Interpolating SABR Parameters Versus SABR LMM for Interest Rate Options

# SABR LMM vs no-arbitrage term structure of SABR parameters


# SABR LMM vs no-arbitrage term structure of SABR parameters












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?

## Answer by BEQuant (score 2, accepted)

https://quant.stackexchange.com/a/71179

I am guessing that the first model you are referring to is the one from Rebonato: Linking caplets and swaptions prices in the LMM-SABR model (2009)? If yes, then I would say that your approach is a simplification of his model. Assuming that you are still able to calibrate to a set of swaptions that are of interests with your method, I would say that your method allows for a faster calibration than calibrating using the two hump-shaped parametric functions that he is using. However, you won't capture the time-homogeneity that he is referring to. In other words, in an ideal model I would expect a 5 years caplet to have a similar future smile 3 years into the future than a 2 years caplet now. This might be an important factor when pricing options where forward volatilities are significant. It's your choice if you want to make this tradeoff. As a suggestion, in order to keep some time-homogeneity, maybe you can use piece-wise constant functions that depends on $T_i - t$ instead of the hump-shaped functions? This way you would avoid simultaneous calibration of the caplets (calibration of caplets can be done in cascade).