This paper focuses on studying the multilevel Monte Carlo method recently introduced by Giles [8] and significantly more efficient than the classical Monte Carlo one. Our aim is to prove a central limit theorem of Lindeberg Feller type for the multilevel Monte Carlo method associated to the Euler discretization scheme. To do so, we prove first a stable law convergence theorem, in the spirit of Jacod and Protter [15], for the Euler scheme error on two consecutive levels of the algorithm. This leads to an accurate description of the optimal choice of parameters and to an explicit characterization of the limiting variance in the central limit theorem of the algorithm. We investigate the application of the Multilevel Monte Carlo method to the pricing of Asian options, by discretizing the integral of the payoff process using Riemann and trapezoidal schemes. In particular, we prove stable law convergence for the error of these second order schemes. This allows us to prove two additional central limit theorems providing us the optimal choice of the parameters with an explicit representation of the limiting variance. For this setting of second order schemes, we give new optimal parameters leading to the convergence of the central limit theorem. Complexity analysis of the Multilevel Monte Carlo algorithm were processed.