Abstract This paper deals with the numerical solution of Black–Scholes option pricing partial differential equations by means of semidiscretization technique. For the linear case a fourth-order discretization with respect to the underlying asset variable allows a highly accurate approximation of the solution. For the nonlinear case of interest modeling option pricing with transaction costs, semidiscretization technique provides a competitive numerical solution with respect to others recently given in [B. Düring, M. Fournier, A. Jüngel, Convergence of a high order compact finite difference scheme for a nonlinear Black–Scholes equation, Esaim–Math. Modelling Numer. Anal.–Modelisation Mathematique et Analyse Numerique 38 (2004) 359–369; B. Düring, Black–Scholes type equations: mathematical analysis, parameter identification & numerical solution, Dissertation, University Mainz, July 2005].