black scholes - 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

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.