The constant elasticity of variance (CEV) model is a practical approach to option pricing by fitting to the implied volatility smile. However, pricing American options is computationally intensive because no analytical formulas are available. In this talk, we present numerical methods to find the optimal exercise boundary with respect to an American put option under the CEV model. This problem corresponds to the free boundary (also called an optimal exercise boundary) problem for a partial differential equation. We use a transformed function that has Lipschitz character near the optimal exercise boundary to determine the optimal exercise boundary. After determining the optimal exercise boundary, we calculate American put option values under the CEV model using the finite-difference method. Finally, we use the proposed numerical method to obtain several results that are then compared with the results of other methods.