Analytical and numerical results for American style of perpetual put options through transformation into nonlinear stationary Black-Scholes equations

摘要

We analyze and calculate the early exercise boundary for a class ofstationary generalized Black-Scholes equations in which the volatility functiondepends on the second derivative of the option price itself. A motivation forstudying the nonlinear Black Scholes equation with a nonlinear volatilityarises from option pricing models including, e.g., non-zero transaction costs,investors preferences, feedback and illiquid markets effects and risk fromunprotected portfolio. We present a method how to transform the problem ofAmerican style of perpetual put options into a solution of an ordinarydifferential equation and implicit equation for the free boundary position. Wefinally present results of numerical approximation of the early exerciseboundary, option price and their dependence on model parameters.