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

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.