The conservative Kurganov–Tadmor (KT) scheme has been successfully applied to option-pricing problems by Germán I. Ramírez-Espinoza and Matthias Ehrhardt. These included the valuation of European, Asian and nonlinear options as Black–Scholes partial differential equations, written in the conservative form, by simply updating fluxes in the black box approach. In this paper, we describe an improvement of this idea through a fully vectorized algorithm of nonoscillatory slope limiters and the efficient use of time solvers. We also propose the application of second-order extensions of KT to option-pricing problems. Our test problems solve one-dimensional benchmark and convection-dominated European options as well as digital and butterfly options. These demonstrate the robustness and flexibility of the pricing methods and set a basis for complex problems. Further, the computation of option Greeks ensures the reliability of these methods. Numerical experiments are performed on barrier options, early exercisable American options and two-dimensional fixed and floating strike Asian options. To the authors’ knowledge, this is the first time American options have been priced by applying the early exercise condition on the semi-discrete formulation of central-upwind schemes. Our results show second-order, nonoscillatory and high-resolution properties of the schemes as well as computational efficiency.