Asian options are popular path-dependent options and it has been a long-standing problem to price them efficiently and accurately. Since there is no known exact pricing formula for Asian options, numerical pricing formulas like lattice models must be employed. A lattice divides a certain time interval into n time steps and the pricing results generated by the lattice (called desired option values for convenience) converge to the theoretical option value as n → ∞. Since a brute-force lattice pricing algorithm runs in subexponential time in n, some heuristics, like interpolation method, are used to strike the balance between the efficiency and the accuracy. But the pricing results might not converge due to the accumulation of interpolation errors. For pricing European-style Asian options, the evaluation on the major part of the lattice can be done by a simple formula, and the interpolation method is only required on the minor part of the lattice. Thus polynomial time algorithms with convergence guarantee for European-style Asian options can be derived. However, such a simple formula does not exist for American-style Asian options. This paper suggests an efficient range-bound algorithm that bracket the desired option value. By taking advantages of the early exercise property of American-style options, we show that part of the lattice can be evaluated by a simple formula. The interpolation method is required on the remaining part of the lattice and the upper and the lower bounds option values produced by the proposed algorithm are essentially numerically identical. Thus the theoretical option value is said to be obtained practically when the range bound algorithm runs on a lattice with large number of time steps.