Accurate prediction of the roll response is of significant practical relevance not only for ships but also ship type offshore structures such as FPSOs, FLNGs and FSRUs. This paper presents a new body-exact scheme that is introduced into a nonlinear direct time-domain based strip theory formulation to study the roll response of a vessel subjected to moderately large amplitude incident waves. The free surface boundary conditions are transferred onto a representative incident wave surface at each station. The body boundary condition is satisfied on the instantaneous wetted surface of the body below this surface. This new scheme allows capturing nonlinear higher order fluid loads arising from the radiated and wave diffraction components. The Froude-Krylov and hydrostatic loads are computed on the intersection surface of the exact body position and incident wave field. The key advantage of the methodology is that it improves prediction of nonlinear hydrodynamic loads while keeping the additional computational cost small. Physical model tests have been carried out to validate the computational model. Fairly good agreement is seen. Comparisons of the force components with fully linear and body-nonlinear models help in bringing out the improvements due to the new formulation.