A multi-name credit derivative is a security tied to an underlying portfolio of corporate bonds or other credit-sensitive securities. It enables investors to buy and sell protection against the default losses in the portfolio. The value of a multiname derivative depends on the distribution of portfolio loss at multiple horizons. Intensity-based models of the loss point process that are specified without reference to the portfolio constituents determine this distribution in terms of few economically meaningful parameters, and lead to tractable credit derivatives valuation relations that can be addressed by a variety of efficient methods. This paper proposes random thinning to extend the reach of these models beyond the portfolio level. Random thinning decomposes the portfolio loss process into the sum of its constituent loss processes, and allocates aggregate portfolio risk to sub-portfolios. We show that any loss process can be thinned, and that the associated thinning process is a probabilistic model for the next-to-default. We derive a formula for the constituent default probability in terms of the thinning process and the portfolio intensity, and show

Correlated default risk plays a significant role in financial markets. Dynamic intensity-based models, in which a firm default is governed by a stochastic intensity process, are widely used to model correlated default risk. The computations in these models can be performed by Monte Carlo simulation. The standard simulation method, which requires the discretization of the intensity process, leads to biased simulation estimators. The magnitude of the bias is often hard to quantify. This paper develops an exact simulation method for intensity-based models that leads to unbiased estimators of credit portfolio loss distributions, risk measures, and derivatives prices. In a first step, we construct a Markov chain that matches the marginal distribution of the point process describing the binary default state of each firm. This construction reduces the original estimation problem to one involving a simpler Markov chain expectation. In a second step, we estimate the Markov chain expectation using a simple acceptance/rejection scheme that facilitates exact sampling. To address rare event situations, the acceptance/rejection scheme is embedded in

We follow a long path for Credit Derivatives and Collateralized Debt Obligations (CDOs) in particular, from the introduction of the Gaussian copula model and the related implied correlations to the introduction of arbitrage-free dynamic loss models capable of calibrating all the tranches for all the maturities at the same time. En passant, we also illustrate the implied copula, a method that can consistently account for CDOs with different attachment and detachment points but not for different maturities. The discussion is abundantly supported by market examples through history. The dangers and critics we present to the use of the Gaussian copula and of implied correlation had all been published by us, among others, in 2006, showing that the quantitative community was aware of the model limitations before the crisis. We also explain why the Gaussian copula model is still used in its base correlation formulation, although under some possible extensions such as random recovery. Overall we conclude that the modeling effort in this area of the derivatives market is unfinished, partly for the lack of an operationally attractive

This paper presents a dynamic multi-portfolio default model for consistent and arbitragefree pricing synthetic CDO tranches that reference a bespoke portfolio. In order to incorporate standard tranche price information, we assume that the bespoke portfolio has name overlapping with some index portfolios. Dividing the total portfolio (parent) into non-overlapping sub-portfolios (children), and assuming homogeneity for both the parent and the children, we use a top-down dynamic default intensity model for the parent, and specify the conditional probability of default in the children given imminent default in the parent. We consider two fundamental cases which are building blocks of more complex applications: (a) the parent is an index and the bespoke is a child; and (b) the bespoke is the parent that contains one or more indices as children. When the parent is the index, the parent default process is uniquely determined independent of the children, and the child conditional default probability distribution is calibrated to the spreads of the children. When the bespoke is the parent and one or more children are indexes, we simultaneously calibrate the parent default intensity model and the child default probability to the standard tranches and child portfolio spreads. The model is designed to establish consistency between the pricing of standard tranches and the pricing of bespoke tranches. Application may include Portfolio enlargement where the bespoke tranche references a “global ” portfolio that contains “regional ” indexes as sub-portfolio. For example, tranches referencing CDX.NA.IG and iTraxx Europe. Portfolio thinning where the bespoke tranche references a sub-portfolio of an index. Combination of portfolio enlargement and thinning. For example, tranche referencing a subset of CDX and a subset of iTraxx. * Risk Management, The Depository Trust & Clearing Corporation.

We propose a bottom-up dynamic credit modelling framework. To achieve a non-trivial coupling, the marginal survival probability processes are multiplied by a common exponential martingale process. Still being a factor coupling, this approach relies on convolution of the conditionally independent random variables. However, due to the much better analytical tractability, this approach allow getting rid of the traditional recursions as convolution methods, and it does not require tuning the factor quadrature, as the factor integration step is not present. Also the model can be entirely speci…ed only in terms of the local moment surface of the common factor process, with di¤erent moments a¤ecting di¤erent segments of the loss distribution.