tail dependence functions describe, respectively, the tail probabilities and conditional tail probabilities of a copula at various relative scales. The properties as well as the interplay of these two functions are established based upon their homogeneous structures. ..."
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Cited by 23 (5 self)
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Tail dependence and conditional tail dependence functions describe, respectively, the tail probabilities and conditional tail probabilities of a copula at various relative scales. The properties as well as the interplay of these two functions are established based upon their homogeneous structures. The extremal dependence of a copula, as described by its extreme value copulas, is shown to be completely determined by its tail dependence functions. For a vine copula built from a set of bivariate copulas, its tail dependence function can be expressed recursively by the tail dependence and conditional tail dependence functions of lowerdimensional margins. The effect of tail dependence of bivariate linking copulas on that of a vine copula is also investigated.
V.: Inhomogeneous dependence modeling with timevarying copulae
 J. Bus. Econom. Statist
, 2009
"... Measuring dependence in a multivariate time series is tantamount to modelling its dynamic structure in space and time. In the context of a multivariate normally distributed time series, the evolution of the covariance (or correlation) matrix over time describes this dynamic. A wide variety of applic ..."
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Cited by 22 (5 self)
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Measuring dependence in a multivariate time series is tantamount to modelling its dynamic structure in space and time. In the context of a multivariate normally distributed time series, the evolution of the covariance (or correlation) matrix over time describes this dynamic. A wide variety of applications, though, requires a modelling framework different from the multivariate normal. In risk management the nonnormal behaviour of most financial time series calls for nongaussian dependency. The correct modelling of nongaussian dependencies is therefore a key issue in the analysis of multivariate time series. In this paper we use copulae functions with adaptively estimated time varying parameters for modelling the distribution of returns, free from the usual normality assumptions. Further, we apply copulae to estimation of ValueatRisk (VaR) of a portfolio and show its better performance over the RiskMetrics approach, a widely used methodology for VaR estimation.
FAST SIMULATION FOR MULTIFACTOR PORTFOLIO CREDIT RISK IN THE tCOPULA MODEL
, 2005
"... We present an importance sampling procedure for the estimation of multifactor portfolio credit risk for the tcopula model, i.e, the case where the risk factors have the multivariate t distribution. We use a version of the multivariate t that can be expressed as a ratio of a multivariate normal and ..."
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Cited by 21 (2 self)
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We present an importance sampling procedure for the estimation of multifactor portfolio credit risk for the tcopula model, i.e, the case where the risk factors have the multivariate t distribution. We use a version of the multivariate t that can be expressed as a ratio of a multivariate normal and a scaled chisquare random variable. The procedure consists of two steps. First, using the large deviations result for the Gaussian model in Glasserman, Kang, and Shahabuddin (2005a), we devise and apply a change of measure to the chisquare random variable. Then, conditional on the chisquare random variable, we apply the importance sampling procedure developed for the Gaussian copula model in Glasserman, Kang, Shahabuddin (2005b). We support our importance sampling procedure by numerical examples.
Pricing multiname credit derivatives: heavy tailed hybrid approach
, 2002
"... In recent years, credit derivatives have become the main tool for transferring and hedging credit risk. The credit derivatives market has grown rapidly both in volume and in the breadth of the instruments it offers. Among the most complicated of these instruments are the multiname ones. These are in ..."
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Cited by 20 (1 self)
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In recent years, credit derivatives have become the main tool for transferring and hedging credit risk. The credit derivatives market has grown rapidly both in volume and in the breadth of the instruments it offers. Among the most complicated of these instruments are the multiname ones. These are instruments with payoffs that are contingent on the default realization in a portfolio of names. The modeling of dependent defaults is difficult because there is very little historical data available about joint defaults and because the prices of those instruments are not quoted. Therefore, the models cannot be calibrated, neither to defaults nor to prices. In this paper, we present a methodology for the estimation, simulation, and pricing of multiname contingent instruments. Our model is a hybrid of the wellknown structural and reduced form approaches for modeling defaults. The dependence structure of our model is of a tcopula that possesses nontrivial tail dependence. The tcopula allows for more joint extreme events, which have a big impact on the prices of multiname instruments, e.g. n thtodefault baskets and CDOs. We demonstrate this impact with n thtodefault baskets.
Portfolio credit risk with extremal dependence: Asymptotic Analysis and Efficient Simulation
, 2006
"... We consider the risk of a portfolio comprised of loans, bonds, and financial instruments that are subject to possible default. In particular, we are interested in performance measures such as the probability that the portfolio incurs large losses over a fixed time horizon and the expected excess los ..."
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Cited by 19 (1 self)
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We consider the risk of a portfolio comprised of loans, bonds, and financial instruments that are subject to possible default. In particular, we are interested in performance measures such as the probability that the portfolio incurs large losses over a fixed time horizon and the expected excess loss given that large losses are incurred during this horizon. Contrary to the normal copula that is commonly used in practice (e.g., in the CreditMetrics system), we assume a portfolio dependence structure that is semiparametric, does not hinge solely on correlation, and supports extremal dependence among obligors. A particular instance within the proposed class of models is the socalled tcopula model that is derived from the multivariate Student t distribution and hence generalizes the normal copula model. The size of the portfolio, the heterogenous mix of obligors, and the fact that default events are rare and mutually dependent makes it quite complicated to calculate portfolio credit risk either by means of exact analysis or naïve Monte Carlo simulation. The main contributions of this paper are twofold. We first derive sharp asymptotics for portfolio credit risk that illustrate the implications of extremal dependence among obligors. Using this as a stepping stone, we develop importance sampling algorithms that are shown to be asymptotically optimal and can be used to efficiently compute portfolio credit risk via Monte Carlo simulation.