Unlike the European options pricing, the closed-form solution generally does not exist due to the early exercise feature of American options. Hence, numerical approximation methods are normally employed to solve them. Presented in this paper is a new numerical method to price American bond options. To numerically solve the resulting partial differential complementarity problem (PDCP), we develop a class of finite volume method for the spatial discretization, coupled with the stable fully implicit time stepping scheme of the partial differential equation (PDE). Then, the resulting linear complementarity problems (LCPs) are solved by using an efficient iterative method, the modulus-based matrix splitting iteration method, where the $H_{+}$-matrix property of the system matrix guarantees its convergence. Numerical experiments are implemented to verify the accuracy, efficiency and robustness of the new method.