(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: