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.

  • $\begingroup$ @dm63: any suggestion? $\endgroup$ – Gordon Jul 21 '16 at 14:57
  • $\begingroup$ Gordon, I think Mark Joshi can answer your question without any formula $\endgroup$ – user16651 Jul 21 '16 at 17:26
  • 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. $\endgroup$ – Gordon Jul 21 '16 at 17:43
  • 2
    $\begingroup$ From memory I think this is addressed in Blyth and Uglum "Rates of Skew" in Risk Magazine from about 10 years ago. $\endgroup$ – dm63 Jul 21 '16 at 19:31
  • 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. $\endgroup$ – dm63 Jul 21 '16 at 20:11

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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.