# How to derive the half slope rule from asymptotic relation between implied and local volatility?

In their paper(https://www.sciencedirect.com/science/article/pii/S0764444200017493) BERESTYCKI, BUSCA, and FLORENT proved an asymptotical relation(for short maturity) between implied and local volatity: implied vol is the harmonic mean of the local volatility.

This result is used in https://www.math.nyu.edu/faculty/avellane/Avellaneda.pdf. to derive equation (16) from (11). Are they just saying that the harmonic mean is close to the arithmetic mean?

A similar result is demonstrated in the Gatheral book(lemma page 88) but it is in terms of implied and local variance not implied and local volatilty. https://www.amazon.com/Volatility-Surface-Practitioners-Guide/dp/0471792519/ref=sr_1_1?keywords=gatheral&qid=1567079638&s=gateway&sr=8-1

Thanks