(11a)
where and
(11b)
Proof: See the appendix.
The risky valuation in Proposition 2 has a backward nature. The intermediate values are vital to determine the final price. For a discrete time interval, the current risky value has a dependence on the future risky value. Only on the final payment date , the value of the contract and the maximum amount of information needed to determine the risk-adjusted discount factor are revealed. The coupled valuation behavior allows us to capture wrong/right way risk properly where counterparty credit quality and market prices may be correlated. This type of problem can be best solved by working backwards in time, with the later risky value feeding into the earlier ones, so that the process builds on itself in a recursive fashion, which is referred to asbackward induction . The most popular backward induction valuation algorithms are lattice/tree and least square Monte Carlo.
For an intuitive explanation, we can posit that a defaultable contract under the unilateral credit risk assumption has an embedded default option (see Sorensen and Bollier (1994)). In other words, one party entering a defaultable financial transaction actually grants the other party an option to default. If we assume that a default may occur at any time, the default option is an American style option. American options normally have backward recursive natures and require backward induction valuations.
The similarity between American style financial options and American style default options is that both require a backward recursive valuation procedure. The difference between them is in the optimal strategy. The American financial option seeks an optimal value by comparing the exercise value with the continuation value, whereas the American default option seeks an optimal discount factor based on the option value in time.
The unilateral CVA, by definition, can be expressed as
(12)
Proposition 2 provides a general form for pricing a unilateral defaultable contract. Applying it to a particular situation in which we assume that all the payoffs are nonnegative, we derive the following corollary: