One-factor Gaussian Copula Model Research Articles

Introducing an analytical solution and an improved one-factor gaussian copula model for the pricing of heterogeneous CDOs

This paper introduced an analytical solution and improved one-factor Gaussian copula models to the pricing of tranches of a Collateralized debt obligations (CDO) portfolio. Prices of CDO tranches are calculated and compared using the analytical model and different one-factor Gaussian copula models including a two-category heterogeneous model and a completely heterogeneous model that uses individual rate parameter and correlation coefficient for each reference entity in a CDO portfolio. When correlation among reference entities is low, the price calculated from the analytical model matches very well with the one-factor Gaussian copula models. However, as the correlation among reference entities increases, prices calculated using both the analytical solution and the homogeneous or two-category one-factor Gaussian copula models significantly deviate from the completely heterogeneous one-factor Gaussian copula model. This result verifies our belief that uniform parameters cannot completely capture all the heterogeneities in a CDO portfolio. Completely heterogeneous one-factor Gaussian copula model using individual rate parameters and correlation coefficients for each reference entities provides more reliable and accurate prices for structured securities.

An analysis of simultaneous company defaults using a shot noise process

During subprime mortgage crisis, it became apparent that incumbent models had underestimated company default correlations. Complex models that attempt to incorporate default dependency are difficult to implement in practice. On the contrary, practical models, such as One-Factor Gaussian Copula model, greatly underestimated simultaneous default probabilities. In this article, we develop a model for a company asset process and based on this model, we calculate simultaneous default probabilities using option-theoretic approach. Our model focuses on one industry and includes a shot noise process in the asset model directly. The risk factor driving the shot noise process is common to all companies in the industry but the shot noise parameters are assumed to be company-specific; therefore, every company responds to this common risk factor differently. Apart from the shot noise process, the asset model includes company specific Brownian motion. Compared to commonly used geometric Brownian motion asset model in option-theoretic approach, our model predicted higher simultaneous default probabilities for Citigroup Inc. in 2008, and for all company combinations for the years of 2009 and 2010. Our model is easy to implement and can be extended to analyze any finite number of companies without greatly increasing computational difficulty.

Copula-based factor model for credit risk analysis

A standard quantitative method to access credit risk employs a factor model based on joint multivariate normal distribution properties. By extending a one-factor Gaussian copula model to make a more accurate default forecast, this paper proposes to incorporate a state-dependent recovery rate into the conditional factor loading, and model them by sharing a unique common factor. The common factor governs the default rate and recovery rate simultaneously and creates their association implicitly. In accordance with Basel III, this paper shows that the tendency of default is more governed by systematic risk rather than idiosyncratic risk during a hectic period. Among the models considered, the one with random factor loading and a state-dependent recovery rate turns out to be the most superior on the default prediction.

On CDO Tranche Pricing When Copula is Nearly Comonotone

Pricing of a CDO base tranche is considered when copula is comonotone and approximations are derived for the base tranche expected loss, obtained with a one-factor Gaussian copula model, when correlations are sufficiently close to 100%. Numerical examples are provided showing that a two-term approximation reproduces the true value with no more than approximately 5% relative error, uniformly across detachments and average default probabilities for correlations between 95% and 100%.

Levy Subordinator Model: A Two Parameter Model of Default Dependency

The May 2005 crisis and the recent credit crisis have indicated to us that any realistic model of default dependency needs to account for at least two risk factors, firm-specific and catastrophic. Unfortunately, the popular Gaussian copula model has no identifiable support to either of these. In this article, a two parameter model of default dependency based on the Levy subordinator is presented accounting for these two risk factors. Subordinators are Levy processes with non-decreasing sample paths. They help ensure that the loss process is non-decreasing leading to a promising class of dynamic models. The simplest subordinator is the Levy subordinator, a maximally skewed stable process with index of stability 1/2. Interestingly, this simplest subordinator turns out to be the appropriate choice as the basic process in modeling default dependency. Its attractive feature is that it admits a closed form expression for its distribution function. This helps in automatic calibration to individual hazard rate curves and efficient pricing with Fast Fourier Transform techniques. It is structured similar to the one-factor Gaussian copula model and can easily be implemented within the framework of the existing infrastructure. As it turns out, the Gaussian copula model can itself be recast into this framework highlighting its limitations. The model can also be investigated numerically with a Monte Carlo simulation algorithm. It admits a tractable framework of random recovery. It is investigated numerically and the implied base correlations are presented over a wide range of its parameters. The investigation also demonstrates its ability to generate reasonable hedge ratios.

