Evaluating the Hagan SABR Approximation with Negative Beta
Summary
The document asks whether the Hagan implied volatility approximation remains appropriate when SABR beta is negative, as can arise when beta is estimated from the relationship between at-the-money volatility and the forward price. The answers do not establish a universal validity result. Instead, they recommend judging the approximation by its intended use and by the properties of the resulting implied volatility surface.
One proposed check is whether option prices derived from the surface imply a valid terminal probability density without negative values across strikes. A separate concern is whether the calibrated surface produces sensible volatility behavior when passed into models for more complex products, including forward volatility behavior. Another answer links the beta choice to the assumed backbone and response of the surface to forward price shocks. The document gives qualitative criteria but no derivation, parameter ranges, numerical tests, or proof, so it does not settle when negative-beta calibrations are reliable.
Key ideas
- The answers assess negative-beta Hagan approximations according to their intended application rather than giving a blanket validation.
- A useful option pricing check is whether the implied terminal density remains nonnegative across strikes.
- A surface that fits European options may still produce unsuitable behavior when used in exotic pricing models.
- The beta choice affects the volatility surface backbone and its response to forward price changes.
- The document supplies qualitative checks, not a proof or numerical validation.
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Full text
# Approximate Hagan formula for SABR model with negative beta
# Approximate Hagan formula for SABR model with negative beta
While looking into fixing the $\beta$ parameter (based the following regression: $\text{ln } \sigma^{ATM}_t = \text{ln } \alpha - (1-\beta)\text{ln }F_t$, as explained in West (2004), page 6) before calibration of SABR to equity option market data, I found that inferred $\beta$'s are often negative. This was discussed in an existing question earlier (SABR beta range), and got some useful comments. Conclusion: SABR is unconditionally valid as long as $\beta<1$.
My question is; although SABR model is valid, is the Hagan approximate formula also still valid to use for negative $\beta$, why (not)?
## Answer by Kiann (score 1)
https://quant.stackexchange.com/a/42012
my two cent's worth. It depends ultimately what you are trying to achieve, and calibrate to.
The key tests for implied vols on option prices, relate to their pdf (probability density functions).
The Hagan expansion allows us to have a analytical form, by which one can compute the implied vols, and subsequently feed that into a Black76 equation.
If the test is for a non-arbitrage condition (i.e. no negative pdf densities across the strikes) for an European terminal distribution, and if given a certain beta, one can recover a pdf that does not violate negative pdfs, I would propose that is fine.
The 2nd key test though, is when the implied vol surface is calibrated such, is it consistent with the volatility behaviour as it will be fed into other models, and more exotic products?
For example, we always calibrate an exotic model to match the European options first (whether ATM or ITM/OTM). Now, if that is successful, is the vol surface and parameters then fed into other more complicated models such as a short-rate model with vol-skew?
If yes, but SABR negative-beta creates wrong (usually forward vols) behaviour, then that's when one has a problem.
## Answer by Yanyi Yuan (score 0)
https://quant.stackexchange.com/a/48752
Given the variation, ATM vol = alpha * F ^(beta-1), if your stochastic process for forward price dF= alphaF^beta dW, that means your effective beta, CEV, is 1. This gives horizontal backbone of the vol surface. I think it all depends on whether this is what you expect to see - the vol surface is stickey under shocked price scenarios.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.