ATM Skew Between Bachelier and Black-Scholes Implied Volatility
Summary
This exchange clarifies a local relationship between normal and lognormal implied volatility when both models are used to represent the same option prices. The question presents a conversion factor involving the forward or spot level and moneyness, then asks why differentiating it creates a nonzero Black-Scholes volatility slope near at-the-money even when normal volatility is comparatively flat. The answer notes that the derivative of the moneyness conversion factor at the at-the-money point is negative one half, yielding the stated local slope after scaling by normal volatility and the underlying level.
It also confirms that lognormal volatility is the Black-Scholes implied volatility, so the comparison is between Black-Scholes and Bachelier implied-volatility skews. This is a local conversion result, not evidence that either model captures an observed market surface universally. The brief answer does not derive the full conversion, specify conventions in detail, or discuss how rates, forwards, or market dynamics affect skew away from at-the-money.
Key ideas
- A moneyness conversion factor links normal implied volatility to lognormal implied volatility.
- Its derivative at the at-the-money point gives a negative one-half contribution to the local slope.
- Black-Scholes implied volatility is the lognormal volatility in this comparison.
- The result describes a local model conversion and does not establish a general market skew pattern.
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Full text
# Why is Bachelier implied volatility more skewed than the Black-Scholes implied volatility?
# Why is Bachelier implied volatility more skewed than the Black-Scholes implied volatility?
I found the following explanation in a paper by Grunspan (see attached paper page 6) but have trouble understanding it:
> By differentiating Formula (3) with respect to m, it turns out that the Black-Scholes skew $\frac{\partial\sigma_{LN}}{\partial m}$ at the money ($m = 1$) generated by the Bachelier model is $\frac{\partial\sigma_{LN}}{\partial m} = -\frac{1}{2}\frac{\sigma_N}{S}$ ($\sigma_{LN}$ is by definition the implied lognormal volatility). Therefore, the Bachelier model is highly skewed ATM (a slope of $−50\%\times\frac{\sigma_N}{S}$). Another way to explain this feature is that given call prices, when we use the BS model, the function $\sigma_{LN}$ is a decreasing and convex function of $m$, i.e., it generates a skew, while the function $\sigma_N$ is a rather flat function of $m$. Thus, normal volatility is most suited for products such as swaptions for instance.
I am not sure what Formula (3) is, but it might be $\sigma_{LN} = \frac{1}{S}\frac{\ln m}{m-1}\sigma_N$.
My two questions are:
- How does he get the formula above, i.e. $\frac{\partial\sigma_{LN}}{\partial m} = -\frac{1}{2}\frac{\sigma_N}{S}$ and more importantly what does this tell us about the two skews?
- Doesn't this concern the slope of the Black-Scholes IV, since the slope of the log-normal volatility is equal to that?
> Therefore, the Bachelier model is highly skewed ATM (a slope of $−50\%\times\frac{\sigma_N}{S}$).
Here is the paper: Grunspan Paper
## Answer by Mike (score 1)
https://quant.stackexchange.com/a/58917
- You can take a derivative $\frac{\partial}{\partial m}\frac{\ln m}{m-1}$ at point $m=1$, so you will get $-\frac{1}{2}$.
- Yes, Black-Scholes volatility is log-normal volatility. In other terms it's comparison of Black-Scholes IV and Bachelier IV.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.