Valuation and VaR Computation for CDOs Using Stein’s Method

Collateralized debt obligations (CDOs) are an innovation in the structured finance market that allow investors to invest in a diversified portfolio of assets at different risk attachment points to the portfolio. The basic concept behind a CDO is the redistribution of risk: some securities backed by a pool of assets in a CDO will be higher rated than the average rating of the portfolio and some will be lower rated. Generally, CDOs take two forms, cash flow or synthetic. For a cash flow vehicle, investor capital is used directly to purchase the portfolio collateral and the cash generated by the portfolio is used to pay the investors in the CDO. Synthetic CDOs are usually transactions that involve an exchange of cash flow through a credit default swap or a total rate of return swap. The CDO basically sells credit protection on a reference portfolio and receives all cash generated on the portfolio. In these types of transaction, the full capital structure is exchanged and there is no correlation risk for the CDO issuer. In this study, we are primarily interested in valuing (synthetic) single tranche CDO. It is very important to note that these products are exposed to correlation risk. In practice the CDO issuer sells protection on a portion of the capital structure on a reference portfolio of names. In exchange, he receives a running spread, usually paid quarterly, which value depends on the risk of the individual issuers in the reference portfolio and on a correlation hypothesis between those names. For liquid reference portfolios (indices) like Trac-X and iBoxx there exists now a liquid market for these single tranche CDOs and as a consequence for the correlation.