Swaption Price with Negative Swap Rate

Question

To price Swaptions, I use the Black '76 model. I'm trying to update the model to handle negative interest rates. One such approach to doing this is detailed here. In particular I'm interested in the "shifted lognormal" approach.

Here, they replace the strike rate k and swap rate r with k-c and r -c, effectively shifting the model's lower bound from 0 to -c. This new lower bound means that it can accommodate negative interest rates.

If I have understood this model correctly, then we are essentially changing nothing about the model except shifting the parameters, in other words we can take the vanilla Black '76 model, change k and r to k-c and r-c, and then we'll be able to use it for negative interest rates.

Why does this hold theoretically? "k - c" is not the same as "k", so I would think that the model has been fundamentally changed and is not giving a fair price for a swaption with swap rate r and strike k if that is the derivative of interest.

Second, how do we decide the shift parameter c? Different values of c will give different prices, and the authors do not provide clarification on how to calibrate the shift parameter, or when to adjust it dynamically as interest rates evolve.

Answer 1

The point is that the shifted model is calibrated to keep the atm option price the same (equal to the market price ). Specifically , c is selected (somewhat arbitrarily) to represent a level that swap rates cannot go below. Typically it might be -0.50pct or -1pct. Then, the vols are changed to keep atm option prices the same. And now, you have a model that is still calibrated to the market but gives different prices for out of the money options than before.