When I calibrate a short rate model to market swaption vols, what curve am I getting when I plug in the calibrated parameters into the analytical formulae (assuming they exist for the model I'm looking at)?
Since swaption vols apply to a swap that is projected and discounted off different curves, I can't get my head around whether the resulting short rate model simulates the discount curve, projection curve or somehow the "total" swap.
I'm basically trying to figure out how to price a swap using simulated curves produced using a short rate model in this new dual curve world. I need projections and discount factors, and I don't know if they should come from one calibrated short rate model, or if I should be somehow have two short rate models (one for the projection curve and one for discount, both maybe calibrated using market swaption vols but different term structures)?
Any help would be appreciated.