We calibrate SABR on each expiry and tenor combination using market data. (e.g. 1mx10y, 3mx10y etc.) Then how about the non-standard expiry like 2.5mx10y? Do I linear interpolate the alpha, beta, rou parameters from 1mx10y and 3mx10y? Thanks
This is a fair question. In my experience with most implementations, once you build the SABR parameter (expiry/swap-tenor) matrices by calibrating them to the ATMs/supplied skew data, what you end up with is a SABR interpolated vol cube object. This object is then fed into your pricer and when you price a non-standard expiry it's the SABR parameters that are interpolated. This makes sense since what this gives you is the entire smile section for the non-standard expiry (extracted from the supplied market/skew data), not just a single vol point.
However it's worth mentioning that the ATM implied vols the above approach outputs may be different to what u'd get if you interpolated the points on the ATM swaption matrix directly - especially for long-dated expiries. So perhaps the approach of interpolating the ATMvols instead of alpha mentioned by Antoine Conze would be one way to address this. But bear in mind this doesn't make one approach "more correct" than another - after all vol surface interpolation is a highly subjective area. Ultimately, if you're employing SABR to begin with then it's the parameters that you're really after, interpolated or not.
For non-standard expiries or tenors, interpolating the SABR parameters is one approach. However, there are some considerations to keep in mind:
Interpolation Method: Linear interpolation is straightforward and commonly used, but it might not always capture the nuances of the market. Other interpolation methods, such as spline interpolation, can provide smoother transitions between points.
Interpolation of Volatility Surface vs. Parameters: Instead of interpolating the SABR parameters directly, another approach is to interpolate the implied volatility surface generated by the SABR model for the given expiries and tenors. Then, for the non-standard expiry/tenor, you can calibrate the SABR model to the interpolated volatility surface to obtain the parameters. This method ensures that the interpolated volatilities are consistent with market observations.
Stability of Parameters: The β parameter is often kept fixed (e.g., β=0.8 for swaptions) based on market conventions, so you may not need to interpolate it. For the other parameters, ensure that the interpolated values lead to a stable and well-behaved volatility surface.
Market Dynamics: Always keep in mind the market dynamics and conditions when interpolating. During periods of high market stress or significant shifts in the yield curve, simple interpolation methods might not capture the true market sentiment