Factor models for credit correlation

Stewart Inglis and Alex Lipton describe dynamic and static factor models for credit correlation, and show how the static model can be calibrated to the market and used for the pricing of standard and bespoke tranches including tranchelets

Credit default swaps (CDSs) represent insurance contracts on individual obligors, synthetic collateralised default obligations (CDOs) represent various tranches of baskets backed by CDSs, while cash CDOs are backed by corporate bonds, mortgages, and other assets. Originally, the Gaussian Copula approach was used to price CDO tranches (see, for example, Li (2000)). Unfortunately, in its basic form it is incapable of reproducing market prices. Several researchers have tried to generalise it with some degree of success; these attempts are nicely summarised in recent review papers by Andersen & Sidenius (2005) and Burtschell et al (2005). However, one can say that an adequate framework for describing credit correlation is still missing. The reasons for the modelling difficulties are manifold, but high dimensionality of the problem is clearly one of them.

Currently, the so-called base correlation framework is used by most (but not all) investment banks for the pricing and risk management of credit baskets and their tranches. While sufficiently flexible to reproduce the quoted break-even coupons (BECs) for the standard CDX and iTraxx tranches, this framework is conceptually unsound and its usage can result in unexpected (and often unpleasant) P&L surprises. Pricing of bespoke tranches in this framework is even more problematic. However, the true weakness of the base correlation approach becomes apparent when it is applied to the pricing of dynamic products, such as forward-starting tranches and the like.

There exists an obvious temptation to reduce the dimensionality of the correlation problem by assuming the evolution of different credits in the basket depends on just a handful of common factors plus (possibly) some idiosyncratic factors that can be integrated out. This can be done in two complementary ways: (a) by modelling the dynamics of hazard rates (see Duffie & Garleanu (2001), Mortensen (2006), Chapovsky et al (2006)); or (b) by modelling the evolution of the so-called latent factors (see for example, Baxter (2007), among others).

Here, we propose a single latent factor model that provides a satisfactory description of the evolution of individual names as well as their collective behaviour. We also build a static version of the model that we apply to solving problems static in nature, such as pricing of standardised and bespoke tranches, etc.

The paper is organised as follows. We briefly describe a structural model for a single-name CDS, which we use as a prototype for developing a multi-name model. Then, we present a dynamic model for solving the pricing problem for a tranche of a credit basket and other structured credit products. We then introduce a static version of the model and discuss its numerical implementation, and calibrate the static model to the market for standardised CDX and iTraxx tranches. This is followed by a discussion of possible applications of the static model to pricing bespoke tranches and a brief summary of our findings.

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