Asymptotic Expansions of the Lognormal Implied Volatility : A Model Free Approach

TL;DR

This paper derives explicit asymptotic expansions for lognormal implied volatility in various regimes by inverting the Black-Scholes formula, providing a model-free approach with practical computational methods.

Contribution

It introduces a systematic method to compute asymptotic expansions of implied volatility across different market regimes, including explicit formulas and an inductive procedure.

Findings

Explicit first five terms of asymptotic expansions

A recursive method to compute all expansion terms

Closed-form power series for at-the-money implied volatility

Abstract

We invert the Black-Scholes formula. We consider the cases low strike, large strike, short maturity and large maturity. We give explicitly the first 5 terms of the expansions. A method to compute all the terms by induction is also given. At the money, we have a closed form formula for implied lognormal volatility in terms of a power series in call price.