American options and semilinear parabolic partial differential equations in weighted Sobolev spaces

To value an American option as a function of time t and price of the underlying asset S is currently a major research problem in both Financial Markets and for academic purposes. Options and more general financial derivatives (also known as contingent claims) are now an important tool in risk management. One of the earliest models used in pricing derivatives is the Black-Scholes model, for which the movement in the price of the underlying asset on which the claim is based is modeled by geometric Brownian motion. Other models used for theoretical and numerical analysis of American options include the free boundary problem method, linear complimentarity problem method, and variational inequality methods. And there are others based on the Cox, Ross, and Rubinstein binomial approach.;Despite the existence of these methods, there is a strong practical demand to create new methods which firstly are more computationally efficient and make explicit the mathematical framework involved. Partial differential equations (PDEs) of parabolic type have fundamental applications to modelling processes with diffusion and uncertainty. The pricing of European options can be reduced to the calculation of certain solutions of parabolic equations, often called backward Kolmogorov's equations and obtained through Ito's lemma. In this work we are dealing with American options and we transform the Black-Scholes equation into a nonlinear parabolic equation in the entire space variable. The initial condition might be unbounded and so we strive to show the existence of the solution in some weighted Sobolev space.