Hi all,

I would like to price some non-ATM swaptions.

Therefore, I need the appropriate Black-76 implied volatility for non ATM strikes.

Assuming that I only have ATM swaptions implied volatilities from the market, is it possible (and sound) to calibrate a SABR model from these ATM quotes ONLY to infer the ITM/OTM swaption implied volatilities?

Thank you in advance.

Anything is possible...  Is a "calibration" along the lines you have described likely to yield something useful, that is the question.

How do you calibrate the SABR model if you have only ATM swaptions?

Btw, in the current market situation (i.e. small rates / negative rates), if I had only ATM swaption vols, I would use a Hull-White with a piecewise constant volatility term structure model to extrapolate the vols.

How would determine the "bendy-ness" of the smile if you only have one point (ATM) ?

How do you calibrate the SABR model if you have only ATM swaptions?

Btw, in the current market situation (i.e. small rates / negative rates), if I had only ATM swaption vols, I would use a Hull-White with a piecewise constant volatility term structure model to extrapolate the vols.

You have a close-form solution of the ATM volatility under the SABR model so provided you have more than 3 ATM market data points you should be able to calibrate alpha, beta and rho.

Regarding the Hull-White approach, the underlying assumption would be to have a surface constant in the strike/moneyness direction and a varying vol in the years to exercise dimension for a specified underlying swap tenor, is that correct?

swskewgraph.gif

How would determine the "bendy-ness" of the smile if you only have one point (ATM) ?

Maybe not in the case of a single slice indeed, but in general if you have additional constraints on arbitrage-freeness of your whole surface, why not? Furthermore, the ATM level does not correspond to a constant strike as it depends on the years to exercise so you would not even have aligned vol points but sparse points instead.

The key ability of the Hull White model is to apportion volatility as a function of time, and it's useful for deals which depend on how volatility is is distributed in time. Many options, especially Europeans, are only sensitive to the total amount of volatility up to the exercise date, and not to how the volatility is distributed in time. Being a Gaussian model, the implied normal volatility predicted by the HW model is likely to be almost flat ...

The SABR model is intended to consolidate the smile risks into three components (vega --- risk to all vols going up and down, vanna --- risk to the skew increasing or decreasing, and volga --- risk to the smile increasing or decreasing). Without implied vols at at least three strikes, one cannot calibrate it from the market. The best one can do is to use proxies to get the SABR parameters ... ie find another market (which has the smile information available), and map the SABR parameters to the market of interest ... usually means taking the same rho and volvol (and beta) values from the proxy instruments. E.g., maybe caplet or bond option markets can be used to suss out the swaption SABR parameters.
WARNING: In principle, this means that one would use the proxy instruments to hedge the vanna and vega risks ,,,

How do you calibrate the SABR model if you have only ATM swaptions?

Btw, in the current market situation (i.e. small rates / negative rates), if I had only ATM swaption vols, I would use a Hull-White with a piecewise constant volatility term structure model to extrapolate the vols.

You have a close-form solution of the ATM volatility under the SABR model so provided you have more than 3 ATM market data points you should be able to calibrate alpha, beta and rho.

Regarding the Hull-White approach, the underlying assumption would be to have a surface constant in the strike/moneyness direction and a varying vol in the years to exercise dimension for a specified underlying swap tenor, is that correct?

If I understand correctly, you want to use constant SABR parameters for the entire vol cube? In this case, I'm not sure that the extrapolated surface will be meaningful, but I may be wrong.
Regarding the HW approach, the underlying assumption is [$]dr(t) = [\theta(t) - \kappa r(t)] dt + \sigma(t) dW(t)[$], where [$]\theta[$] is given by the HJM framework, [$]\kappa[$] can be calibrated in the constant vol case, then inputed to bootstrap the function [$]\sigma[$]. In the bootstrap, you find [$]\sigma_{i-1}=\sigma(t_{i-1} < t \leq t_i)[$] that minimize the errors on all swaptions maturing at [$]t_i[$]. If you want to extrapolate swaption prices only at a given tenor, you can replace the minimizations by rootfinders to match the ATM swaption prices with the same tenor.

It depends what you are trying to do. A lot of people in this thread think sell-side and for that purpose, good luck trying to find a meaningful calibration.

On the other hand if you have access to the history of forward rates and ATM vol, I believe you can get quite creative and you might end up with something interesting, if not useful.