Pricing American call options using the Black-Scholes equation with a nonlinear volatility function

In this paper, we 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. We 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 socalled Gamma variational inequality with a new variable depending on the Gamma of the option. We apply a modified projected successive over-relaxation method in order to construct an effective numerical scheme for discretization of the Gamma variational inequality. Finally, we present several computational examples of the nonlinear Black-Scholes equation for pricing American-style call options in the presence of variable transaction costs.