# 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)?

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.

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.