Why an Implied Volatility Smile Has a Sharper Center at Fixed ATM Volatility

Summary

The note explains the intuition behind Hull’s statement that a U-shaped implied volatility smile corresponds to a distribution with more probability in both small and extreme moves than a lognormal distribution. Higher implied volatility away from at-the-money is associated with heavier tails, while the lower volatility near the center can seem to imply fewer small moves.

The key qualification is that the comparison holds ATM volatility, and thus the distribution’s standard deviation, fixed. Adding tail probability while keeping that spread unchanged requires the distribution to become thinner through the intermediate range; this can leave relatively more probability concentrated near the center as well as in the tails. The explanation is qualitative and brief: it does not derive the result mathematically or specify a particular distribution, and the interpretation depends on comparing distributions at the same ATM volatility.

Key ideas

• A U-shaped implied volatility smile is associated with greater likelihood of both small and extreme moves relative to a lognormal benchmark.
• The comparison assumes ATM volatility, and therefore overall dispersion, is held constant.
• Heavier tails at fixed dispersion require less probability in intermediate moves.
• The explanation offers intuition rather than a formal derivation or calibrated distribution.

Tags

# Why does IV smile imply a more "peaked" distribution than lognormal?


# Why does IV smile imply a more "peaked" distribution than lognormal?












Below, Hull claims that a U-shaped volatility smile suggests that "both small and large movements in the [underlying] are more likely than with the lognormal distribution, and intermediate movements are less likely."

I see why the smile suggests that the implied distribution has fatter tails than the lognormal distribution, since the higher tail IV indicates that the market assigns a higher probability to extreme underlying price movements.

However, I don't see why the U-shaped smile suggests that the implied distribution also has a higher peak, with more probability assigned to small movements. By the previous logic, since the smile dips for $K/S_0\approx 1$, doesn't it mean that the market assigns a lower probability to small underlying price movements?

## Answer by dm63 (score 2)

https://quant.stackexchange.com/a/81521

If you increase the probability of the tail events while reducing the probability of small moves, you will have increased the standard deviation of the distribution. The passage in Hull assumes that you need to maintain the same standard deviation (because you want to calibrate to the same ATM volatility). This latter implies that the distribution must get very skinny in the middle to make up for the fatter tails.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.