  We analyze and calculate the early exercise boundary for a class of
stationary generalized Black-Scholes equations in which the volatility function
depends on 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 including, e.g., non-zero transaction costs,
investors preferences, feedback and illiquid markets effects and risk from
unprotected portfolio. We present a method how to transform the problem of
American style of perpetual put options into a solution of an ordinary
differential equation and implicit equation for the free boundary position. We
finally present results of numerical approximation of the early exercise
boundary, option price and their dependence on model parameters.
