Bachelier and Black-Scholes Fits of the Volatility Surface, what about SABR?

I always wondered if Bachelier was really worse than Black-Scholes in practice. As an experiment I fit various implied volatility surfaces with Bachelier and Black-Scholes and look at the average error in implied volatility by slice.

In theory Bachelier is appealing because slightly simpler: log returns are a bit more challenging to think about than returns. And it also takes indirectly into account the fact that OTM calls are less likely than OTM puts because of default risk, if you assume absorbing probability at strike 0.

Bachelier
Black-Scholes
The error is in general smaller for Black-Scholes for short expiries, and higher for long expiries when compared to Bachelier. Interestingly, in theory, the difference of the models is more pronounced for longer expiries. One would have imagined that for very short expiries Bachelier would be equivalent to Black, it is not the case in term of fitting the smile.

Displaced diffusion (mixing both Bachelier and Black linearly) allows to gain x2 accuracy.

Displaced Diffusion
What about SABR? Let’s look at lognormal SABR, usually used for equities.
Lognormal SABR (beta=1)
The fit is much better for long expiries, but not so great for a few outliers for long maturities, it can be actually worse than the simple displaced diffusion. A normal SABR might fix that.

Normal SABR (beta=0) using Hagan lognormal formula
If one relies on the standard lognormal formula, the beta=0 SABR behaves very badly.
Normal SABR using Hagan normal formula
This is fixed using the normal (bpvol) Hagan formula directly. The fit is then better overall for long maturities as expected from Black-Scholes vs Bachelier behavior.

If one look in log scale, the conclusion is not so obvious: beta=1 produces a better fit for a majority even for long expiries, but worse for a few (30% in my case) outliers.

What’s clear however, is that one should never use the Black implied volatility Hagan formula with beta=0. This leaves a question on displaced SABR. Is displaced SABR is better suited than SABR with varying beta?

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