In this work, the influence of an external magnetic field on the refractive index changes and optical absorption coefficients of a typical GaAs/AlxGa1-xAs coaxial cylindrical quantum well wire is studied. Energy eigenvalues and eigenvectors in the presence of a magnetic field are calculated using the fourth-order Runge-Kutta method and optical properties are obtained using the compact density matrix approach. It is found that the resonant peak values of the linear and the third-order nonlinear refractive index changes as well as the optical absorption coefficients of a cylindrical quantum wire decrease with increasing the magnetic field. However, these optical properties of a coaxial cylindrical quantum well wire are nonmonotonic functions of the magnetic field, due to the anticrossing of the subbands. The anticrossing effect leads to a large increase of the resonant peak values of these optical properties. Results also indicate that the resonant peak values of these optical properties decrease with increasing the barrier width.