Finite difference methods for pricing American put option with rationality parameter: Numerical analysis and computing

In this paper finite difference methods for pricing American option with rationality parameter are proposed. The irrational exercise policy arising in American options is characterized in terms of a rationality parameter. The model is formulated in terms of a new nonlinear Black–Scholes equation that requires specific numerical methods. Although the solution converges to the solution of the classical American option price when the parameter tends to infinity, for finite values of the parameter the classical boundary conditions cannot apply and we propose specific ones. A logarithmic transformation is used to improve properties of the numerical solution that is constructed by explicit finite difference method. Numerical analysis provides stability conditions for the methods and its positivity. Properties of intensity function are studied from the point of view of numerical solution. Concerning the numerical methods for the original problem we propose the θθ-method for time discretization, thus including explicit, fully implicit and Crank–Nicolson schemes as particular cases. The nonlinear term is treated by a Newton method. The convergence rate is illustrated by numerical examples.