The earliest type of a collateralized debt obligation (CDO) takes the form of a cash CDO. It relies on the cash flows generated from a portfolio of risky financial assets (such as corporate loans, mortgage loans or emerging market corporate bonds) to pay returns to investors holding CDO notes. The cash flows associated with the reference portfolio are sold to investors in different tranches to cater their different risk-return profiles. Each tranche has a different default risk profile as the subordinated tranches bear the first losses associated with the defaults in the reference portfolio.
Rapid growth and liquidity in the credit derivatives market led to the emergence of a second type of CDO, synthetic CDOs. Instead of referencing assets that can generate cash flows, synthetic CDOs reference credit derivatives, usually credit default swaps (CDSs). Again, the cumulative loss on the pool is divided into different tranches. A given tranche covers only a portion of the total potential losses of the portfolio. However, its payoff is associated with the cumulative default loss distribution of the underlying portfolio of reference contracts. The seller of protection on that tranche, i.e. the holder of that tranche (who is then the taker of default risk) receives regular cash flow payments as a percentage of the remaining balance of that tranche. In return, he must pay the buyer of protection any losses on that tranche that are incurred through defaults.
An example: The holder of the 3%-7% tranche gets quarterly cash flow payments as a percentage of the balance. When the total loss of the pool exceeds 3%, the balance of 3%-7% starts to reduce. When the total loss reaches 7%, the balance of that tranche is depleted.
To price CDOs, one has to proceed along the following steps:
(1) Extract the implied default probability from individual CDS prices.
(2) From (1), build the joint distribution function by assuming some correlation structure.
(3) Either use simulation or construct the distribution function of the cumulative loss analytically, and get CDO prices.
(4) Use (3) to further calibrate the correlation parameters so that in the end the model prices match with the market prices (by optimization).