Uniform bounds for Black--Scholes implied volatility

TL;DR

This paper derives uniform bounds for Black--Scholes implied volatility using optimization representations and symmetries, providing new insights and reproofs of asymptotic behaviors at extreme strikes and maturities.

Contribution

It introduces novel uniform bounds for implied volatility based on optimization problems and symmetries, enhancing understanding of extreme strike and maturity behaviors.

Findings

Derived upper and lower bounds for implied volatility

Reproved asymptotic formulas at extreme strikes/maturities

Utilized symmetries to generate new bounds from existing ones

Abstract

In this note, Black--Scholes implied volatility is expressed in terms of various optimisation problems. From these representations, upper and lower bounds are derived which hold uniformly across moneyness and call price. Various symmetries of the Black--Scholes formula are exploited to derive new bounds from old. These bounds are used to reprove asymptotic formulae for implied volatility at extreme strikes and/or maturities.