The prices and hedging strategies in the real financial  market models are often described by fully nonlinear versions of the standard Black-Scholes equation. We concentrate on two classes of models: first, nonlinear Black-Scholes equations in which the volatility depends on  second space derivatives of the price(=solution) and then on regime-switching models described by systems of semilinear parabolic equations with exponential nonlinearities. The following characteristic  properties of these parabolic problems are typical: unbounded domain, boundary degeneration, maximum-minimum principle and nonnegativity preservation. We develop effective discretizations that reproduce these properties.