On the Implied volatility of European and Asian call options under the Stochastic Volatility Bachelier Model

• Authors: Elisa Alòs, Eulàlia Nualart and Makar Pravosud.

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 Itô’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 stochastic alpha-beta-rho (SABR), fractional Bergomi and local volatility models, and provide some numerical simulations that confirm the accurateness of the asymptotic formula for the skew.