shifted SABR - ATM vol

Question

quick question guys.

I know that for Shifted SABR (or any other Shifted model), we simply model the underlying price process (lets say the forward interest rate F), as F' = F + x, x being the shift.

1) To calibrate the alpha (or the ATM vol parameter) of the model, do we calibrate to an option where the strike K is equal to F or K is equal to F + x? (i.e. does the ATM point shift by the shift parameter)?

2) How do we reconcile this to the fact that in the real world, the actual ATM option is the one whose strike is equal to F? (i.e. its kind of like the shifted model pricing a real world ATM call option as an ITM option..)

Thanks!

Answer 1

Let $\sigma(F,K)$ be the SABR implied vol.

In the shifted model, the formula essentially becomes $\sigma(F+x,K+x)$ (you have to shift the strike as well).

So to answer your question in the ATM vol calibration you take $K=F$ in order to have $F+x=K+x$.

There is no need to "reconcile" anything as it is just a model. Once you have your model, you have to calibrate it and if it matches your smile then you're happy.