I'm implementing B-K trinomial interest rate tree for CMS related product with peice-wiselinear volatility parameters.On B-K interest rate model, there is no closed form solution for thebond price. So I think I should calculate long term bond price on short terminterest rate tree.But it takes too much time to calculate 30Year CMS rate on 3m based B-K tree.For example, 10NC3m bermudan style cancelable CMS swap (30Year CMS Rate vs 3m Libor+spread),it is almost impossible to price.Are there any tips on how to estimate Long term rate on short term interest rate tree?Thanks in advance,H.G. Kim

Two pieces of advise1. Do not use BK. If you insist oin short-rate models, use one-factor Gaussian (aka Hull-White)2. Do not use 1-factor models to price derivatives on spreads between CMS and Libor ratesGood luck-V

How about model suggestions ?BGM ?

Kim, for the bermudan on a CMS-Libor swap, using a short-rate tree should also be possible and it should not be unuseably slow. Since there is no closed-form solution for the CMS rate on any node of the tree, it must be computed by pricing the swap on the sub-tree conditonal upon that node being reached; this is the time-consuming part. I suggest that you use some programming tricks, for instance, don't compute the CMS rate when it is not necessary. Spend some time on deciding on the exercise boundary, one thought I have is that you select five nodes from the tree, being top, mid, bottom and the two between (top and mid) and (mid and bottom). For these five nodes, compute the conditional value of the two indices (CMS and libor). Now you have the early exercise value for these five points. Fit a function of these five values vs the short rate r and identify where this function cuts below zero; this level of r separates the tree into two parts and you only have to compute the intrinsic values for the part which you should exercise early. You can also experiment if you can repeat the above process to speed up the remaining computation time for intrinsic value calculation, again, by using functional form to represent the CMS and LIBOR rate at each node.Although the above is more tedious in the programming but it should speed up the computation by avoiding spending time calculating values which are not going to contribute to the final result.Finally, I agree with Piterbarg that it is not good to use a 1-factor model for pricing derivatives on spread; but for your case, it is less critical because it is an option with one leg being CMS and one leg libor. DO NOT use 1-factor model to price option with one leg being the difference between 30y CMS and 2y CMS and the other leg being libor.Phil

QuoteOriginally posted by: philthegreekFinally, I agree with Piterbarg that it is not good to use a 1-factor model for pricing derivatives on spread; but for your case, it is less critical because it is an option with one leg being CMS and one leg libor. DO NOT use 1-factor model to price option with one leg being the difference between 30y CMS and 2y CMS and the other leg being libor.PhilI disagree; in this case it is critical not to use a 1-factor model. The value of an option to exchange a Libor leg for a CMS leg is driven primarily by the changes in the slope of the interest rate curve, and it is very hard to get high enough volatility of the slope from a 1-factor model-V

In that case, would you also discourage the use of short-rate model for bermudan swaption?

Piterbarg,Could you please recommend any model fit for this CMS-Libor case? or any paper. Thanks.

BGMP.

QuoteOriginally posted by: philthegreekIn that case, would you also discourage the use of short-rate model for bermudan swaption?A properly-calibrated 1-factor model is acceptable for Bermudans. Bermudans, however, are very different from options on the difference between CMS and Libor legs. For Bermudans, correlations of core swap rates observed on different dates (their respective fixing dates) are important, and these correlations can be well-appropximated in a 1-factor model. For the original deal in question, correlations of different rates on the same date are required, and these cannot, in general, be recovered in a (reasonable) one-factor model-Vladimir

QuoteOriginally posted by: PaolosBGMP.Second that-V

QuoteOriginally posted by: domilar04Piterbarg,Could you please recommend any model fit for this CMS-Libor case? or any paper. Thanks.This paper could be of some help-V

Gaussian HW and LN BK models have very different skew features. Depending on your deal specs, you may have to take this into consideration. Separately, given that the implied periods in CMS rates tend to overlap with your exercise points, it is helpful to dynamically build yield curve on an exercise point and figure out CMS from there. Again, this offers nothing new except some programming tricks...