Hi, I have a question regarding the default probabilities and dependence/independence in general. Part(ii) So for this we assume that both firm's probability of default (and default intensities) are independent right? So we just rearrange the formula and solve for the intensities? Is this correct regarding my statement on independence? Part(iii) I follow the solution but I tried to calculated risk-neutral prob of double default by saying P(Double Default) = P(Default company A and company B) = P(Default A)*P(Default B) = (1-exp(-lambda,A*T)*(1-exp(-lambda,B*T). So I've assumed here that the prob of default of each firm is independent, but the solution says that these events are dependant on each other hence the risk-neutral expression for pricing the derivative is used to find P(Double Default). Then find intensities after. Have I understood this correctly? I kind of understand why you'd assume P(Double Default) is not an independent event for company A and B (as they are joint investors), but then why can we assume in part(ii) they are independent? To summarise I guess what I'm trying to say is we assume independent for part (ii) and dependant for (iii) and (iv) and why is this the case? Many thanks, Darragh
All we've got here is that the intensity rates in this question are constant. At this stage they can be treated separately because the bonds are issued by difference companies and because there's no reason to consider them together. The intensities are driven entirely by the price of each bond. We know the individual default intensities, but now we need to consider the double-default situation. The lambdaA and lambdaB aren't going to help here because the behaviour of the bonds might be connected. The solution employs the general risk-neutral pricing formula where the only non-zero payout from the derivative occurs when both bonds default. The double-default intensity (and probability) is driven entirely by the price of the derivative on offer. In part (iii) they might be connected, but we don't even have that assumption. By considering the joint behaviour we avoid the issue. Dependence is only explicitly required in part (iv) because this leads to the maximum possible default rate. The argument says that double-default is most likely to occur when (and if) the defaulting behaviour of the bonds is perfectly correlated - ie when one defaults so does the other one. This is well covered here.
Hi Steve, Thanks for your help on this, my comments below in bold with regard to part (iii) which is only last bit I'm confused on: I don't think it does. The solution employs the general risk-neutral pricing formula where the only non-zero payout from the derivative occurs when both bonds default. Ok so I calculated the probability of double default by saying P(double default) = P(Firm A and Firm B defaulting) = P(Firm A defaulting)*P(Firm B defaulting) = (1-exp(-lambda,A*(0.75))*(1-exp(-lambda,B*(0.75)) as prob of default for firm A and B is independent? Why does this not give me the same solution to using the Risk-Neutral formula? Thank you, Darragh
Hi Sorry, I missed your latest post. I've revised my earlier response to try and clarify. I didn't make a clear distinction between when we need to consider the defaults as being dependent (parts (iii) & (iv)) and when it doesn't matter even if they are (part (ii)). If the issuer of the derivative believed that the defaults were independent then your formula would be perfect and the price of the derivative wouldn't be $7,900.
