An off-center shift of the magnetic hysteresis loop, usually known as exchange bias, develops in an asymmetric nanodisk due to the Dzyaloshinskii-Moriya (DM) interaction. Results exhibit the onset of bias in a system without material interfaces, originating from the relation between the chirality defined by the DM interaction and the geometry. In addition, programmable magnetization bias by the variation of an external field emerges as a possibility. Our simulations, carried out using Mumax3 code, also yield double hysteresis loops, evidencing a magnetization reversal driven by vortex nucleation and annihilation. Bias in a homogeneous nanodisk and its control seem promising for applications.