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Shifted SABR ATM Calibration Uses the Unshifted Forward and Strike

Article Quant Q&A · Author: user4226384
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Summary

The document answers how to select the at-the-money strike when calibrating the volatility level in a shifted SABR model. The shift is added to both the forward and strike in the model’s implied-volatility calculation. Consequently, an option that is at the money in market terms, with strike equal to the original forward, remains at the money after both are shifted by the same amount.

The response therefore uses the market ATM option to calibrate the ATM volatility parameter; it does not move the calibration strike to forward plus shift. The shift is a modeling device, and the resulting model should be judged by how well it fits the observed volatility smile. The explanation is concise and does not cover parameter estimation procedures, alternative model conventions, or quantitative fit evidence.

Key ideas

  • A shifted SABR calculation applies the shift to both the forward and the strike.
  • The market ATM strike remains equal to the unshifted forward when calibrating ATM volatility.
  • Shifting both quantities by the same amount preserves their equality and ATM status.
  • Model quality is assessed by how well its calibrated smile matches observed market data.

Tags

OptionsVolatilityDerivatives pricing
Full text
# shifted SABR - ATM vol


# shifted SABR - ATM vol












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 by Dark (score 1, accepted)

https://quant.stackexchange.com/a/25669

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.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.