Ebook Stochastic Intensity Modelling for Structured Credit Exotics
In the last several years the market for credit derivatives experienced an explosive growth both in terms of volume and innovation. The market in standard tranche CDOs became liquid (ITX, CDX, and others) and provides possibilities for hedging correlation risk. At the same time new exotic products are traded over the counter.
These can be split broadly in two categories: default derivatives with complex payoffs (bespoke tranches, CDO2 and others) and derivatives with payoffs that depend of spread levels and mark-to-market (options on tranches, leveraged super-senior, credit CPPI, etc). While the former require modelling of the defaults of individual single name credits, the latter require dynamical modelling of defaults and spread levels. This growth produces a need for new models for valuing and managing the risk.
The Gaussian copula model, became the industry standard for the valuation of CDO. Emergence of a skew market in CDO of standard portfolios then gave rise to a number of extensions of this model that attempt to account for the structure of observed prices (for example). All these approaches are similar in that they model loss distribution of a basket of credits at a given time horizon starting from default probabilities of single names through some copula function. These models have had variable success in calibrating to observed prices, depending on the complexity and flexibility of the copula and some became popular. Their main advantage is that the properties of single name credits are explicit inputs, which allows one to model more complicated derivatives such as CDO2. The main drawback is that explicit intensity dynamics is absent from these models, which makes it impossible to use these models for modelling of more exotic credit derivatives, such as option on tranches.
Recently several authors proposed models that directly model the dynamics of the loss distribution of a given portfolio. Forward loss distribution is an input into the model which makes it possible to calibrate exactly to the structure of observed prices by construction. The model has dynamics as well so it possible to price options on tranches. The main drawback of this model is that single name information is not an explicit input into the model and that makes it difficult to price bespoke basket CDOs or certain types of exotic structures such as CDO2.
Direct modelling of stochastic credit default intensities for the valuation of CDO transactions was proposed by Duffie and Garleanu. The main advantage of this approach is that the dynamics of default intensities for each credit can be specified which allows one to deal with CDO, CDO2 and options on tranches within a single model, at least in principle. For a long time the perception was that this class of models is too complex and requires the use of Monte-Carlo methods for their implementation making efficient calibration impossible and deterring practitioners from using them in industrial applications. This motivated us to develop a modelling framework under which the default intensities of each single-name are modelled individually, yet the framework is still simple enough to allow for efficient calculations suited for use by practitioners. Very recently similar approaches were proposed in the literature.
The remainder of the paper is organised as follows. In Section 2 we described the basic setup of the model. Then in Section 3 we discuss parametrisation and calibration of the model to the CDO tranche market. In Section 4 we extend the framework to the pricing of option-like exotic credit derivatives. Finally we conclude in Section 5.
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