There exists a LIBOR Market Model with stochastic volatility for pricing and hedging exotic (e.g. path-dependent) interest rate options with smile. However let us consider the following approach:
- calibrate standard SABR to vanilla options with available tenors
- interpolate (either linearly or more sophisticatedly) SABR parameters between calibration tenors. i.e. make model parameters functions of time
- use interpolation to obtain volatility smiles on arbitrary intermediate tenors
- check the resulting surface for no-arbitrage and do some sort of smoothing if necessary
What are the drawbacks of this approach for pricing and hedging Asian (average rate) options versus SABR LMM?
EDIT: I'm working on a very illiquid market where there are no swaptions and maturity grid of quoted caps/floors is very sparse. Suppose that I want to price 11M x 1Y caplet with daily averaging. In order to do so I need a set of daily caplet volatility smiles, however the market quotes only 11M and 1Y smiles. Is it valid to calibrate 11M and 1Y smiles separately and then do some sort of no-arbitrage interpolation inbetween in order to obtain all the intermediate smiles needed for daily averaging? Is it conceptually different from Rebonato volatility and volatility of volatility parametrizations in SABR LMM?