Monte Carlo path simulations are common in mathematical and computational finance as a way of estimating the expected values of a quantity such as a European put option, which is functional to the solution of a stochastic differential equation (SDE). The computational complexity of the standard Monte Carlo (MC) method grows quite large quickly, so in this thesis we focus on the Multilevel Monte Carlo (MLMC) method by Giles, which uses multigrid ideas to reduce the computational complexity. We use a Euler-Maruyama time discretisation for the approximation of the SDE and investigate how the convergence rate of the MLMC method improves the computational times and cost in comparison with the standard MC method. We perform a numerical analysis on the computational times and costs in order to achieve the desired accuracy and present our findings on the performance of the MLMC method on a European put option compared to the standard MC method.