# 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:

1. calibrate standard SABR to vanilla options with available tenors
2. interpolate (either linearly or more sophisticatedly) SABR parameters between calibration tenors. i.e. make model parameters functions of time
3. use interpolation to obtain volatility smiles on arbitrary intermediate tenors
4. 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?

• Your approach is the SABR model with interpolation on tenor. This is valid for a specific expiry. For the Asian option, you need an average over expiry, but your approach won't fit the volatility smile on shorter expiries. I don't think your approach is suitable for the Asian (or any path-dependent) option. Jun 7 at 1:25
• @jChoi can you please elaborate a little bit more on it? The aforementioned approach will fit the volatility smile of vanilla options in all available tenors b/c on each tenor it is the standard SABR model. The interpolation is needed for intermediate tenors not quoted on market. AFAIK this is how parametric forms of SABR LMM work as well -- one need to choose some parametrization of model parameters to describe the dynamics away from calibration tenors. Jun 7 at 8:55
• Say, you want to price an Asian option on 10y swap rate over 3 months. Then, you need to calibrate to 1m10y, 2m10y, and 3m10y swaption smiles. But your approach, as I understand, is to calibrate 3m1y, 3m5y, 3m10y (or any fixed expiry instead of 3m). Jun 7 at 13:15
• @jChoi please see the edit. Jun 8 at 8:35
• I understand your question better. So you need to price an Asian option averaging 11m3m, (11m+1d)3m, ... , 1y3m rates. Given 11m and 1y is just one month apart, the SABR coefficient interpolation should be OK. Even if you know the exact SABR parameter for all caplets expiring 11m and 1y, however, you still need a model for the dynamics of the evolution of the caplet rates in order to price the Asian (or any path-dependent) option. Just knowing the SABR parameters is not enough IMHO. Jun 11 at 1:08

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).