Smile asymptotic for Bachelier Implied Volatility

TL;DR

This work advances the understanding of Bachelier implied volatility by transferring the regular variation framework of Benaim and Friz from Black–Scholes to the Bachelier model. It derives explicit wing asymptotics for the Implied Volatility $I(\kappa)$ in terms of the tail behaviour of the underlying returns, and links these asymptotics to the analyticity strip of the returns’ characteristic function. Under tail-integrability assumptions (IR) and (IL), the right and left wings satisfy precise relations involving $\ln \bar{F}(\kappa)$ and $\ln F(-\kappa)$, respectively; moreover, if the CF is analytic in a strip $\Im(\xi)\in(-\lambda_-,\lambda_+)$ (Condition I), the wings’ growth becomes linear with slope determined by $\lambda_\pm$. These results generalize Lee-type moment relationships to the Bachelier setting and apply to common finance models (e.g., Heston, Tempered Stable, Meixner), providing a practical bridge between tail/CF properties and the wings of the Bachelier implied volatility smile.

Abstract

We investigate the asymptotic behaviour of the Implied Volatility in the Bachelier setting, extending the framework introduced by Benaim and Friz for the Black-Scholes setting. Exploiting the theory of regular variation, we derive explicit expressions for the Bachelier Implied Volatility in the wings of the smile, linking these to the tail behaviour of the underlying's returns' distribution. Furthermore, we establish a direct connection between the analyticity strip of the returns' characteristic function and the asymptotic formula for the Implied Volatility smile at extreme moneyness.

Theorems & Definitions (18)

• Lemma 2.1
• proof
• Definition 2.2
• Theorem 2.3: cf., [bingham1987regular, Th.4.12.10, p.255]
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Theorem 3.3: Right wing formula
• proof