In the paper Bally and Pagès (2000) an algorithm based on an optimal discrete quantization tree is designed to compute the solution of multi-dimensional obstacle problems for homogeneous -valued Markov chains (Xk)0[less-than-or-equals, slant]k[less-than-or-equals, slant]n. This tree is made up with the (optimal) quantization grids of every Xk. Then a dynamic programming formula is naturally designed on it. The pricing of multi-asset American style vanilla options is a typical example of such problems. The first part of this paper is devoted to the analysis of the Lp-error induced by the quantization procedure. A second part deals with the analysis of the statistical error induced by the Monte Carlo estimation of the transition weights of the quantization tree.