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 α_{L} to α_{H}. The variables α_{L} and α_{H} are known as the attachment point and detachment point, respectively. If α_{L} is 8% and α_{H} 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 ...

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