A Note on the Equivalence between the Normal and the Lognormal Implied Volatility : A Model Free Approach

TL;DR

This paper establishes a model-free equivalence between normal and lognormal implied volatilities, linking them through mathematical functions and expanding their relationship across different strikes and models, with implications for hedging strategies.

Contribution

It generalizes the known SABR model result by formulating a model-free equivalence between implied normal and lognormal volatilities for any strike and model.

Findings

Implied normal volatility is linked with the incomplete Gamma function.

An expansion of implied normal volatility in terms of European call option time-value.

The equivalence between implied normal and lognormal volatilities is applicable across models.

Abstract

First, we show that implied normal volatility is intimately linked with the incomplete Gamma function. Then, we deduce an expansion on implied normal volatility in terms of the time-value of a European call option. Then, we formulate an equivalence between the implied normal volatility and the lognormal implied volatility with any strike and any model. This generalizes a known result for the SABR model. Finally, we adress the issue of the "breakeven move" of a delta-hedged portfolio.