Hi, I understand the Black model framework used to price a cap assumes lognormality of the libor, while the Black model framework used to price a swaption assumes lognormality of the swap rate.They cannot both be correct, since a basket of lognormals is not lognormal. Here I am reading a text by Rebonato. I quote him:"Whatever the true distributions might be, as long as the lognormal distributions are matched, as they are by the pricing procedure, to the first two moments, the impact of this inconsistency is quite small."I don't understand this statement.Firstly, doesn't the 1st momemt--mean--not play any part at all in the Black pricing framework?Secondly, the 2nd moment--vol-- are independently quoted for caps and for swaptions. How are they matched?Thanks for any enlightenment.

A bit cramped for time here so only can throw out a very quick observation to stimulate your thinking. How is a term swap rate a basket of lognormals? What do you mean by a basket?

I guess the statement goes roughly in the direction " if you have lognormal Libors, then swap rates being weighted sums of these forwards (with stochastic weights) won't be longormal; but whatever the implied distribution of the swap rates might be, if you match a lognormal one via mean and variance, this will give almost the same swaption prices as in the Libor model"

Farshid Jamshidian wrote cogently on this issue a long time ago.