Pricing & Risk Management of Synthetic CDOs

The purpose of this paper is to analyze the risks of synthetic CDO structures and their sensitivity to model parameters. In order to measure these sensitivities, I also introduce the latest techniques in the pricing and risk management of synthetic CDOs. I show how to model the conditional and unconditional default distributions of a typical synthetic deal using a simple mathematical framework. Strictly speaking, the findings of this paper are only directly applicable to synthetic structures, however; many of the modeling and riskmanagement insights discussed apply to structures involving a waterfall. ___________________ *Jaffar Hussain a senior risk analyst at the Capital Markets Division of the Saudi National Commercial Bank. Introduction The purpose of this paper is to introduce the latest techniques in the pricing and risk management of synthetic CDOs and measure the risks and model sensitivities of a typical synthetic deal. Although these techniques are only directly applicable to synthetic structures, many of the modeling and risk-management insights discussed in the paper apply to structures involving a waterfall. A synthetic collateralized debt obligation, synthetic CDO, is made up of a portfolio of credit default swaps. The arranger of a synthetic CDO distributes the credit risk of the portfolio by creating and selling tranches to investors. Every tranche has an attachment and detachment point that determines the amount of loss, and correspondingly the number of defaults, the tranche can absorb. For example, the first tranche, known as the equity tranche, might be responsible for portfolio credit losses between 0% and 3%, the next tranche would then be responsible for portfolio losses that exceed 3% up to the size of the tranche, and so on. The least risky tranche of a CDO is known as the senior tranche, or super-senior tranche. Tranches between the equity tranche and the most senior tranche are known as mezzanine tranches. The challenge in pricing a synthetic CDO lies in the difficult task of formulating a model for the joint default behavior of the underlying reference assets. Understanding and modeling the joint default dynamics of the reference assets are important in order to compute the expected losses for each tranche. The expected losses, in turn, determine the fair spread of the tranche. In fact, once the joint default distribution of the reference assets has been specified, we can price any tranche that references these assets. In the following section, I will present a simple approach to computing the joint default distribution of a reference portfolio. The approach is based on a recursive procedure and requires no Monte Carlo simulations. I will also compare the results of this recursive approach with the results obtained using a Monte Carlo procedure that simulates the default times of the reference assets and the corresponding losses in the portfolio. The Monte Carlo approach is computationally time-consuming as it requires a large number of simulations in order to produce enough defaults that can impact the most senior tranches of a CDO. Computing the Distribution of Default Losses Pricing a synthetic CDO boils down to computing the joint distribution of defaults of the reference portfolio. Computing the default distribution, in turn, depends crucially on the default probabilities of the reference credits and the pairwise correlation between every pair of credits. The correlation among the assets will drive the joint default behavior of the assets. The model we use here is a one-factor model whereby the defaults are driven by one factor which we take to represent a common economic driver of credit events. Default losses are then calculated conditional on the state of this economic factor. This procedure will result in computing the conditional default distribution. The next step is to integrate the conditional default distribution over the common factor to arrive at the