I am working on the calibration of Libor Market Models to the swaption markets. I have chosen a piecewise constant time structure for the forward rates. To calibrate this term structure to the swaption market, I have implemented the cascade calibration developed by Brigo and Mercurio in their book. Unless I have performed a smoothing to the swaption data, I obtain negative and imaginary volatilities with this procedure. Anyone knows how to go around this problem?

yesdon't use the cascade calibration-V

Thank you very much for your reply. I have two other approaches to calibrate the lognormal libor market model:1) Time homogeneous volatility. The volatility is a function of the time to maturity. I have tried to calibrate to the caps and swaptions relevant for pricing, but I get poor results with this approach.2) Time piece-wise constant calibration. A more general procedure than the cascade calibration that allows to fit the libor volatilities to caps and/or swaptions. Regularity conditions must be applied to the volatility function in order to get a smooth behaviour. I am already working on this, it is much slower than the cascade calibration, but allows for more flexibility and an exact calibration to the relevant instruments to price a certain exotic.I am taking the correlation as an input not involved in the calibration process.What kind of volatility function and calibration approach is more usual in the market?Thanks.

Hi,The second one in your list is by far preferred, and, as far as I can tell, is used by all serious LMM users. As far as the speed is concerned, it is indeed slower than the cascade calibrattion, but considering that the cascade calibration does not work, it is a minor disadvantage. For your benchmarking, a calibration to a 40y swaption grid (ie all swaptions/caps with total maturity <= 40y) takes about 1 minute for us. Of course this is very implementation and hardware dependent. I am sure you can make it faster if you really tried.-V

Hiwould you mind providing more details of the second procedure? I understand that it is a non-parametric method with some regularity-preserving conditions. Do you include away-from-the-money caps in the calibration set or do you separate deterministic and stochastic LMM calibration?Best regards - Dunbar

See some comments on this here (see Chapter 7)-v

I have tried to calibrate LFM with different procedures according to Brigo-Mercurio's book obtaining the following results (referred to the Euribor market):1) With the simple piece-wise time Constant calibration (Formulation 3 in Brigo-Mercurio's book) you can fit perfectly caplet volatilities and obtain automatically a smooth behaviour. However you don't take into account any information from the swoption market2) The piece-wise time homogenous volatility (Formulation 2 in Brigo-Mercurio's book) brings to negative volatilities after a certain period of time (tipically 10-12 years). This is because the term structure of volatility is decreasing too steeply3) The piece-wise joint calibration (Formulation 5 in Brigo-Mercurio's book) allows you to fit jointly cap & Swoption but -as Busyant has pointed out - this requires regularity conditions in the penalty function to obtain a smooth behaviour. This is my favourite one.4) The cascade calibration has exactly the same problems that Busyant reported (negative, imaginary or inconsistent volatilities). Maybe the reason is a temporal misalignament in the swoption matrix due to different times of update of the different entries of the matrixP.S. About the correlation matrixEven if it's possible to use swoption prices to calibrate the correlation I prefer to use an exogenous matrix. So I use historical data to determine the parameters of a parametric form (for example Rebonato (1998) or Schoenmaker & Coeffey). See the Paper of Brigo-Morini for referenceP-

I was told recently there is a new paper by Morini where the cascade calibration is analyzed in detail. It seems the interpolation of the missing rows of the swaption matrix and the rank of the exogenous correlation matrix can be chosen so as to have this calibration work in all cases without noisy outputs. Have a look to make sure I understood correctly. The paper has many examples and is posted athttp://ssrn.com/abstract=552581 Brigo, Damiano and Morini, Massimo, "An Empirically Efficient Analytical Cascade Calibration of the LIBOR Market Model Based Only on Directly Quoted Swaptions Data" (January 2004).

Basically this one of Brigo and Morini is a somewhat different algorithm. It is similar in spirit to the cascade calibration of brigo and mercurio, but takes into account that some of the maturities are missing, rather than making some usual replacement. It seems it can avoid the problems of negative volatilities typical of brigo and mercurio tests.