This study investigates the irrational behavior of American put options holders that results in exercising options at non-optimal times. Investors usually react to market information and consequently market movements. These emotional reactions lead to exercising options strategy at a time that might not be optimal. In this situation, we consider irrational behavior in the option pricing problem. For this, we used the proposed intensity-based models with stochastic intensity parameters. Under these models, the option pricing problem leads to a nonlinear parabolic partial differential equation (PDE) with an additional term to the PDE of the American option under rational strategy (classical American option with optimal exercise strategy) due to the intensity functions of models. In this paper, we are interested in finding a stable solution for the resulting PDE using a finite element method. For this, we show the stability of the proposed finite element method by proving some theoretical results. Our numerical experiments demonstrate the accuracy and efficiency of the proposed method to obtain fast solutions for the pricing problem of American put options under irrational behavior.