Hello,At pp7 of the well known paper of Graeme West : "Calibration of the SABR model in illiquid markets", we find the cubic that we need to solve to get alpha parameter from ATMvol (assuming that we have already solved rho and u through daily calibration) .This is very important tool as it allows us to account for 1st factor (level) movements of the smile when we mark to market our trading book. Now, that we have moved to negative rates many traders have started using normal dynamics for interest rate swaps.I wonder what is the corresponding formula to find alpha if we are at the Normal Case (beta=0) of the SABR model.

I looked briefly. Since the cubic contain beta, what is the issue? Just set it to 0 and solve for your root.

It depends if you mark against shifted lognormal vols or against normal (bp) vols. For shifted lognormal, Alan is more or less correct. For normal, the formula is slightly different.The normal cubic polynomial is detailed in the following paper:http://papers.ssrn.com/abstract=2467231

Thank you for the paper logos01. Could you please indicate if the normal cubic polynomial is the formula 38? I am a little bit confused as the formula contains beta, which is supposed to be zero as we are at the Normal Case of SABR. I would greatly appreciate it if Alan could also make a short comment here regarding the formula 38.Thank you both in advance.