Pricing options on CDO tranches

rleeuk · May 4th, 2007, 9:23 pm

Can someone recommand a good reading on this topic? I'm interested to know if there has been any research published by the banks. The only systematic treatment I've seen so far is from Hull & White..Thanks a lot!

scholar · May 7th, 2007, 1:51 pm

check out defaultrisk.com, there are many papers out there on tranche options

rleeuk · May 7th, 2007, 2:57 pm

Thanks for the pointer, I saw some papers there which just describe briefly the pricing of tranche options as an application of their particular model. I was looking for the CDO equivalent to Schönbucher's paper "A Measure of Survival" on CDS options, which covers the fundamentals. Any recommendations? ("Forward and European Options on CDO Tranches" by Hull & White is closer to what I'm looking for)Basically I want to understand why lognormal specification for the spread movement is not such a great choice, as implied by a couple of people I've talked to.

scholar · May 7th, 2007, 3:48 pm

Unlike a CDS option, a tranche option does not knock out upon default, so there is no need to use a survival measure, a usual risk-neutral one will do. As for the lognormal dynamics, the simplest answer is that the option price is a function of two variables, realized portfolio loss and future tranche spread at option maturity (or future MTM of the tranche), not one. The use of Black Scholes for tranche option seems hard to justify.

rleeuk · May 7th, 2007, 4:58 pm

So for the spread dynamics, do you basically mean that lognormal is unrealistic? I'm just thinking, since there is nothing wrong with the change of numeraire argument, can't we just use the value of the premium leg as numeraire, and reduce the problem to modelling spread movement only? then we can price the option if (a big "if") we use the correct dynamics of the spread _under the new measure_, which may not be lognormal, because it's "divided" by the portfolio loss.does this make any sense?

rleeuk · May 8th, 2007, 10:35 am

I guess a better question is, where exactly does the argument in http://www.schonbucher.de/papers/cdsoptions.pdf break down for tranche options, if we replace the default time tau by the stopping time at which the tranche is completely wiped out? Hull & White's paper seem to imply that the same formula works..

scholar · May 8th, 2007, 12:53 pm

I think the difference between CDS options and tranche options is precisely due to a multi-name nature of the latter: for a CDS option, we only have to care that CDS survives up to the option maturity, and this is achieved by putting zero weights on scenarii where default occurs prior to the option expiration. For a tranche, the option does not knock out upon defaults in the portfolio. Instead, we have to explicitly model contagion, i.e. impact of defaults on spreads of surviving names. I don't know how this can be achieved with a change of measure trick.

rleeuk · May 8th, 2007, 2:00 pm

Can't we get around this dependency problem between loss and spread by considering everything relative to the loss? I guess my point is, instead of postulating the dynamics of X_t and Y_t separately and worrying about their dependence structure, if we are only interested in X_t / Y_t, or more generally a function of X_t and Y_t, why not just model X_t / Y_t in the first place?Maybe lognormal dynamics of spread (X_t / Y_t) would imply unrealistic dynamics for the loss process, Im just guessing..

rleeuk · May 8th, 2007, 2:19 pm

OK then to recover the value of the option in unit cash, you need to value the premium leg consistently with the specification of the spread dynamics, is that where the problem is?

scholar · May 8th, 2007, 7:44 pm

I would guess so. Another problem with a change of measure would be to estimate parameters I think.