We show that for Fe-based alloys the spontaneous volume magnetostriction can be calculated from first principles in good agreement with experiment. Our results are obtained by modeling the paramagnetic state above the magnetic ordering temperature as a state with disordered local moments. Investigating bcc Fe-Co and fcc Fe-Pt(Pd) alloys which exhibit large values of the volume magnetostriction in a certain range of chemical composition and concomitantly an anomalously low thermal expansion coefficient (Invar effect), we find that this behavior is due to changes in the electronic structure induced by the effects of magnetic disorder leading to reduced local atomic moments in the paramagnetic state. The large values of the spontaneous volume magnetostriction found in Invar alloys scale with the equally observed large changes in the local moments.