Semismooth Newton methods with domain decomposition for American options

In this paper, we develop a class of parallel semismooth Newton algorithms for the numerical solution of the American option under the Black-Scholes-Merton pricing framework. In the approach, a nonlinear function is used to transform the complementarity problem, which arises from the discretization of the pricing model, into a nonlinear system. Then, a generalized Newton method with a domain decomposition type preconditioner is applied to solve this nonlinear system. In addition, an adaptive time stepping technique, which adjusts the time step size according to the initial residual of Newton iterations, is applied to improve the performance of the proposed method. Numerical experiments show that the proposed semismooth method has a good accuracy and scalability.