This thesis deals with stochastic portfolio optimization. In particular, we focus on two crucial aspects: The representation of uncertainty through the creation of a free-of-arbitrage scenario tree and the resolution of the portfolio optimization model by the L-shaped decomposition method. For the tree construction, we investigate and implemented a method called moment matching, comparing against the plain Monte-Carlo method that sample from Geometric Brownian Motion. Furthermore, we focus on how to exclude arbitrage opportunities during the generation of the scenario tree. For the resolution of the model, we use an L-shaped decomposition. All the methods presented have been implemented and tested on some instances, showing the effectiveness of both the moment matching method and the L-shaped decomposition.