  The purpose of this paper is to analyze and compute the early exercise
boundary for a class of nonlinear Black--Scholes equations with a nonlinear
volatility which can be a 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
new method how to transform the free boundary problem for the early exercise
boundary position into a solution of a time depending nonlinear parabolic
equation defined on a fixed domain. We furthermore propose an iterative
numerical scheme that can be used to find an approximation of the free
boundary. We present results of numerical approximation of the early exercise
boundary for various types of nonlinear Black--Scholes equations and we discuss
dependence of the free boundary on various model parameters.
