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Say we plot the implied volatility against strike price and moneyness for some options. As the implied volatility depends on the option pricing model it is reasonable to expect some differences here.

What do the curves look like for the Bachelier model and the Black-Scholes model, respectively? What is the difference, where is it located (in terms of term and moneyness) and why is it there?

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As @nimbus3000 mentions, the shape of the vol curve differs by markets so I won't comment on that here. I'll restrict my comments to the Black(-Scholes) vs. Bachelier section of the question.

You can approximate Normal (Bachelier) vols from Black vols by (there is a second order effect related to the product of the square of the Black vol and the maturity but ignored here):

$$ \sigma_N = \sigma_B \sqrt{F\times K} $$

Where $F$ and $K$ are the forward and strike, respectively. Since you're interested in moneyness, consider $K = F\times k$ for some %-moneyness $k$. Then $$ \sigma_N = \sigma_B \times F \sqrt{k} $$

From this I have 2 observations:

  1. Bachelier vols are not independent of the level of the underlying (unlike Black vols).
  2. The transformation is almost linear in F, so the shape of the Bachelier vol skew for a given maturity will roughly mimic the shape of the Black curve.
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For the Black Scholes model, the smile depends on the asset is what I've found. The smile is also different for equity and equity index options. For equity index, the part of the smile is on the downside, lower than the current price, is more or less a straight line and on the upside resembles the definition of smile that you see.

For equity options, the smile for most times is fairly simple as in the definition would say, but I have seen times when the ATM vols were higher than the vols on either side of the curve.

This of course is true for the market I trade, different markets would have their own idiosyncrasy.

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