This paper deals with the calculation of the elastic properties for single-walled carbon nanotubes (SWCNTs) under axial deformation and hydrostatic pressure using the atomistic-based continuum approach and the deformation mapping technique. A hyperelastic model based on the higher-order Cauchy-Born (HCB) rule being applicable at finite strains and accounting for the chirality and material nonlinearity is presented. Mechanical properties of several carbon nanotubes (CNTs) are computed and compared with the existing theoretical results and a good agreement is observed. Moreover, by comparison with atomistic calculations, it is found that the present model can reproduce the energetics of axially deformed CNTs. The model is then adopted to study the dependence of the elastic properties on chirality, radius and strain which yields an upper bound on the stability limit of axially and circumferentially stretched nanotubes. The influence of chirality is found to be more prominent for smaller tubes and as the diameter increases, the anisotropy induced by finite deformations gets nullified. It is discerned that the constitutive properties of the CNT can vary with deformation in a nonlinear manner. It is also found that the CNT displays a martial softening behavior at finite tensile strains and a hardening behavior at slightly compressive strains.