We study the robustness of the sensitivity with respect to parameters in expectation functionals with respect to various approximations of a Lévy process. As sensitivity parameter, we focus on the delta of an European option as the derivative of the option price with respect to the current value of the underlying asset. We prove that the delta is stable with respect to natural approximations of a Lévy process, including approximating the small jumps by a Brownian motion. Our methods are based on the density method, and we propose a new conditional density method appropriate for our purposes. Several examples are given, including numerical examples demonstrating our results in practical situations.