  The purpose of this survey chapter is to present a transformation technique
that can be used in analysis and numerical computation of the early exercise
boundary for an American style of vanilla options that can be modelled by class
of generalized Black-Scholes equations. We analyze qualitatively and
quantitatively the early exercise boundary for a linear as well as a class of
nonlinear Black-Scholes equations with a volatility coefficient which can be a
nonlinear function of the second derivative of the option price itself. A
motivation for studying the nonlinear Black-Scholes equation with a nonlinear
volatility arises from option pricing models taking into account e.g.
nontrivial transaction costs, investor's preferences, feedback and illiquid
markets effects and risk from a volatile (unprotected) portfolio. We present a
method how to transform the free boundary problem for the early exercise
boundary position into a solution of a time depending nonlinear nonlocal
parabolic equation defined on a fixed domain. We furthermore propose an
iterative numerical scheme that can be used in order to find an approximation
of the free boundary. In the case of a linear Black-Scholes equation we are
able to derive a nonlinear integral equation for the position of the free
boundary. We present results of numerical approximation of the early exercise
boundary for various types of linear and nonlinear Black-Scholes equations and
we discuss dependence of the free boundary on model parameters. Finally, we
discuss an application of the transformation method for the pricing equation
for American type of Asian options.
