The Numerical Methods Of American Options'pricing Problems

The Black-Scholes model assumes constant volatility for the price of the underlying asset. For stock options this simple asset price model does not lead to prices which would be consistent with the observed market prices.In the real market we have to consider some options ,whose price is composed by the backward stochastic differential equation and martingale methods.The high dimensional Black-Scholes equation have the form asfollows:Many different generalizations giving more consistent prices have been proposed.One such approach is to assume the volatility to be also stochastic in addition to the stochastic return component of the simple asset price model,the popular case is Heston's model.For European options the price is obtained by solving a parabolic partial differential equation (PDE).The spatial operator is of convection diffusion type with varying coefficients an a second-order cross-derivative.In order to have a numerical solution this two-dimensional spatial operator is discretized using the PDE methods,like finite element method ,finite difference method and finite volume method.We should take great notice to the approximations of the convection and second-order cross-derivative terms,since they can influence the schemes'property ,such as whether or not satisfying the max principle theorem. As well as we known,If a scheme satisfies the max principle theorem,it will have good steady property.Howerer,in the researches before ,there are few schemes of high dimensional PDEs have this property. American options can be exercised anytime during the life time of the option contract and therefore it is more flexible than the European one .This early exercise possibility of an American option leads to a lower bound for the option price.By combining this early exercise constraint with the PDE ,a time-dependent linear complementarity problem(LCP) is obtained for the prices of American options.Our aim...