# SABR: implied vol formulas

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):

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

The wikipedia formulas are also different from Hagan's 2014 paper:

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.