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Why the Forward Should Stay Fixed in SABR Calibration

Article Quant Q&A · Author: Zac Likes Vol
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Summary

The document asks whether a SABR calibration for caps, floors, and options on SOFR futures should treat the forward as a free parameter. The author uses a fixed beta and compares calibrating rho and nu (then deriving alpha) with fitting all three SABR parameters. Moving the forward appears to improve the fit to market implied volatilities, whose shape does not match the model’s usual curvature around the forward.

The response argues that the forward is a market observable and an input to SABR, not a model parameter to adjust for fit. Changing it changes the quantity being modeled, much like altering a measurement to suit a limited instrument. The post raises concern that this could affect Greeks and other risk measures but does not derive those effects or provide numerical evidence. It is a brief conceptual answer, so it leaves the practical calibration setup and risk consequences unexplored.

Key ideas

  • The forward is an observable input to SABR rather than a parameter to tune for a better fit.
  • Changing the forward alters the market quantity represented by the model.
  • The post compares two parameter-fitting approaches while holding beta fixed.
  • The response does not quantify the effect of a shifted forward on Greeks or risk measures.

Tags

OptionsVolatilityDerivatives pricingFixed income
Full text
# Calibrating SABR -- Can I calibrate the forward like any other parameter?


# Calibrating SABR -- Can I calibrate the forward like any other parameter?












Essentially the title to the above. I am using SABR to price caps and floors (as well as options on SOFR futures). I currently have two calibration techniques, the first calibrates based on rho and nu (using these to calculate alpha), and the second calibrates all 3 at the same time. I also use a fixed beta.

As I understand, the forward is an input but not a parameter of the model, and I have noticed I can get better fits when I alter the forward. Therefore I am wondering what would happen in practice if I calibrate/move the forward to generate better fits; does this have effects mathematically I have not considered on delta/vega/other risk metrics? The market I am looking at has forwards/ATMs that don't lend themselves to the normal SABR model's natural shape, i.e. generating a parabolic shape around the forward.

Ignore the legend, but the green is SABR, red is implied market vols, and yellow is ATM vol.

## Answer by user35980 (score 1)

https://quant.stackexchange.com/a/78315

I think this is a bit like measuring someone's height by asking them to kneel because your tape measure is too short.

As you say, the parameters are numerical variables input to the model. Forwards, ATM vol, skew ...etc are market observables you are trying to calibrate your model to. Changing the forward will change what you're attempting to model.

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.