When I am looking at my nice convex iTraxx tranche position, I can imagine writing options to monetise some of that convexity (at the expensive of gains in volatile markets). But without a real model that describes the effect of volatility on tranched indices, I worry that I am missing something.There is a serious need for a synthetic tranched CDO model, or an nTD (nth to default) model, that takes into account the volatility of spreads and that takes into account spread-spread correlation.If the model exists and is widely known and used, please excuse my ignorance! And please PM me a link to a paper.It is well known that there exists a framework for pricing options on CDS. People often refer to the work of Schoenbucher. Presumably this framework could be extended to allow spread-spread correlations, and therefore one should be able to price an option on a basket of CDSs without much difficulty.Is it not possible to generalize this pricing framework further to price nTD and synthetic tranched indices with volatile underlyings?Where is the the difficulty? Would it make it easier to assume that default time correlations are the same as spread-spread correlations? Then at least you would have fewer parameters flying around.Naively (health warning – I am a trader, not a quant!), from the standpoint of a simulation, here is how I think of these things.In equity worst-of-n puts, you simulate the forward prices of all the underlyings, using very much the same machinery that is used in a standard gaussian copula simulation. In fact, unknown to most equity derivs guys, they are implicitly using the gaussian copula assumption in their calculations. In verbal pseudcode here is worst-of-n put pricing:(1) using correlation matrix and cholesky decomposition, produce correlated (0,1) randoms(2) adding in drift and volatility, convert to correlated forward simulation(3) measure the payoff function(4) repeat until reasonably convergedI would imagine that CDS basket option pricing would go more or less along those lines, but with a few more complications. And here is current synthetic CDO simulation pricing:(1) using correlation matrix and cholesky decomposition, produce correlated (0,1) randoms(2) convert to uniform correlated randoms(3) using individual spreads and recoveries determine correlated default times(4) measure the price of each tranche(5) repeat until reasonably converged Maybe what is needed is more of a day by day (week by week?) simulation in which you add in ingredients like this:(n) generate random CDS levels for all underlyings using spread correlations and spread volatilities(n+1) check for defaults(n+2) if T=contract maturity do all accounting and measure price of each tranche Could that work? Why is this so difficult? Or is it “easy”, but unbelievably computationally intensive?Any links appreciated.

I've always thought that pricing nTD and tranches by looking only at default correlation was silly for market makers. If you're gonna hold onto the trade (unhedged) until maturity then OK, but that's not what we're doing is it? If you're dynamically rehedging your deltas, then you mostly care about CDS spread correlation and volatility...not defaults. In a 5 name investment-grade FTD, you probably don't care at all about defaults. How volatile the names are and the correlation of their spreads will determine your PnL...so why don't we model this? Am I wrong in thinking that I'd rather own (delta-hedged) protection on a FTD made up of volatile names than names with the same spread and correlation with lower spread volatility? However, these two FTDs will be priced the same if the credit curves and (default) correlations are identical.Is this what you're talking about erstwhile?

amen - exactly my point. in another thread (and many times on the phone talking to credit brokers) i've used the example of delta hedging a large investment grade FTD, exactly as you say. the models being used (at least in public domain) are inappropriate. the main risk parameters are not even inputs in the model!

the models come from securitization...where you do care mostly about losses and don't delta hedge. I don't think a model like the one you're suggesting would be too difficult to implement...you just need 2 more inputs for every name: vol and correlation of spreads.

if you can do it, go for it! maybe you'll be one of the first guys to arb FTDs against single name options, or even CDS basket options! i wonder what trade you could put on to maximally exploit any naive pricing? it might involve the Hi Vol index... probably more convex derivatives like 2TD and 3TD...?it might also be interesting to see a massively oversimplified model for the purposes of gaining intuition. for example, the large pool model might be extended in a way that all the credits have the same vol and pairwise correlation. i wonder if that might still result in something that calculated reasonably quickly?

