Flat Volatility Skew and the Risk-Neutral Distribution

Summary

The document asks whether a flat implied-volatility skew implies a normal risk-neutral distribution when option prices are converted into a density using the Breeden–Litzenberger relationship. It notes that the density inferred from its stated formula appears to have increasingly positive skewness and high kurtosis as the chosen moneyness range widens. The author asks whether restricting that range is appropriate and what bounds would yield correct distribution moments.

No answer or calculation method is supplied, so the text mainly identifies a conceptual and numerical issue. It does not provide option-price data, maturity details, or explain how the density is normalized over the available strikes. The bounds concern matters because observed option prices cover a finite strike range, while tail probabilities and higher moments are sensitive to assumptions outside it. A flat volatility input by itself does not settle how the full distribution should be recovered from finite market observations; the document leaves that question unresolved.

Key ideas

• The document asks whether constant implied volatility leads to a normal risk-neutral density.
• It invokes the Breeden–Litzenberger relationship to infer a density from option prices.
• It reports that estimated skewness and kurtosis change as the moneyness range is widened.
• It raises the issue of finite strike coverage and its effect on tail-sensitive moments.
• It provides no bounds or resolution for the density estimation question.

Tags

# Does a flat volatility skew imply a normal risk-neutral distribution?


# Does a flat volatility skew imply a normal risk-neutral distribution?












Using Breeden-Litzenberger to convert a flat volatility skew to the risk-neutral distribution, would I obtain a normal risk-neutral distribution?

I use the formula $$p(K) = \frac{n(d_2)}{K\sigma \sqrt{T-t}}$$

Therefore, if I obtain a normal risk-neutral distribution, I should obtain zero skewness and kurtosis if I calculate it off the risk-neutral probabilities.

However, when working with a flat volatility skew, if I extend the range of the moneyness bounds and obtain a longer range of risk-neutral probabilities, I get a more positive skew and much higher kurtosis for the new risk-neutral probability distribution.

Main Question: Is it correct to limit the moneyness bounds in this case (and what would be the correct bounds if there was one) if I wish to calculate the "correct" skewness and kurtosis of the risk-neutral distribution?

Please let me know if I have any conceptual errors and a link to any relevant articles would be very much appreciated as well.

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