This thesis is based on two articles devoted to optimal stopping problems of American type options.
In article A, 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. An approximate binomial-trinomial tree algorithm for the reselling model is constructed.
In article B, 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 and B.
Multivariate Markov price processes and American type options for such processes with generalpayoff functions with not more than polynomial rate of growth are considered. Convergence results are obtainedfor optimal reward functionals of American type options for perturbed multivariateMarkov processes and payoff functions. These results are applied to approximation tree type algorithmsfor American type options for exponential diffusion type priceprocesses including mean-reverse stochastic processesused to model stochastic dynamics of energy prices.
We consider the problem of optimal reselling of Europeanoptions. A bivariate exponential diffusion process is used todescribe the reselling model. In this way, the reselling problem isimbedded to the model of finding optimal reward for American typeoption based on this process. Convergence results are obtained foroptimal reward functionals of American type options for perturbedmulti-variate Markov processes. An approximation bivariate treemodel is constructed and convergence of optimal expected reward forthis tree model to the optimal expected reward for the correspondingAmerican type option is proved