Abstract: Lecture presents a survey of results from a new book with an above title, being at the moment in the final stage of preparation. The book is devoted to approximation methods for option rewards of American type options for multivariate modulatd Markov type price processes with discrete time. The classes of price processes under consideration include multivariate modulated Markov chains, modulated random walks, and various autoregressive models of price processes. General convergence results are presented, as well as their applications to space skeleton approximations and tree approximation algorithms. Also, results related to studies of structure for optimal stopping domains are presented as well as results related to option reselling problem. Theoretical results are illustrated by results of experimental studies. Finally, connection with problems of optimal stopping and convergence for multivariate modulatd Markov type price processes with continuous time is discussed.