SABR Smile Dynamics Versus Local Volatility Behavior

Summary

The post examines a motivation for the SABR model: local volatility models are said to predict implied-volatility smile movements opposite to those observed in markets, whereas the original SABR paper is described as moving the smile in the same direction as the forward. The author questions how that claim follows from the SABR implied-volatility expansion. With the backbone parameter held fixed, the at-the-money volatility term is proportional to the forward raised to a negative power when beta is below one, which appears to imply falling volatility as the forward rises. The author further suggests that negative spot-volatility correlation could reinforce that decline if the volatility parameter changes with the forward.

This is a conceptual question about interpreting model dynamics, not a resolved derivation. It highlights that conclusions may depend on what parameters are held fixed and how the volatility level responds to a forward move. The post offers no numerical illustration or answer, so it does not settle whether the apparent contradiction reflects a misunderstanding, different assumptions, or a distinction between smile translation and volatility-level change.

Key ideas

• The post contrasts local-volatility smile dynamics with the same-direction smile shifts attributed to SABR.
• The author questions that interpretation using the SABR at-the-money volatility backbone.
• Holding the SABR volatility parameter fixed can imply lower at-the-money volatility as the forward rises when beta is below one.
• Negative correlation may affect the volatility parameter as the forward changes, complicating the comparison.
• The document poses the issue without supplying a derivation or definitive resolution.

Tags

# Hagan et. al original argument for SABR


# Hagan et. al original argument for SABR












In the original SABR paper (Hagan et al 2002 ), the introduction of the famous model is motivated by the observation that local volatility models spot dynamics work the wrong way. As the spot increases the implied vol decreases- and conversely- which is at odd with markets. Thus the necessity of an alternative parametrization. In the introduction they write

"The dynamic behaviour of smiles and skews predicted by local vol model is exactly opposite the behavior observed in the marketplace: when the price of the underlying asset decreases local vol models predict that the smiles shifts to higher prices; when the price increases, these models predict that the smile shift to lower prices. In reality, asset prices and market smiles move in the same direction"

In contrast, according to the authors (pag 15) :

..The SABR model also predicts that whenever the forward price $f$ changes, the the (sic) implied volatility curve shifts in the same direction and by the same amount as the price $f$.

I do not understand where in the paper this claim is supported. Actually, what I see from the implied vol expansion is that the backbone of the ATM volatility $\sigma(f,f)$, which is $\alpha/f^{1-\beta}$, for fixed $\alpha$ ,decreases as $f$ increases. Analogoulsy all the other volatilities, in direct contrast with the claim. What is more, is that if $\alpha$ can move, and as logical we are in the situation $\rho <0$ because of the vol skew/leverage, then as $f$ increases $\alpha$ will have a tendency to decrease, depressing further the ATM ivol.

Am I getting this wrong? Can somebody explain?

https://www.researchgate.net/profile/Patrick_Hagan3/publication/235622441_Managing_Smile_Risk/links/59e4d89f458515250246e626/Managing-Smile-Risk.pdf

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