The kth default time distribution and basket default swap pricing

We propose an alternative method for finding the kth default time distribution in a homogeneous portfolio with dependency. Analysing order statistics of default times with a one-factor Gaussian copula model, we explicitly derive the probability distribution. Moreover, we compute the prices of basket default swaps such as the kth to default swaps and m out of n default swaps within our framework. To test the efficiency and accuracy of our method we compare the theoretical prediction with existing methods.

Pricing of CDOs based on the multivariate Wang transform

This paper extends the one-factor Gaussian copula model, the standard market model for valuing CDOs, based on the multivariate Wang transform. Unlike the existing models, our model calibrates the parameter associated with a risk adjustment for default threshold, not correlation parameter, which always exists and is unique for any market price of CDO tranche. A Student t-copula model is also considered within the same framework to describe a fat-tail distribution observed in the actual market. Through numerical experiments, it is shown that our model provides a better fit to the market data compared with the existing models.

Levy Subordinator Model of Default Dependency

This article presents a model of default dependency based on Levy subordinator. It is a tractable dynamical model, computationally structured similar to the one-factor Gaussian copula model, providing easy calibration to individual hazard rate curves and efficient pricing with Fast Fourier Transform techniques. The subordinator is an alpha=1/2 stable Levy process, maximally skewed to the right, with its distribution function known in closed form as the Levy distribution. The model provides a reasonable fit to market data with just two parameters to assess dependency risk, a measure of correlation and that of the likelihood of a catastrophe.

Accurate and Fast Integration Over the Market Factor in One-Factor Gaussian Copula CDO Model

We propose and test a new algorithm for the numerical integration of the conditional total portfolio loss distribution over the market factor in the one-factor Gaussian copula model. It has higher precision than the popular numerical schemes which use the same or even higher number of sampling points. The consequent reduction of the number of sampling points allows us to reduce significantly the time of pricing CDO tranches or model calibration. We show that the popular Gauss-Hermite quadrature is unsuitable for this task and comment on the appropriateness of several integration methods for the calculation of tranche correlation sensitivities.

Modeling of CDO Squareds

This article is an extract of a more complete and detailed work on a specific class of credit derivatives, namely Collateralized Debt Obligations made out of Collateralized Debt Obligations, widely known as CDO Squareds (CDO2). This asset class has become quite popular in recent years as it deserves investors9 appetite for higher credit spread via leverage when investing in bonds. This article focuses more on the modeling aspects of plain vanilla structures than on factors which would allow for tailoring a CDO2 for specific investors9 risk appetites. First we give a brief introduction to CDO2 and recall important specifies which have to be taken into account for modeling. The following section intends to put John Hull9s ideas of pricing CDO2 which he proposed during the ICBI Global Derivatives Conference of May 2005 in concrete and formal mathematical terms. First, we will model the total loss conditional on the market factor on the respective underlying portfolios via a double-t factor copula model. This allows us to calculate the correlation between these underlying entities and to put the values into a correlation matrix. We will refer to this step as “capturing the second dimension, i.e. the cross dependence structure”. With the help of this matrix, it is now possible to calculate—conditional on the commonality factor—the total loss distribution of the CDO2 via a one-factor Gaussian Copula Model and numerical methods. The obtained CDO2 loss distribution is used for pricing a CDO2 tranche inspired by Credit Default Swap (CDS) pricing methods. <b>TOPICS:</b>CLOs, CDOs, and other structured credit, simulations, performance measurement

A Note on the Correlation Smile

The correct modeling of default dependence is essential for the valuation of multiname credit derivatives. However for the pricing of synthetic CDOs a one-factor Gaussian copula model with constant and equal pairwise correlations, default intensities and recovery rates for all assets in the reference portfolio has become the standard market model. If this model were a reflection of market opinion there wouldn’t be the implied correlation smile that is observed in the market. The purpose of this paper is to explain the structure of the smile by discussing the influence of dierent correlation matrices on CDO spreads.

Deriving the Dependence Structure of Portfolio Credit Derivatives using Evolutionary Algorithms

Even if the correct modeling of default dependence is essential for the valuation of portfolio credit derivatives, for the pricing of synthetic CDOs a one-factor Gaussian copula model with constant and equal pairwise correlations for all assets in the reference portfolio has become the standard market model. If this model were a reflection of market opinion, there wouldn't be the implied correlation smile that is observed in the market. The purpose of this paper is to derive a correlation structure from observed CDO tranche spreads. The correlation structure is chosen such that all tranche spreads of the traded CDO can be reproduced. This implied correlation structure can then be used to price off-market tranches with the same underlying as the traded CDO. Using this approach we can significantly reduce the risk to misprice off-market derivatives. Due to the complexity of the optimization problem we apply Evolutionary Algorithms.