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Hedger

Buying an option from a risky counterparty

July 23rd, 2001, 8:01 pm

I am trying to build a pricing model for a short position in a forward contract with a risky counterparty. I am breaking the forward into a long put and a short call - the long put position as bought from a risky counterparty but the short call priced as from a risk-free one.How should I use BS for the option bought from a risky counterparty, for example one whose unsecured obligations trade at a spread S above the risk-free rate? Should I use the risk-free rate + S in the BS formula, which makes quite a big difference to the price, or should I use the ordinary risk-free rate to calculate the (forward) risk-neutral expected value of the option at expiry and then discount it to start date with the risky rate, namely risk-free + S? This would be equivalent to lending the premium to a risky counterparty for the duration of the option.
 
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Aaron
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Joined: July 23rd, 2001, 3:46 pm

Buying an option from a risky counterparty

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.
 
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Paul
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Joined: July 20th, 2001, 3:28 pm

Buying an option from a risky counterparty

July 24th, 2001, 7:39 am

The question (and answer) throws up lots of interesting points...more questions and answers. Here are a few random thoughts.Choose your default model: It sounds like you are going for the 'reduced-form' model. But when does default happen? Only at expiry or can it happen before?If the BS eqn is V_t + 0.5 vol^2 S^2 V_SS + r S V_S - r V = 0 then the reduced form version is justV_t + 0.5 vol^2 S^2 V_SS + r S V_S - (r + p) V = 0 where P is some probability of default. Notice that only one of the interest rates has been modified, the one that represents the discounting.If the contract is European, as Aaron says, just divide up into the two contracts and PV the negative cashflow differently. However, you may want to have more prob of default for low S, where counterparty owes you the most. Hence p(S). But then you will have to solve the BS eqn numerically, no nice formulae. (Oh, BTW, I'm using different notation my S is asset price, p is the spread, your S!)If the contract is American things get more interesting. Will the counterparty somehow declare default before expiry? If you exercise early can they still default? Whatever the situation you are unlikely to be able to divide up the contract into two pieces. The trick would be something like solving the BS equation with p(V). I.e. the risk of default depends on how much money the counterparty stands to lose. Numerically there are no real difficulties. P