On the harmonic mean representation of the implied volatility

TL;DR

This paper reveals that arbitrage-free implied volatility can be represented as a harmonic mean of a positive function, linked to local volatility, and introduces a new coordinate system where short-dated implied volatility approaches the arithmetic mean.

Contribution

It provides a model-free harmonic mean representation of implied volatility and explores its relation to local volatility and a new coordinate system based on Fukasawa's map.

Findings

Implied volatility equals the harmonic mean of a positive function for any fixed maturity.

A new coordinate system using $f_{1/2}$ makes implied volatility approach the arithmetic mean of local volatility.

Explicit formula for volatility swap derived in the SSVI parameterization.

Abstract

It is well know that, in the short maturity limit, the implied volatility approaches the integral harmonic mean of the local volatility with respect to log-strike, see [Berestycki et al., Asymptotics and calibration of local volatility models, Quantitative Finance, 2, 2002]. This 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 $f_{1/2}$ [Fukasawa, The normalizing transformation of the implied volatility smile, Mathematical Finance, 22, 2012], 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. As an illustration, we consider the case of the SSVI parameterization: in this setting, we obtain an explicit formula for the volatility swap from options on realized variance.