In this paper, the authors investigate a nonlinear generalization of the Black–Scholes equation for pricing American-style call options, where the volatility term may depend on both the underlying asset price and the Gamma of the option. They propose a numerical method for pricing American-style call options that involves transforming the free boundary problem for a nonlinear Black–Scholes equation into the so-called Gamma variational inequality with a new variable depending on the Gamma of the option. They apply a modified projected successive over-relaxation method in order to construct an effective numerical scheme for discretization of the Gamma variational inequality. Finally, they present several computational examples of the nonlinear Black–Scholes equation for pricing American-style call options in the presence of variable transaction costs.