I am looking to build a swaption vol cube using SABR for fitting skew along different swaption expiries. For example, I fit SABR parameters for expiry = 1yr and expiry = 2yr. Now if my swaption expiry is 1.5 yr, how do I interpolate the SABR formulae? Lesniewski Vol Cube paper talks about using a hump shaped curve to fit ATM vols (H(t) = (alpha + beta * t) exp(-lambda t) + mu). However there is no discussion on how to interpolate other parameters (SABR alpha, rho, assuming beta is fixed) in an arbitrage free way. I have searched on this forum and there is no clear answer on questions on this issue... Is there some accepted way? OR do people simply use ad hoc ways such as linear interpolation of SABR params between maturities....

SABR/LMM (there are several attempts by different authors, I am aware of one by Hagan/Lesniewski) is perhaps the "correct" way to go about it.... or am missign something. In this type of SABR/LMM there seems to be a ton of parameters to fit..so it doesn't come across as parsimonious. It is the price to be able to match all the observed swaption quotes I guess. Would appreciate any insights on potential parsimonious parametrizations of the correlation structures that arise - Libor/Libor and Libor/vol and vol/vol correlations.