Pricing Perpetual Put Options by the Black-Scholes Equation with a   Nonlinear Volatility Function
  We investigate qualitative and quantitative behavior of a solution of the mathematical model for pricing American style of perpetual put options. We assume the option price is a solution to the stationary generalized Black-Scholes equation in which the volatility function may depend on the second derivative of the option price itself. We prove existence and uniqueness of a solution to the free boundary problem. We derive a single implicit equation for the free boundary position and the closed form formula for the option price. It is a generalization of the well-known explicit closed form solution derived by Merton for the case of a constant volatility. We also present results of numerical computations of the free boundary position, option price and their dependence on model parameters. 
Maria do Rosario Grossinho, Yaser Kord Faghan, Daniel Sevcovic
q-fin.MF