In this paper we develop a thorough survey of the European option pricing under exponential Lévy models. We sweep all steps from equivalent martingale measures construction to numerical valuation of the option price under these measures. We apply the Esscher transform technique to provide two examples of equivalent martingale measures: the Esscher martingale measure and the minimal entropy martingale measure. We numerically compute the option price using the fast Fourier transform. The results are detailed with an example of each exponential Lévy class. The main contribution of this paper is to build a comprehensive study from the theoretical point of view to practical numerical illustration and to give a complete characterization of the studied equivalent martingale measures by discussing their similarity and their applicability in practice.