  In this paper we analyze a nonlinear Black--Scholes model for option pricing
under variable transaction costs. The diffusion coefficient of the nonlinear
parabolic equation for the price $V$ is assumed to be a function of the
underlying asset price and the Gamma of the option. We show that the
generalizations of the classical Black--Scholes model can be analyzed by means
of transformation of the fully nonlinear parabolic equation into a quasilinear
parabolic equation for the second derivative of the option price. We show
existence of a classical smooth solution and prove useful bounds on the option
prices. Furthermore, we construct an effective numerical scheme for
approximation of the solution. The solutions are obtained by means of the
efficient numerical discretization scheme of the Gamma equation. Several
computational examples are presented.
