I've been researching the SABR model and one of the main benefits it seems is that you can obtain a closed for solution of the implied BS volatility in certain cases.

In all the papers I've read, I have not found any proofs/reasoning as to where this solution actually comes from.

Does anyone know/can link me to a derivation of it?

This is the formula I am referring to.



2 Answers 2


It comes from Heat Kernel expansion and differential geometry.

See Theorem 6 and Section 8 of http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1717676&download=yes

  • $\begingroup$ The link is broke. Could you please list the exact title and author of the paper, as well as a working link to it? Thank you. $\endgroup$
    – Hans
    Commented Mar 11, 2018 at 0:00
  • 1
    $\begingroup$ New link: papers.ssrn.com/sol3/papers.cfm?abstract_id=1717676 $\endgroup$
    – Idonknow
    Commented May 7, 2020 at 9:40

The derivation is in "Managing Smile Risk" by Pat Hagan et al. A copy is here:


It is not closed form, but rather an approximation based on expansions.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

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