The following question arises in stochastic programming: how can one approximate a noisy convex function with a convex quadratic function that is optimal in some sense. Using several approaches for constructing convex approximations we present some optimization models yielding convex quadratic regressions that are optimal approximations in $L_1$, $L_\infty$ and $L_2$ norm. Extensive numerical experiments to investigate the behaviour of the proposed methods are also performed.