Sensitivities of options via Malliavin calculus: applications to markets of exponential Variance Gamma and Normal Inverse Gaussian processes

This paper presents new sensitivities for options when the underlying follows an exponential Lévy process, specifically Variance Gamma and Normal Inverse Gaussian processes. The calculation of these sensitivities is based on a finite-dimensional Malliavin calculus and finite difference methods via Monte-Carlo simulations. In order to compare the real performance of this method we use the inverse Fourier method to calculate the exact values of the sensitivities of European call and digital options written on the S&P 500 index. Our results show that variations of the localized Malliavin calculus approach outperform the finite difference method in calculations of the Greeks and the new sensitivities that we introduce.