I noticed that the approximate implied volatilities in Hagan’s 2002 paper (see B69.a and BC9.c, please refer to the paper for the definition of the parameters): enter image description here

are different from the formulas on Wikipedia (under Asymptotic solution):

enter image description here enter image description here

The wikipedia formulas are also different from Hagan's 2014 paper: enter image description here

Does anybody know where these formulas on wikipedia come from? A reference would be nice.

The wiki page mentions F_mid, which can either be the geometric or the arithmetic average. How to decide which average to use? It seems that Hagan's paper uses the arithmetic average.

Thanks in advance!


  • $\begingroup$ Which formula on Wikipedia? Can you be specific? $\endgroup$ Oct 24 '18 at 0:34
  • $\begingroup$ @JaehyukChoi: i have updated my post. I hope it is more clear now. $\endgroup$
    – Nele
    Oct 24 '18 at 6:33
  • $\begingroup$ I found at that the wiki formula's are in fact in Hagan's 2002 paper, in appendix C. $\endgroup$
    – Nele
    Oct 31 '18 at 14:35

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