Transformation methods for evaluating approximations to the optimal exercise boundary for linear and nonlinear Black-Scholes equations

摘要

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