On the implied volatility of Inverse options under stochastic volatility models

Abstract

In this paper we study short-time behavior of the at-the-money implied volatility for Inverse European options with fixed strike price. The asset price is assumed to follow a general stochastic volatility process. Using techniques of the Malliavin calculus such as the anticipating It^o's formula we first compute the level of the implied volatility of the option 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 also show that our results extend easily to Quanto-Inverse options. 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. Finally, we provide an empirical application using Bitcoin options traded on Debirit to show how our theoretical formulas can be used to model real market data of such options.