I am trying to calibrate the 1 factor Hull White model to ATM swaptions. The strategy which I use is to minimise the sum of squared difference between model and market prices for the swaptions on the diagonal of the swaption matrix. I am using only swaption maturities till 10 years. So the swaptions which I am using for the calibration are 10x1, 9x2, ..., 1x10. And I am running a DE algorithm for jointly calibrating both the mean reversion parameter and volatility parameter (both assumed constant).
Now I understand that a lot of practitioners fix the value of the mean reversion parameter and only calibrate for the volatility parameter. However I have not been able to find any references for how a "best value" for the mean reversion parameter is decided. So it would be great if someone could shed some light on what exactly are the steps to decide on a "best value" for the mean reversion.
The currency for which I am trying to calibrate is GBP.