Approximating Lognormal Implied Volatility from Normal Volatility

Summary

The document asks how to derive an approximation that maps at-the-money implied volatility to volatility at another strike under a lognormal distribution. The relationship is framed as a strike-dependent scaling rule, using the ratio of the at-the-money strike to the target strike under a square root. It identifies the target as a way to obtain a lognormal volatility input when the market’s rate distribution is understood to be closer to normal.

The answer points to an external derivation attributed to earlier work on interest-rate options and gives the historical motivation: traders used lognormal pricing models even when they regarded normal dynamics as a better description. The document offers intuition and a reference rather than a derivation or validation. The approximation’s assumptions, applicable strike range, and accuracy are not discussed, so it should not be treated as a general volatility-surface rule without further analysis.

Key ideas

• The proposed approximation scales at-the-money volatility by a square-root function of the strike ratio.
• The conversion was motivated by using lognormal pricing models for rates believed to behave more nearly normally.
• The answer references an earlier derivation rather than reproducing its steps.
• The document does not state the approximation’s range of validity or accuracy.

Tags

# Log-normal Volatility Approximation


# Log-normal Volatility Approximation












In a comment to this question, it is mentioned that, under the log-normal distribution, \begin{align*} vol(k) \approx vol(atm) \times \sqrt{\frac{atm}{k}}. \end{align*} Here, $k$ is the strike, $atm$ is the at-the-money strike, and $vol(k)$ is the implied volatility corresponding to strike $k$. I have difficulty to derive this approximation. Any suggestion is appreciated.

## Answer by dm63 (score 3, accepted)

https://quant.stackexchange.com/a/28234

Page 3 of this document ad-co.com/analytics_docs/ALevin_QP_2012.pdf shows the result, originally given in Risk Magazine by Blyth and Uglum.

The intuition for the formula is given in my comment above. The original motivation for such a formula was for interest rate options in the 1990s. Everyone had a lognormal pricing model, but traders understood that the distribution of interest rates may be closer to normal. Hence we needed a formula to plug in the right lognormal vol into our models.

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.