This paper generalizes the results about the discretization error in simulation of extrema of one-dimensional Brownian motion to the case of one dimensional alpha-stable process. Our main result shows that the discretization error has a weak order of convergence of precisely 1/alpha for alpha in (0; 2], and a strong order of convergence of precisely 1/alpha for alpha in (1; 2]. The asymptotic distribution of the discretization error is described by the path decomposition of an alpha-stable process at the time of its maximum or equivalently can be described by the alpha-stable process conditioned to stay positive and negative. We also find the asymptotic distribution for the discretization error in simulation of the extrema by observing the process under randomly distributed discretetime points. The expectation of the asymptotic distribution is obtained to improve the approximation of the expectations, which usually are functionals of the alpha-stable process. Numerical simulations are also provided for nine asymptotic expectations and nine histograms of normalized discretization error for the stability parameter alpha= 1:1; 1:5; 1:9 and the skewness parameter beta= 0:9; 0; 0:9