This paper analyzes and values an American barrier option with continuous payment plan written on a dividend paying asset under the classical Black-Scholes model.The integral representation of the initial premium along with the delta hedge parameter for an American continuous-installment down-and-out call option are obtained by using the decomposition technique.This offers a system of nonlinear integral equations for determining the optimal exercise and stopping boundaries,which can be utilized to approximate the option price and delta hedge parameter.The implementation is based on discretizing the quadrature formula in the system of equations and using the Newton-Raphson method to compute the two optimal boundaries at each time points.Numerical results are provided to illustrate the computational accuracy and the effects on the initial premium and optimal boundaries with respect to barrier. This paper analyzes and values ​​an American barrier option with continuous payment plan written on a dividend paying loan under the classical Black-Scholes model. The integral representation of the initial premium along with the delta hedge parameter for an American continuous-installment down-and- out call option are obtained by using the decomposition technique. This provides a system of nonlinear integral equations for determining the optimal exercise and stopping boundaries, which can be utilized to approximate the option price and delta hedge parameter. The implementation is based on discretizing the quadrature formula in the system of equations and using the Newton-Raphson method to compute the two optimal boundaries at each time points. Numerical results are provided to illustrate the computational accuracy and the effects on the initial premium and optimal boundaries with respect to barrier.