Smile asymptotics for Bachelier implied volatility

TL;DR

This paper analyzes the asymptotic behavior of Bachelier implied volatility in extreme strike regimes, linking it to the tail decay of the underlying distribution and the analyticity of the characteristic function.

Contribution

It extends large-strike asymptotics from Black-Scholes to Bachelier models using regular variation theory, providing explicit formulas and connections to distribution tails.

Findings

Implied volatility in the wings relates to the tail decay of returns.

Linear growth of implied variance implies exponential tail decay.

Analyticity of the characteristic function is linked to asymptotic slope of the smile.

Abstract

We investigate the asymptotic behaviour of the implied volatility in the Bachelier setting, extending the large-strike results established for the Black-Scholes framework. Exploiting the theory of regular variation, we derive explicit expressions for the Bachelier implied volatility in the wings of the smile, directly linking them to the tail decay of the underlying returns' distribution. Furthermore, we establish a rigorous connection between the analyticity strip of the characteristic function and the asymptotic slope of the volatility smile. Moreover, we show that if the implied variance grows linearly for large absolute moneyness degree, the underlying returns' distribution must exhibit exponential tail decay, and the corresponding characteristic function is analytic in a horizontal strip of the complex plane. These findings characterizes valid models based solely on the observable asymptotic behaviour of the smile.