Hi,Does anyone know how to hedge an option on a tranche in a synthetic CDO?Cheers,Riz

QuoteOriginally posted by: chdryraHi,Does anyone know how to hedge an option on a tranche in a synthetic CDO?Cheers,RizI think it depends on the option and where the tranche is in the capital structure. Can you be a little more specific??

Apologies.Say you have a CDO with 100 names (CDS's) in it. Maturity is 5 years. Let's say 3 tranches with attachments 0-2%, 2-10% and 10-100%. I sell the 2-10% tranche which pays k bps to an investor together with a put option on the tranche struck at k bps and maturity 6 months (ie tranche is effectively cancellable on the first payment). The tranche put option is essentialy a call option on protection. How would I hedge the put option?Cheers,Riz

QuoteOriginally posted by: chdryraApologies.Say you have a CDO with 100 names (CDS's) in it. Maturity is 5 years. Let's say 3 tranches with attachments 0-2%, 2-10% and 10-100%. I sell the 2-10% tranche which pays k bps to an investor together with a put option on the tranche struck at k bps and maturity 6 months (ie tranche is effectively cancellable on the first payment). The tranche put option is essentialy a call option on protection. How would I hedge the put option?Cheers,RizOkay, so the decision on whether or not to put is based on whether the equivalent tranche should pay more or less for credit protection at exercise. I assume that you are trying to hedge this risk rather than the value of the put from month 0 to 6. You have your protection, if the guy decides to put you still want to be able to put on protection at the same level you did to him. If he doesn't put you don't want to have this tranche hedged. The determinants of where a given tranche should price out for the most part are going to be correlation and absolute level of credit risk. I'm thinking you could figure out your deltas to protect the tranche, as though you were going to retain it and hedge it, but use CDS options instead of straight swaps, or use cancelable CDS, however with the cancelable CDS you would be paying for protection 2x during the period of the option. That should cover you on the absolute credit risk side. Shifts in correlation I'm not sure what product you would use to hedge those. I guess keep checking back, there are some pretty bright people on here who may have experience on this, they're all busy making money right now though.

first consider the case where your tranche is actually the entire portfolio cap structure (0-100%). You have the option to sell portfolio protection, say with expiration in 3 months, at some strike. If the portfolio itself is a traded security (e.g. iboxx/tracx), then you can delta hedge with a position in the index. If it does not trade, then you will have to hedge with delta positions in the underlying credits. For the full cap structure, you really just care about the average spread, so calculating your deltas is not too hard unless you have an extremely barbelled distribution of spreads.Now, lets relax the assumption that your option is to buy protection on the full cap structure, and let it be for any tranche with attachment points a and b.The underlying can be quite sensitive to the distribution of spreads in the portfolio as well as the average spread. In addition, it is also sensitive to correlation. While it is theoretically possible to get deltas and corr01 numbers on each reference credit and hedge using those values, I wouldn't recommend it. You will have all kinds of "cross-gamma" effects and sensitivities to changes in the shape of the portfolio spread distribution that don't even have greek letters. I do not know of any market consensus on how to hedge these options (notice how much massive volume there is in tranche options.) One possibility is factor-based hedging approach, i.e. allow spreads to evolve as linear combinations of a small number of factors and hedge the factor deltas. This has the advantage of giving you deltas that are relatively stable with respect to small movements in the underlying, and of giving you some flexibility in how to actually implement the hedge in terms of single name cds. It has the disadvantage of forcing you to estimate a lot of parameters (factor exposures) which ups the noise level considerably.

Thanks for the nice summary. My question is whether anyone has experience with possible model errors for pricing and hedging due to a misspecification of the factor copula (Gaussian versus t versus Archimedean etc) for options on tranches.

The Gaussian copula is flat out wrong for pricing tranche options unless it is a conditional Gaussian with a term structure of correlation. Simply put, the Gaussian copula has a memory wrt correlation so the forward tranche value will be inconsistent with the spot if you put both into the same Gaussian copula.However, that is the least of your problems. Think about it. In order to get arbed because you specified the wrong copula, some one has to be able to trade the spot tranche, the forward tranche, and an option on the tranche, and still get arb after absorbing all the transaction costs.As an academic exercise, however, I think it would be interesting to compare a Gaussian copula with non-time-varying correlation, a Gaussian with a term structure of corr, and a different family without a memory property (perhaps a form of archimedian.)

Cheers.

JabairuStork,Can you elaborate or point reference articles that explain the "memory" of the gaussian copula that you ention below. I think what you're saying is that if you price a default leg from t1 to t2 as the difference between the leg from (0,t2) minus the leg from (0,t1), then you get a different answer than if you do it directly as the leg from (t1,t2). This is all assuming I use the same flat correlation for all 3 pricings. This sounds like the same argument to defend base correlation versus "forward" correlation. But what is the precise explanation of this "memory" effect?

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