Hi,I have a lognormal floor pricer and a shifted-lognormal floor pricer.The lognormal pricer is bad for low strikes, because the prices of the floor converges to 0 when strikes goes to 0% (ie it neglects the possibility of negative interest rates).So that for the moment I use the lognormal pricer when strike>0,5% and the shifted-lognormal pricer otherwise.What I am trying to do is an unique pricer which gives coherent prices for all strikes (no gap in the floor prices).I thought of doing a convex combination of the two pricers in a particular range of strikes (around 0,5%), but I wanted to know if someone already faced this problem? Or if anybody has any idea?

I don't think you have actually stated all your requirements.For example, why doesn't the shifted lognormal work for you for all strikes?

Hi,Thanks for replying.You are right, I forgot that I would like the prices to be near the old prices (ie lognormal prices) when the strike is >0,5% (ideally equal to the lognormal prices from a certain strike).So that the users of the pricer do not note any change in the prices for large strikes.

Isn't that enough, that you just calibrate your shifted lognormal vols to produce prices equal to those you obtain from a lognormal model?

What product do you want to price, hedge with this model? For example in the specific case of IR swaptions and the SABR models there are many ways to tweak the distribution near zero (and for high strikes) depending on your belief about zero floor prices.

Hi,Thanks for replying. To be more precise on my requirements, I would like the prices (1) to be near the lognormal prices when strike is large AND (2) to be near the shifted-lognormal prices when strike is low.I think I stated all requirements now. Sorry for unclarity.So, Pimpel, your solution is great, but I think it does not fulfill (2), right?Zhulian : I just want to price floors. Sorry but I don't understand what do you mean by "tweak the distribution".

It depends on your asset. For example some approximations are valid with low volatilities only, etc.

My underlying is the LIBOR rate for USD, EUR and CHF.My first idea was to keep the same prices for k<alpha (shifted lognormal) and the same prices for k>beta (lognormal). And for k into [alpha, beta] to take some convex combination of the two pricers. One problem is that alpha and beta will be determined arbitrarily...

Shifted lognormal and lognormal both have the problem of a lower limit. You might want to consider a Black-normal model; in this model, rates are normally distributed. The SABR model with beta=0 corresponds to this situation, and Hagan et. al. explicitly cover this example in their SABR papers. We briefly looked at this a year ago in an attempt at pricing the implicit 0% strike floors embedded in some swaps. There is a decent asymptotic expansion of this in "Probability Distribution in the SABR Model of Stochastic Volatility" by Hagan et. al. (the formula for sigma_n on page 4). The expansion in their earlier paper "Managing smile risk" had some bad behavior - I think it had to do with zeta(K) or xHat(zeta(K)) steeply jumping up to 1 from nearly 0 as K approached 0 from above. As I recall, prices for strikes away from 0 are not bad as well under this model, but I only looked at this problem for less than a week, so I can't vouch for that. At any rate, the prices for 0% strike floors ended up being pretty close to the premium that banks were charging for the embedded floor.