Natural exponential families (NEFs) are an important part of statistics’ theory. Indeed, many well-known parametric models, such as Poisson, Gaussian, inverse Gaussian or gamma are part of them. After a reminder of NEfs’ framework, we will focus on two important related functions : the variance function, which is the writing of the variance in terms of the mean and the generalized variance function, which is defined by taking its determinant. We will see how these functions can give informations about the underlying NEF. In particular, we will ask if the generalized variance function can characterize a particular NEF and with what restriction. We will also realize that this problem of characterization is related to the uniqueness of solutions of some Monge-Ampère equations.