Before revision: In this paper , we study a parabolic free boundary problem which shows that the solutions of this free boundary problem are increasing functions. Furthermore, we provide a rigorous verification for that the free boundary for this problem is convex .
After revision: In this paper , we study a parabolic free boundary problem arising from pricing American-style put options on an asset whose index follows a time-homogeneous diffusion process. The time-homogeneous diffusion process in this paper includes the geometric Brownian motion process, the CEV process, the mean-reverting Gaussian process or the Vasicek model, and the mean-reverting square root process or the CIR model. The contributions are to provide rigorous proofs of following facts. The value of an American-style put option increases with an increase in the time-to-maturity and decreases with an increase in the underlying asset index. The early exercise boundary is a strictly decreasing convex function of the time-to-maturity under the given conditions .