On the implied volatility of Inverse options under stochastic volatility models

Abstract

In this paper, we study the short-time behavior of at-the-money implied volatility for inverse European options with a fixed strike price. The asset price is assumed to follow a general stochastic volatility process. Using techniques from Malliavin calculus, such as the anticipating Itô's formula, we first compute the implied volatility of the option as its maturity approaches zero. Next, we derive a short-maturity asymptotic formula for the skew of the implied volatility, which depends on the roughness of the volatility model. We also demonstrate that our results can be easily extended to Quanto-Inverse options. We apply our general findings to the SABR and fractional Bergomi models and provide numerical simulations that confirm the accuracy of the asymptotic formula for the skew. Finally, we present an empirical application using Bitcoin options traded on Deribit, showing how our theoretical formulas can be applied to model real market data for such options.