On the implied volatility of European and Asian call options under the stochastic volatility Bachelier model

TL;DR

This paper analyzes the short-time behavior of implied volatility for European and Asian call options under a stochastic volatility Bachelier model, deriving asymptotic formulas and validating them with numerical simulations.

Contribution

It introduces a novel asymptotic analysis of implied volatility skew for options under stochastic volatility, including rough volatility models, using Malliavin calculus techniques.

Findings

Derived short-maturity asymptotic formulas for implied volatility and skew.

Validated formulas with numerical simulations for SABR and fractional Bergomi models.

Abstract

In this paper we study the short-time behavior of the at-the-money implied volatility for European and arithmetic Asian call options with fixed strike price. The asset price is assumed to follow the Bachelier model with a general stochastic volatility process. Using techniques of the Malliavin calculus such as the anticipating Ito's formula we first compute the level of the implied volatility when the maturity converges to zero. Then, we find a short maturity asymptotic formula for the skew of the implied volatility that depends on the roughness of the volatility model. We apply our general results to the SABR and fractional Bergomi models, and provide some numerical simulations that confirm the accurateness of the asymptotic formula for the skew.