Numerical methods for dynamic Bertrand oligopoly and American options under regime switching

Abstract

Bertrand oligopolies are competitive markets in which a small number of firms producing similar goods use price as their strategic variable. In particular, each firm wants to determine the optimal price that maximizes its expected discounted lifetime profit. The oligopoly problem can be modeled as nonzero-sum games which can be formulated as systems of Hamilton–Jacobi–Bellman (HJB) partial differential equations (PDEs). In this paper, we propose fully implicit, positive coefficient finite difference schemes that converge to the viscosity solution for the HJB PDE from dynamic Bertrand monopoly and the two-dimensional HJB system from dynamic Bertrand duopoly. Furthermore, we develop fast multigrid methods for solving these systems of discrete nonlinear HJB PDEs. The new multigrid methods are general and can be applied to other systems of HJB and HJB-Isaacs PDEs arising from American options under regime switching and American options with unequal lending/borrowing rates and stock borrowing fees under regime switching, respectively. We provide a theoretical analysis for the smoother, restriction and interpolation operators of the multigrid methods. Finally, we demonstrate the effectiveness of our method by numerical examples from the dynamic Bertrand problem and pricing American options under regime switching.