July 23rd, 2001, 11:54 pm
I think the answer depends on the nature of the risk.The most important factor is whether the risk is correlated to the price of the underlying. A particular case would be if your option is likely to cause counterparty bankruptcy if it goes in the money. Obviously, any positive correlation of option value and counterparty financial health increases the value of the option, any negative correlation is reduces it.Assuming independence of counterpary risk and underlying price movement, the other possible complication is the term structure of the counterparty risk. If the counterparty's credit deteriorates, you might find it optimal to exercise early to avoid default. Or, if the term structure were non-uniform (say the counterparty is likely to default only when a note comes due in March 2002) you might want to exercise right before the bad period. In general, the worst term structure would be a binary event immediately after you buy the option (either the counterparty credit becomes perfect or worthless), the best would be the same binary event one nanosecond before expiry (in which case you exercise, or not, two nanoseconds before expiry). A uniform term structure of credit risk would be intermediate. If credit derivatives are traded on this counterparty, it should be possible to deduce a default probability curve and solve for an option price backward from expiry.If you have a European option, or if default has the worst possible term structure, then it is correct to adjust the BS price as you suggest: take the future value and discount at the counterparty’s borrowing rate. This will overpenalize an American option written by a counterparty with a uniform term structure of credit risk.