On the Harmonic Mean Representation of the Implied Volatility

It is well known that, in the short-maturity limit, the implied volatility approaches the integral harmonic mean of the local volatility with respect to log-strike; see [H. Berestycki, Busca, and Florent, Quant. Finance, 2 (2002), pp. 61--69]. This short paper is dedicated to a complementary model-free result: An arbitrage-free implied volatility in fact is the harmonic mean of a positive function for any fixed maturity. We investigate the latter function, which is tightly linked to Fukasawa's invertible map $f_{1/2}$ [M. Fukasawa, Math. Finance, 22 (2012), pp. 753--762], and its relation with the local volatility surface. It turns out that the log-strike transformation $z = f_{1/2}(k)$ defines a new coordinate system in which the short-dated implied volatility approaches the arithmetic (as opposed to harmonic) mean of the local volatility.