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SABR Beta and Normal Versus Lognormal Implied Volatility

Article Quant Q&A · Author: gb4
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Summary

The document clarifies that normal and lognormal implied volatility refer to different option pricing conventions, not necessarily to the distribution assumed for the underlying in the SABR model. A normal implied volatility is the value used with Bachelier pricing to reproduce an option price; lognormal implied volatility is the value used with Black pricing. The answer points readers to the relevant SABR appendix formulas for these conversions.

It explains that setting beta to zero makes the underlying process conditionally normal, but does not by itself invalidate the lognormal implied volatility formula: that is a separate way of expressing the option price. The response says beta is commonly constrained to the range from zero to one in practice, while suggesting the original paper may allow broader nonnegative values. It does not derive the formulas or settle their precise validity ranges, so readers should consult the cited model equations and use caution when generalizing beyond common calibration conventions.

Key ideas

  • Normal implied volatility is defined through Bachelier option pricing, while lognormal implied volatility is defined through Black pricing.
  • The volatility quoting convention does not determine the distribution of the underlying in the SABR model.
  • At beta zero, the underlying is conditionally normal, but Black implied volatility can still be used to represent the option price.
  • The answer describes beta from zero to one as a common practical range, while noting possible broader theoretical scope.

Tags

OptionsVolatilityDerivatives pricing
Full text
# SABR Implied Vol: Normal Approximation vs Log-Normal Approximation


# SABR Implied Vol: Normal Approximation vs Log-Normal Approximation












I am having trouble understanding the difference between the normal and log-normal implied volatilities from Hagans SABR model: http://web.math.ku.dk/~rolf/SABR.pdf.

As far as i understand the main result presented by Hagan is the implied volatility formula given by equation (2.17a) in that paper. However, upon reading the appendices i have become confused at the difference between the normal implied volatility and the lognormal implied volatility and how the value of $\beta$ effects this. The main result presented by Hagan is the Black implied volatility obtained using the SABR option price formulas and the black option pricing formulas. There are a few things i don't understand:

- For what values of $\beta$ is the Black (log normal) implied volatility formula for SABR option prices presented by Hagan valid for?

- What is the normal implied volatility formula and for what values of $\beta$ is the normal implied volatility valid for?

- Hagan also presents implied $\textit{normal}$ volatility for Black's model, but i thought Black's model was for log normal?

- Are the Black (log normal) implied volatilities and normal implied volatilities valid for the same range of values of $\beta$, or different ones?

Initially, i thought that Hagan's lognormal approximation of the implied (Black) volatility was valid for $0 < \beta \leq 1$ due to the fact that Hagan says if $\beta = 0$ this represents the "stochastic normal model". But i am not sure anymore.

In general, i am confused at the difference between the normal and log normal implied vols and what role beta plays in determining these. Any help in understanding this would be great, thank a lot.

## Answer by Sanjay (score 2, accepted)

https://quant.stackexchange.com/a/46743

I am not going to answer all of your questions, but let me give it a go.

- I don't have a qualified answer to this one. In practice however $\beta$ is always bounded such that $\beta \in [0,1]$ (0 and 1 both included). see SABR chapter in Derman & Miller (2006). But as far I know, $\beta$ is not bounded in the original paper so in theory it can take any non-negatve value and the pricing formula should hold.

- Set $\epsilon=1$ and see equation A.67!

- The implied NORMAL volatility is that level of volatility that will generate the option price when you use the Bachellier pricing formula. see equation A.54a.

Now for the last part of your post:

Don't confuse distribution of the asset with the implied volatilities. When $\beta=0$ then the asset is stochastic normal conditioned on the volatility process.

"...if one were to apply the log normal implied vol formula but with $\beta=0$ does it then become stochastic normal instead?"

Once agian, these two things have nothing to do with each other! The implied log-normal volatility is simply a phrase we use because it is connected to Black-Scholes/Blacks model where the asset is a GBM hence log-normal. So yes, the formula for implied log-normal volatility also holds for $\beta=0$

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.