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We discuss the general optimization problem of choosing a copula with minimum entropy relative to a specified copula and a computationally intensive procedure to solve its dual. These techniques are applied to constructing an empirical copula for CDO tranche pricing. The empirical copula is chosen to be as close as possible to the industry standard Gaussian copula while ensuring a close fit to market tranche quotes. We find that the empirical copula performs noticeably better than the base correlation approach in pricing non-standard tranches and that the market view of default dependence is influenced by maturity.

The modeling of credit events is in effect the modeling of the times to default of various names. The distribution of individual times to default can be calibrated from CDS quotes, but for more complicated instruments, such as CDOs, the joint law is needed. Industry practice is to model this correlation using a copula/base correlation approach, which suffers significant deficiencies. We present a new approach to default correlation modeling, where defaults of different names are driven by a common continuous-time Markov process. Individual default probabilities and default correlations can be calculated in closed form. We provide semi-analytic formulas for the pricing of CDO tranches via Laplace-transform techniques which are both fast and easy to implement. The model calibrates to quoted tranche prices with a high degree of precision and allows one to price non-standard tranches in a consistent and arbitrage-free manner. The number of parameters of the model is flexible and can be adjusted to adapt to the set of market data one is calibrating to. More importantly, the model is dynamically consistent and can be used to price options on tranches and other exotic path-dependent products.

Numerous incidents in the financial world have exposed the need for the design and analysis of models for correlated default timings. Some models have been studied in this regard which can capture the feedback in case of a major credit event. We extend the research in the same direction by proposing a new family of models having the feedback phenomena and capturing the effects of regime switching economy on the market. The regime switching economy is modeled by a continuous time Markov chain. The Markov chain may also be interpreted to represent the credit rating of the firm whose bond we seek to price. We model the default intensity in a pool of firms using the Markov chain and a risk factor process. We price some single-name and multi-name credit derivatives in terms of certain transforms of the default and loss processes. These transforms can be calculated explicitly in case the default intensity is modeled as a linear function of a conditionally affine jump diffusion process. In such a case, under suitable technical conditions, the price of credit derivatives are obtained as solutions to a system of ODEs with weak coupling, subject to appropriate terminal conditions. Solving the system of ODEs numerically, we analyze the credit derivative spreads and compare their behavior with the nonswitching counterparts. We show that our model can easily incorporate the effects of business cycle. We demonstrate the impact on spreads of the inclusion of rare states that attempt to capture a tight liquidity situation. These states are characterized by low floating interest rate, high default intensity rate, and high volatility. We also model the effects of firm restructuring on the credit spread, in case of a default.

This paper uses a unique data set of more than 1000 synthetic Collateralized Debt Obligations (CDOs) deals to describe typical structures, their pricing and performance with the aim of identifying the factors behind the spectacular collapse of this important segment of structured credit market in late 2008. The data suggests that mark-to-market losses on many synthetic CDO tranches were much more significant than in case of simpler, lower-rated products despite the former experiencing little or no impairment of the notional. The losses were driven instead by the concentration of relatively limited number of defaults in a short period of time, suggesting that pre-crisis pricing must have seriously underestimated such risk of default clustering. In view of the post-crisis pick-up in synthetic CDO issuance, the paper attempts to heed this lesson and offer a simple factor model of default correlation in the spirit of Marshall–Olkin that is naturally suited to capturing the temporal dimension of default dependencies that have been crucial for synthetic CDOs investors. The model allows building a rich dependence structure capable of consistently fitting standardized iTraxx and CDX index tranches, which makes it ideal for pricing bespoke CDOs.

The objective of this paper is to offer a methodology for sizing credit-sensitive Asset Backed Securities (ABS) used in the prime mortgage lending sector in the U.S. and then to evaluate their relative performance. Using a multi-factor Monte Carlo simulation framework, we perform a four-step analysis. First, we estimate scenario-specific credit losses from a given mortgage pool. We then structure the pool into a "6-pack" subordination structure based on statistically-determined stress economic scenarios. Next, we estimate performance indicators of the tranches to compare risk-adjusted returns. Finally, we report our results in terms of tranche-specific risk-adjusted returns. The results indicate that the middle tranches of ABS, e.g., BBB and BB, possess the lowest risk-adjusted returns. We also find and explain a "cliff" phenomenon in the tranche-level principal cash flows.

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.

We discuss the general optimization problem of choosing a copula with minimum entropy relative to a specified copula and a computationally intensive procedure to solve its dual. These techniques are applied to constructing an empirical copula for CDO tranche pricing. The empirical copula is chosen to be as close as possible to the industry standard Gaussian copula while ensuring a close fit to market tranche quotes. We find that the empirical copula performs noticeably better than the base correlation approach in pricing non-standard tranches and that the market view of default dependence is influenced by maturity.