Hello every one I'm reviewing Rebonato's Formula for swaption in Brigo and Mercurio's book page 283 (or formula 6.67) which is In BM's book the correlation's rhi_ij between two brownian driving Libor rates maturing at T_i and T_j are freezed over time This assumption seems quite unrealistic to me and I think that correlation should depend on time "s" with this in mind this would lead to the following formula :Does this make sense to anyone ? Am i missing something ?Thank's for any comment

i guess rebonato's swaption formula is used only as an approx for faster calibration in LMM, and mostly nowhere else

Well amit7ful, in BM's book, the result is correct but only because of their assumptions which can be sum up by the followingAs correlation are constant by assumption they can extracted from the integral but in some other more realistic framework what I mean is that we can with very little extra cost integrate numericaly cross variations with time dependent correlations (for example if they depend only on time to maturities T_i -s and T_j -s )Moreover after checking in Rebonato's book "Hedger and the fox", he specifies the correlation's as time dependent and include them in the integral ( formulas 20.40 and 20.41 page 661).regards

There are more accurate approximations in books by Brace and Gatarek

This is true Gamal thx

All that matters is the covariance between the logs of the rates across the time interval.

I guess constant correlation is enough cause no real correlation can be implied from market.

Well I wouldn' say that I would rather say that the impact of a change in correlation matrix on a swaption ( a typical element of a calibration portfolio) is very small So when calibrating and trying to get an implied correlation you get a wide range of correlation matrix that make almost no difference on the calibration functionnal that you use. From this fact you say "why bother with that then ?" and you use some parametric form of correlation or historical estimation and calibrate your portfolio with this. But if we had a liquid market with instruments truly dependent on the correlation then it would be another story and it would be unacceptable not to imply your correlation matrix from market quote. That's my opinion on the subject mj, Miner what would be your views on that topic ?

Well it all depends on what you are pricing. For example, the price of a Bermudan swaption seems to be impervious to correlation as long as you fit the vanilla market correctly.

QuoteOriginally posted by: MinerI guess constant correlation is enough cause no real correlation can be implied from market.Supposedly it can be implied from CMS spread options: http://papers.ssrn.com/sol3/papers.cfm? ... id=1418571

to TheBridge, I agree with u, but usually implied correlation is unstable and sometimes correlation cant be stripped from swaptions cause cap and swaption are different markets.

Well Miner I think that it is a bit similar to what is happening in Index option pricing if you have all implied vol of the component of the index You might choose to derive some implied correlation which is ok but might be quite different from historical ones or you might choose to use some statisitacal estimate of correlation then see that there is a spread between implied vol of Index Option re-builded from individual stock Implied Vol and Statstical Correlation and market implied vol of the Index Option and finally conlcude that it is no big deal since those are different markets The additionnal difficulty here interest rate context is that weights on each Libor making a swap rate are random (even though they are quite stable)

I think the point Miner is making is that you can't directly compare cap vols and swaption vols because:There is no arbitrage mechanism between the two markets,and banks tend to be short cap vol but long swaption vol, and funnily enough the vols you get from the swaption market are lower than those from the cap market.As opposed the index options where you could put on a correlation/dispersion trade

as to the original question: I think the thing with the rho_i,j not time dependent was not meant like this. The correlation structures in the book are surely meant to be time homogeneous in the sense that they only depend on the distance between rate i, j and the calendar time. in my experience, to calibrate the correlation you need cms spread options. It seems not too important what correlation structure is chosen (i.e. the simple ones perform just as well as the more complicated w.r.t. calibration error). However, in order to match the current spread options for short and long maturities at the same time, it seems to be necessary to introduce a time inhomogenous correlation structure with low correlations in the short term and higher ones in the long term calendar time. A simplified version of Rebonatos two state approach applied to correlations (instead of volatilities as he does) seems to work quite well.QuoteOriginally posted by: mjWell it all depends on what you are pricing. For example, the price of a Bermudan swaption seems to be impervious to correlation as long as you fit the vanilla market correctly.what drives the price of bermuda swaps? If it is not the correlation of the forward rates at a fixed calendar time, is it the intertemporal correlation of - presumably - the coterminal swap rates (the finite-dimensional distribution of the the relevant coterminal swap rates) ? If so, how is that captured in a libor market model. It seems as if the price of bermuda swaps priced in a hull white 1F model (bootstrapped to the coterminal swaptions) is driven largely by the mean reversion. Does the mean reversion "code" the intertemporal correlation mentioned above in this particular model? How does this translate to the libor market model (if this is all true of course - I do not know, just quick thoughts from my side - but if it is true, it has to translate, because the libor market model just imposes no arbitrage conditions on the dynamics and leaves therefore all possible degrees of freedom (in principle, of course you chose certain subsets of the volatility functions and thereby restrict yourself to subsets of all possible arbitrage free dynamics).Any thoughts on this someone?

if you only calibrate to co-terminal swaptions then correlation does matter. But if you calibrate to co-terminal swaptions and caplets then the price of a Bermudan seems to be factor independent -- this would pin down the mean reversion in a HW model. There are several papers where people have found similar results: eg my paper with Ametrano on calibration to co-terminal swaptions (www.markjoshi.com)