Two-grid algorithms for pricing American options by a penalty method - Koleva, Miglena; Valkov, Radoslav (Proceedings of the Conference Algoritmy, 2016)
In this manuscript we present two-grid algorithms for the American option pricing problem with a smooth penalty method where the variational inequality, associated with the optimal stopping time problem, is approximated with a nonlinear Black-Scholes equation. In order to compute the numerical solution of the latter unconstrained problem we must solve a system of nonlinear algebraic equations resulting from the discretization by e.g. the finite difference or the finite element method. We propose two-grid algorithms as we first solve the nonlinear system on a coarse grid with mesh size $h^c$ and further a linearized system on a fine grid with mesh size $h^f$, satisfying $h^f=\mathcal{O}((h^c)^{2^k}), \; k=1,2,…$, where $k$ is the number of Newton iterations. Numerical experiments illustrate the computational efficiency of the algorithms.