How Black and Bachelier Implied Volatility Curves Relate
Summary
The document explains how implied volatility curves differ when options are expressed using the Black model or the Bachelier normal model. It gives an approximate conversion: normal volatility scales with Black volatility, the square root of forward times strike. Rewriting strike as a multiple of the forward shows that, at a given maturity, the normal-volatility skew roughly follows the Black curve while also depending on the underlying’s price level.
The answers also describe market variation in smile shapes, including differences between equity and equity-index options and examples of downside skew or elevated at-the-money volatility. These are observations from particular markets rather than universal patterns. The conversion omits a second-order effect involving volatility and maturity, and the discussion does not establish a general curve shape across asset classes or expiries.
Key ideas
- Normal implied volatility depends on the forward price level, unlike Black implied volatility.
- An approximate conversion scales Black volatility by the square root of forward times strike.
- At fixed maturity, the normal-volatility skew roughly mirrors the Black-volatility curve.
- Observed smile shapes vary across markets and underlying assets.
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# Is the Implied Volatility Curve different under the Black-Scholes and Bachelier models?
# Is the Implied Volatility Curve different under the Black-Scholes and Bachelier models?
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?
## Answer by crunch (score 5)
https://quant.stackexchange.com/a/32653
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:
- Bachelier vols are not independent of the level of the underlying (unlike Black vols).
- 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.
## Answer by nimbus3000 (score 2)
https://quant.stackexchange.com/a/32650
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.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.