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:

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

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

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

4. 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.

• "Hagan also presents implied normal normal volatility".... can you point out where it is presented? – Sanjay Jul 22 '19 at 13:57
• Sure, equation (A. 63) in appendix A from linked the paper. It has a paragraph before it starting with: "We can obtain the implied normal volatility for Black’s model". – gb4 Jul 22 '19 at 14:19

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

1. 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.

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

3. 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 $$\beta=0$$

• Thank you very much for your insight. For point 1 when you say between 0 and 1, are 0 and 1 inclusive in that range? If so, since $\beta = 0$ corresponds to the stochastic normal process, if one were to apply the log normal implied vol formula but with $\beta = 0$ does it then become stochastic normal instead? I think you make a good point about me confusing the asset distribution with the implied volatilities, this is likely where i am getting stuck. – gb4 Jul 22 '19 at 14:33
• .... se edited version of my answer – Sanjay Jul 22 '19 at 14:54
• In the overall picture, the answer is yes! However. Be carefull with the conclusions about the distribution of the asset because the the asset distribution is conditioned on the volatility. I personally don't know what we can conclude about the distribution when $\beta=1$ or $0$. That is a separate question. But everything else is correct. – Sanjay Jul 22 '19 at 15:36