This paper presents computational techniques that make a certain class of fully dynamic intensity-based models for portfolio credit risk, along the lines of Duffie and Gârleanu (2001) and Mortensen (2006), just as computationally tractable as the static copula model. Compared to previous such models in the literature, we improve the fit to CDX tranche spreads by a factor of around five, by explicitly taking liquidity and modified-restructuring risk into account, and by allowing for a more flexible correlation structure. The resulting model can be used to hedge a wide range of risks in the credit market, such as the risk of changes in correlations, volatilities, or idiosyncratic default risk.

Due to their computational efficiency, simple factor models remain popular in the pricing of credit portfolio derivatives. In this paper, we continue the elaboration on the fundamental structure of factor models initiated in [3], with a special focus on term structure effects. We describe a number of techniques to understand, and to improve control over, portfolio loss distribution term structures and intertemporal loss correlation. As part of our analysis, we introduce an extension of the RFL model ([2]) to incorporate jumps in the systematic factor and in firm residuals. We also numerically test the dependence of forward-starting synthetic CDOs on the correlation of losses across time. Finally, our analysis highlights the fact that several of the models suggested in the literature are essentially equivalent. 1

The purpose of this article is to show that the correlation smile in liquid CDS index tranches can be explained by the same ideas that have proven to explain the volatility smile in equity options. First, we extend a structural model proposed by Luciano and Schoutens [2005] that models firm values by Variance Gamma processes. We show that these extensions can explain the index spread curves and tranche quotes of DJ iTraxx 5 year simultaneously. Second, we extract the resulting dependence structure into a factor copula approach. The resulting VG copula shares the advantages of the well known Gaussian copula that underlies the most important industry models like CreditMetrics and KMV. We apply this approach to weekly spreads of DJ iTraxx 5 year. We show that this approach fits significantly better to the correlation smile than comparable copula approaches. 1 In the 1980s, Collateralized Debt Obligations (CDOs) were introduced for balance sheet risk management. The emergence of credit derivatives in the 1990s offered the possibility of synthetic risk transfer of a portfolio of bonds or loans, too. Since 2003, credit risk of

Credit default events can happen in a variety of ways. Sometimes a credit slides towards default over a period of time, while other credits default instantly and without warning. Sometimes a credit's problems are unique to itself, but at other times many credits are influenced by common trends or events.

This article describes a new approach to the risk-neutral valuation of CDO tranches, based on a general specification of the tranche loss distributions and the index default distribution. The new model is a term-structure model, and the generality with which the basic distributions are specified allows it to be perfectly calibrated to any set of market prices (for any number of tranches and maturities) that is arbitrage-free. The use of the new model is illustrated by testing market prices for the standardized iTraxx index tranches (for all marketed tranches and maturities) to see if they are arbitrage-free. Other examples include the determination of the arbitrage-free range of prices allowed for an unmarketed standardized tranche and the determination of the cost of exiting a tranche position. For the latter example, both arbitrage-free price ranges, and a preferred price, are obtained. Prices for unmarketed maturities and unmarketed non-standard tranches are also obtained by an interpolation and extrapolation procedure. Because the model is an incomplete-market model characterized by many more parameters than market prices, it was essential to develop an efficient optimization approach to valuation. The article also makes use of a new approach to the problem of unequal recovery rates and notionals.

A semi-analytical parametric approach to modeling default dependency is presented. It is a multi-factor model based on instantaneous default correlation that also takes into account higher order default correlations. It is capable of accommodating a term structure of default correlations and has a dynamic formulation in the form of a continuous time Markov chain. With two factors and a constant hazard rate, it provides perfect fits to four tranches of CDX.NA.IG and iTraxx Europe CDOs of 5, 7 and 10 year maturities. With time dependent hazard rates, it provides perfect fits to all the five tranches for all three maturities.

Delayed, hence non-simultaneous, dependent defaults are discussed in a reduced form model. The model is a generalization of a multi-factor model based on simultaneous defaults to incorporate delayed defaults. It provides a natural smoothening of discontinuities in the joint probability densities in models with simultaneous defaults. It is a dynamic model that exhibits default contagion in a multi-factor setting. It admits an efficient Monte Carlo simulation algorithm that can handle heterogeneous collections of credit names. It can be calibrated to provide exact fits to CDX.NA.IG and iTraxx Europe CDOs just as its version with simultaneous defaults.

Abstract. We present a new structural model for single name equity and credit derivatives which we also correlate across reference names to produce a model for bespoke synthetic CDOs. The model captures volatility and outlook risk along with correlation risk for small and for large moves separately. We show that the model calibrates well to both equity structured products and credit derivatives. As examples, we discuss a number of single name derivatives on IBM spanning the credit-equity spectrum and ranging from volatility swaps, to cliquets, CDS options and CDSs on leveraged loans with pre-payment risk. We also use the model to price tranches on the investment grade DJ.CDX.IG index along with tranches on the high yield index DJ.CDX.HY. We show that the model gives consistent and high precision pricing across all these derivative asset classes. We show that this can be achieved consistently, with the very same parameter choices across these diverse derivative assets and making use of only minor explicit time dependencies. 1.

The jump distribution for the default intensities in a reduced form framework is modeled and calibrated to provide reasonable fits to CDX.NA.IG and iTraxx Europe CDOs, to 5, 7 and 10 year maturities simultaneously. Calibration is carried out using an efficient Monte Carlo simulation algorithm that appears to be suitable for both homogeneous and heterogeneous collections of credit names. The underlying jump process is found to relate closely to a maximally skewed stable Lévy process with index of stability α ∼ 1.5. The market standard for pricing credit derivatives sensitive to default dependency is based on the Gaussian copula. As is well-known, this method is inadequate to price nonstandard products. The model as such is not able to explain the correlation smile. Better models addressing these issues have been developed by various authors. Some recent work in this direction in a reduced form framework involves modeling the default intensities as in Joshi and Stacey [2005], Chapovsky, Rennie and Tavares [2006], Errais, Giesecke and Goldberg [2006], Balakrishna [2007]; modeling dependency with simultaneous defaults as in

This article describes a model that appropriately treats the incompletemarket aspects of collateralized debt obligations (CDO’s). The model is a term-structure model that can be calibrated so that it precisely reproduces, in terms of a single set of model parameters, the market values of the prices of contracts on a number of different tranches and having a number of different maturities for each tranche. The calibrated model gives arbitrage-free ranges of prices for derivatives when this is appropriate, as is the case in the pricing of an unmarketed tranche. In this case, risk-neutral pricing is only a first step in the pricing process, which is completed by negotiation in the market. Thus, within a certain price range, tranche prices are market prices that are determined by investor perceptions as influenced by analysts ’ reports, etc (and not by the predictions of a particular copula model). A related conclusion is that the Black-Scholes type hedging of changes in tranche value with time is not appropriate, and this is illustrated with reference to the effects of the downgrade of Ford and General Motors in May 2005. By way of contrast, definite prices for certain other derivatives are found when this is appropriate, as in the case of the pricing of forward-start CDO’s.