MONTE CARLO METHODS AND MULTILEVEL MONTE CARLO METHODS IN PRICING EUROPEAN OPTION by Hong Kit CHAN Thesis Supervisor: Deng DING Department of Mathematics University of Macau Abstract Monte Carlo methods are very effective approach for the estimation of expectations arising from stochastic simulation. On the widespread use of the methods, its computational complexity and the limitations can make sure to get the best numerical result. There we will discuss methods about Multilevel Monte Carlo and Monte Carlo. Accuracy of Multilevel Monte Carlo methods is raised when the cost of generating individual stochastic samples is same to standard Monte Carlo methods as in the case of European option pricing. And Multilevel Monte Carlo is a recently developed which we expect to see significant improvements in our application. In this thesis, we review the ideas behind the Monte Carlo and the Multilevel Monte Carlo methods, and the results of Michael B. Giles’s analysis of the Multilevel Monte Carlo. This thesis results confirm that Multilevel Monte Carlo methods can improve Monte Carlo methods, like convergence, variance etc.