We present a theoretical study of the transport properties of magnetic point contacts under bias. Our calculations are based on the Keldish's non-equilibrium Green's function formalism combined with a self-consistent empirical tight-binding Hamiltonian, which describes both strong ferromagnetism and charging effects. We demonstrate that large magnetoresistance solely due to electronic effects can be found when a sharp domain wall forms inside a magnetic atomic-scale point contact. Moreover we show that the symmetry of the I-V characteristic depends on the position of the domain wall in the constriction. In particular diode-like curves can arise when the domain wall is placed off-center within the point contact, although the whole structure does not present any structural asymmetry.