Just thinking that adding spread volatility and correlation could be done in a fairly simple and natural way. A sort of crude jump-diffusion, where the jump is actually default. But the scheme has problems...Here is the simple scheme:Step 1: Do one sim run, giving you a series of default times - using the standard model (gaussian copula, etc).Step 2: You have a series of default times. Now fill in the "missing spread history" and delta hedge.Step 3: Continue until reasonably convergedStep 4: Do the usual default accounting keeping track of the cost of delta hedgingThe missing spread history is generated using the input spread vols and spread-spread correltions. Starting at t=0, do a weekly delta hedging simulation with spreads evolving along their own forward prices, but with correlated volatile spreads. This is easily done using the same method used in simulating basket options in equity derivatives. Every week you delta hedge, keeping track of positions and the cash balance as in any delta hedging simulation. But where do you get the delta from? Serious problem number 1: you use the delta produced by the standard model. This is obviously a flaw, as the "new model" will get a different value from standard model, and ought to have different deltas! I would argue this is not a fatal flaw, as hedging incorrectly in a risk-neutral simulation will not produce the wrong value, only an average with too high of a standard deviation. The minimum standard deviation is produced by the correct delta hedging scheme.In the simulation, as time progresses, there will be the occasional default. When this happens of course the history for that spread ends.At the end of all the simulation runs, you have default cashflows but also final CDS positions (presumably quite small at maturity!) and a final cash balance due to the delta hedging action.Advantages of this scheme:(1) Your final FTD or synthetic tranche valuation now will depend in a natural way upon spread vol and spread-spread correlation, as well as time-to-default correlation.(2) This simulation is easy to understand, as it is simply a two-step simulation with default times calculated according to the standard model, and spread dynamics added in afterwards in a simple way. There is no complex mathematics.Serious problems with this scheme:(1) As mentioned before, the deltas come from the old standard model.(2) Credits do not anticipate default in any way. They go from any arbitrary value reached in the simulation directly to default. Very unrealistic.(3) Almost all the drawbacks of the standard model + the drawbacks of simulating spread histories using lognormal simulation correlated via a gaussian copula assumption!A thought on how to improve problem (2). One could either:(1) Anticipate default by cheating! As the default times are known before the spread diffusion phase, insert an arbitrary function that anticipates default by spreads widening continuously by some amount for some number of months before the default occurs. Simple, but unsavoury. Extra parameters that are hard to observe.(2) Come up with a spread evolution process which is more realistic than the above diffusion plus jump-to default, and which includes some sort of realistic death-spiral effect, so that defaults are not equally likely to occur when the spread is 30bp or 200bp.How about it, all you clever quants out there? If a mere trader can come up with a vaguely plausible scheme like this, surely one of you can do a proper job of it and build a real model!

Hi,I agree with you guys on the fact that we need to take into account the spread vol and spread-to-spread correlation. But it seems that it requires once again some heavy computations which are not reasonable for anyone of us.Actually a big bank has developped a kind of model such as this one. So it is used by some players in the market. Is it efficient? No idea.Do you have any books reference or papers on the internent related to this model?

I have never seen any references to such a model.I think there is a significant value to such a model, which is why the world has gone silent on it!

You may try these onesduffieDuffieHope that helps.

http://defaultrisk.com/pp_corr_62.htmhave you guys not seen this paper?"Double Default Correlation"by Martijn van der Voort.It describes how to incorporate both spread correlation and default correlation to price a tranche.Including the former only leads to a small correction in the fair spread of the tranche though (a few percent).

Thanks guys!

Check out this paper by John Hull and Allan White:http://www.rotman.utoronto.ca/~hull/Dow ... del.pdfThe paper attempts to simultaneously model default correlation and credit spread dynamicsin the pricing of correlation products.

Check out this paper by John Hull and Allan White:http://www.rotman.utoronto.ca/~hull/Dow ... del.pdfThe paper attempts to simultaneously model default correlation and credit spread dynamicsin the pricing of correlation products.

Thanks for posting this. My only worry is that structural models are very, very far from reality to begin with, so an abstraction of a structural model seems even more removed from real life. I don't mean to dismiss the work of these eminent people - what they are doing is interesting and useful. I don't know the best model, but when I see it, it will smell a lot better than models i have seen so far!

Sorry for jumping in this thread late...Erstwhile, HTFB:your example of the low and high vol IG FTD trade is good. But, I'd expect that a basket of high vol credits with the same spreads as a basket with low vol credits would have lower default correlation. Thus, the standard model may still add some value for these trades, if calibrated properly.