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This thesis consists of an introduction and five articles devoted to optimal stopping problems of American type options. In article A, we get general convergence results for the American option rewards for multivariate Markov price processes. These results are used to prove convergence of tree approximations presented in papers A, B, C and E.In article B, we study the problem of optimal reselling for European options. The problem can be transformed to the problem of exercising an American option with two underlying assets. An approximative binomial-trinomial tree algorithm for the reselling model is constructed. In article C, we continue our study of optimal reselling of European options and give the complete solution of the approximation problem. In the article D, we consider general knockout options of American type. A Monte-Carlo method is used to study structure of optimal stopping domains generated by combinations of different pay-off functions and knockout domains.In article E the American option with knock out domains is considered. In order to show convergence of the reward functional the problem is reformulated in such a way that the convergence results in paper A can be applied.