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.
There are two main approaches to explain the differences between them. The first one relies on the role of the parasitic roots (this is what we usually teach). The second one is more indirect and based on the general definition of stability. Spijker was the first who presented a norm pair in which the midpoint method is not stable. This example can be extended to the general weakly stable case. Finally, we upgrade this latter approach keeping its advantages and eliminating its weak point.