In this paper we study quantitative stochastic homogenization of a nonlinearly elastic composite material with a laminate microstructure. We prove that for deformations close to the set of rotations, the homogenized stored energy function W_{\hom} is C^{3} and that W_{\hom} , the stress tensor DW_{\hom} , and the tangent-moduli D^{2}W_{\hom} can be represented with the help of stochastic correctors. Furthermore, we study the error of an approximation of these quantities via representative volume elements. More precisely, we consider periodic representative volume elements (RVEs) obtained by periodizing the distribution of the random material. For materials with a fast decay of correlations on scales larger than a unit scale, we establish error estimates on the random and systematic error of the RVE with optimal scaling in the size of the RVE and with a multiplicative random constant that has exponential moments.