Synthetic CDOs and Their Valuation
Synthetic collateralized debt obligations (CDOs) consist of tranches where one party (Party A) agrees to make payments to another party (Party B) that are equal to those losses on a specified portfolio of debt instruments that are in a certain range. In return, Party B agrees to make payments to Party A that are a certain proportion of the amount of principal that is being insured.
Suppose that the range of losses for a particular tranche is from αL to αH. The variables αL and αH are known as the attachment point and detachment point, respectively. If αL is 8% and αH is 18%, Party A pays to Party B the losses on the portfolio, as they are incurred, in the range 8% to 18% of the total principal of the portfolio. The first 8% of losses on the portfolio does not therefore affect the tranche. The tranche is responsible for the next 10% of losses and its notional principal (initially 18−8 = 10% of the portfolio principal) reduces as these losses are incurred. The tranche is wiped out when losses exceed 18%. The payments that are made by Party B to Party A are made periodically at a specified rate applied to the remaining notional tranche principal. This specified rate is known as the tranche spread.
The usual assumption is that all the debt instruments in the portfolio have the same probability distribution for the time to default. Define Q(t) as the probability of a debt instrument defaulting by time t. The one-factor Gaussian copula model ...