Tehnike redukcije varijance u Monte Carlo simulacijama s primjenama u financijskoj matematici

The main goal of this thesis is to study some of the basic variance reduction techniques for Monte Carlo methods and apply them on examples from financial mathematics. In the first chapter Monte Carlo integration problem is explained, as well as theorems that it is based on. A detailed analysis of variance reduction techniques follows. Firstly, the Control Variates method, which is based on using the solutions of a known problem similar to the one which is being observed, is introduced. Then comes the Stratified Sampling method, which is based on the strategic variable generating from different subsets. Finally comes the Importance Sampling method, based on the measure alterations which emphasize the importance of rare occurrences. After that, similarities and differences between these methods are briefly discussed. The last chapter deals with the application of the described methods in financial mathematics. Firstly, the basic principles of the derivative pricing theory are explained, and then follow examples for each of the variance reduction techniques. Finaly, an implemented example shows the comparison between a Monte Carlo method variance and a variance resulting from the usage of the variance reduction technique.