International Journal of Theoretical and Applied Finance

This paper considers credit portfolio models based on Levy processes in general, and the gamma model in particular. It describes both single-name and multi-name situations using the gamma model, along with calibration fits and a comparison of various simple Levy models. There is also extensive historical data, including the May 2005 Auto crisis, which can be described in terms of the model. Parameter-based risk management using the gamma model is also discussed along with implementation details.

We extend the common Poisson shock framework reviewed for example in Lindskog and McNeil [15] to a formulation avoiding repeated defaults, thus obtaining a model that can account consistently for single name default dynamics, cluster default dynamics and default counting process. This approach allows one to introduce significant dynamics, improving on the standard "bottom-up" approaches, and to achieve true consistency with single names, improving on most "top-down" loss models. Furthermore, the resulting GPCL model has important links with the previous GPL dynamical loss model in Brigo et al. [6], which we point out. Model extensions allowing for more articulated spread and recovery dynamics are hinted at. Calibration to both DJi-TRAXX and CDX index and tranche data across attachments and maturities shows that the GPCL model has the same calibration power as the GPL model while allowing for consistency with single names.

We propose a class of credit models where we model default intensity as a jump-diffusion stochastic process. We demonstrate how this class of models can be specialised to value multi-asset derivatives such as CDO and CDO² in an efficient way. We also suggest how it can be adapted to the pricing of option on tranche and leverage tranche deals. We discuss how the model performs when calibrated to the market.

A model for large portfolio credit risk is developed by using results on the asymptotic behavior of stochastic networks. An efficient pricing technique is proposed using a newly-introduced quadrature algorithm. Accurate calibration to iTraxx tranche spreads is demonstrated.

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.

In this paper, we present a methodology for pricing and hedging portfolio credit derivatives in a dynamic credit model. Starting with a single-name Marshall–Olkin framework, we build a dynamic top-down version of the model, which is tractable and preserves the intuition of the original setting. In the first part of the paper, we derive analytically the Fourier transform of the loss variable and we study the skew dynamics implied by the model. In the second part, we develop a theory for dynamic hedging of portfolio credit derivatives. Since the market is incomplete, due to the residual correlation risk, perfect replication cannot be achieved. To find the hedging strategies, we use a quadratic risk minimization criterion.

The pricing of exotic portfolio products, e.g. path-dependent CDO tranches, relies on the joint probability distribution of portfolio losses at different time horizons. We discuss a range of methods to construct the joint distribution in a way that is consistent with market prices of vanilla CDO tranches. As an example, we show how our loss-linking methods provide estimates for the breakeven spreads of forward-starting tranches. .

We define forward copula models and introduce the concept of "chaining" such models. We discuss the use of these concepts in the calibration to the term structure of tranche quotes.