This is more a conceptual question around calibration. My objective is to calibrate a 1-factor Hull White model, and my question relates to calibrating a and sigma (both constants) to swaptions.
Let's say that I have a set of ATM swaption prices (this is my market prices) coming from a model based on OIS discounting. When doing the calibration, i.e. minimizing the diff between model (HW) and market prices, should I do a transformation of the swaption prices to get some sort of proxy for prices under, say, LIBOR discounting? The reason I started thinking about this, is because the HW closed-form formulas that I use are based on the fact that fixing and discount curve is identical.