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


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.


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