In this paper we analyze a nonlinear generalization of the Black-Scholes equation for pricing American style call option in which the volatility may depend on the underlying asset price and the Gamma of the option. We propose a novel method of pricing American style call options by means of transformation of the free boundary problem for a nonlinear Black-Scholes equation into the so-called Gamma variational inequality with the new variable depending on the Gamma of the option. We apply a modified projective successive over-relation method in order to construct an effective numerical scheme for discretization of the Gamma variational inequality. Finally, we present several computational examples for the nonlinear Black-Scholes equation for pricing American style call option under presence of variable transaction costs.
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