Abstract A generalized Black-Scholes equation is considered on the semi-axis. It is transformed on the interval (0, 1) in order to make the computa-tional domain finite. The new parabolic operator degenerates at the both ends of the interval and we are forced to use the Gärding inequality rather than the classical coercivity. A fitted finite volume element space approximation is con-structed. It is proved that the time θ-weighted full discretization is uniquely solvable and positivity-preserving. Numerical experiments, performed to illus-trate the usefulness of the method, are presented. Keywords generalized Black-Scholes equation · degenerate parabolic equation · Gärding coercivity · fitted finite volume method · positivity