Pricing American options under irrational behavior in a Markov regime-switching model with a finite-element method

We study the pricing problem for American options under a regime-switching model with the possibility of a nonoptimal exercise policy (ie, an early or late exercise time), referred to as an irrational strategy. To do so, we consider a Markov-modulated model for the dynamics of the underlying asset (as an alternative to the classical Black– Scholes–Merton model) and an intensity-based model for the irrational strategy, in order to provide more realistic results for American option prices under irrational behavior in real financial markets. By applying a partial differential equation (PDE) approach, the pricing problem for American options under a regime-switching model can be formulated as coupled PDEs. As the numerical procedure to solve the resulting systems of PDEs for this model, we apply a finite-element method to the resulting variational inequality. This provides us with the benefits of finite-element methods, such as their efficacy, speed and simplicity in dealing with the boundary conditions of problems with quadrature convergence rates. Under some appropriate but straightforward assumptions, we establish the stability of the method and compare its accuracy against some recent works, illustrating the suitability of the proposed model and the accuracy of the applied numerical method for the pricing of American options under a regime-switching model with irrational behaviors.