Calibrating Shifted Black Pricing for Negative-Rate Swaptions
Summary
The document discusses the shifted lognormal approach to applying Black ’76 swaption pricing when swap rates can be negative. The method shifts both the swap rate and strike by a constant so the model's effective lower bound moves below zero. The question asks why this change can still produce fair prices and how to choose the shift.
The answer describes calibration rather than a parameter-free theoretical guarantee: choose a shift representing a plausible lower rate level, then adjust volatility so at-the-money option prices match market prices. Once calibrated, the shifted model can produce different prices for out-of-the-money options than the unshifted model. The shift is described as somewhat arbitrary, and the exchange does not give a full calibration procedure or rules for changing it as rates evolve. It highlights that matching at-the-money prices does not make all model prices identical.
Key ideas
- A shifted lognormal model offsets both the swap rate and strike to accommodate negative rates.
- The shift parameter sets the model's effective lower rate bound.
- Volatility can be recalibrated so at-the-money prices match observed market prices.
- The calibrated model may still produce different out-of-the-money prices.
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Full text
# Swaption Price with Negative Swap Rate # Swaption Price with Negative Swap Rate 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 by dm63 (score 2, accepted) https://quant.stackexchange.com/a/31229 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.
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