The bottom-up stress event model, proposed by Duffie and Singleton (1999), is a simple and intuitive model for portfolio credit risk. The model is seldom applied in practice since it is generally believed that the default times, as well as the loss distribution, of a portfolio under this modeling framework can only be generated by computationally expensive Monte Carlo simulation. In this note an alternative approach is taken, avoiding Monte Carlo simulations, making the model tractable and leading to efficient calibrations to data. The idea of the stress event model is easy to understand. Besides idiosyncratic default, each firm may default if there is a joint credit event (Duffie and Singleton 1999) or alternatively referred to as stress event (Schönbucher 2003).
This allows correlation through both changes in stress event intensity as well as through the occurrences of the stress events. The formal definition of the default time of a firm is given in Section 3. In Section 4, we develop a new approach to compute the loss distribution of a portfolio for the stress event model. We first identify independence conditions under which defaults of firms are independent. The loss distribution can then be decomposed into a series expansion for which each term admits a closed form expression. It turns out that only the first few terms of the series are needed to accurately approximate the loss distribution since stress events are infrequent. This leads to a very efficient method to compute the loss distribution of a portfolio.
The multi-factor model, a top-down approach model proposed by Long staff and Rajan (2008), provides strong empirical evidence that default dependence of a portfolio is necessarily multi-factor. The stress event model, like other bottom-up approach models, faces significant computational challenges when the number of non-idiosyncratic factors is more than one. This curse of dimensionality comes from the rapid increase of the number of unconditional loss distributions needed to compute the loss distribution. For example, if the number of conditional loss distributions needed to compute in a one-factor model is 100, it is expected that the number of conditional loss distributions needed in a L-factor model would be 100L.
This is not the case for our new approach due to the novel identification of independence conditions which result in important simplifications to the corresponding series expansion of the loss distribution for the stress event model. It turns out that the number of conditional loss distributions needed in our approach only increases mildly with the number of non-idiosyncratic factors. Hence, the increase in computational time due to the additional non-idiosyncratic factors in the stress event model is much smaller than that in other bottom-up approach models. This extra flexibility for adding additional non-idiosyncratic factors in the stress event model leads to a better fit to market data.
We demonstrate the tractability and efficiency of our approach by two calibration examples in Section 5. In the first example, the model is calibrated to the first five tranches of the 5-year CDX.NA.IG series 13 and the 125 underlying CDS spreads simultaneously. All the CDS spreads are matched exactly and the model implied tranche prices are within the bid-ask spread. In the second calibration example, we regard the stress event model as a top-down model and calibrate it to the term structure of the iTraxx Europe series 7 on four different days simultaneously. The 26 parameters of the model are calibrated to the 60 data and the root-mean-square relative error of the model implied tranche prices is 4.25%.
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Multi-Factor Bottom-Up Model for Pricing Credit Derivatives
