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.
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$\begingroup$ @dm63: any suggestion? $\endgroup$– GordonJul 21, 2016 at 14:57
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$\begingroup$ Gordon, I think Mark Joshi can answer your question without any formula $\endgroup$– user16651Jul 21, 2016 at 17:26
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1$\begingroup$ Thanks @BehrouzMaleki. This question (quant.stackexchange.com/questions/25407/…) and this question (quant.stackexchange.com/questions/7761/…) provide a few clues. But there is such approximation as in the question. $\endgroup$– GordonJul 21, 2016 at 17:43
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2$\begingroup$ From memory I think this is addressed in Blyth and Uglum "Rates of Skew" in Risk Magazine from about 10 years ago. $\endgroup$– dm63Jul 21, 2016 at 19:31
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2$\begingroup$ The intuition is that if there is a constant normalized local vol s, then the local lognormal vol is initially s/A, where A is the current at the money level. The local lognormal vol around the strike is s/K. So on a path that starts at the current market price and ends around the strike, the geometric average local lognormal vol is s/sqrt(AK). Since the current atm lognormal vol , sigma= s/A, the above is equal to sigma*sqrt(A/K). The key issue to be proved is that the average over all paths is the same as the specific path mentioned. $\endgroup$– dm63Jul 21, 2016 at 20:11
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1 Answer
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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.
