Exercise boundary of American-style Asian option

In this paper, we study the smoothness of the optimal exercise boundary (i.e. free boundary) of American-style arithmetic average Asian option with floating strike price in finite horizon case (i.e. parabolic case). Applying stochastic analysis the authors of [G. Peskir, N. Yus, On Asian options of American type, Exotic Option Pricing and Advanced Levy Models, Wiley, Eindhoven, 2004, pp. 217–235; G. Peskir, A. Shiryaev, Optimal stopping and free boundary problems, Lectures in Mathematics ETH Zurich, Birkhauser, 2006, pp. 416–436] proved that the optimal exercise boundary is a continuous and monotonic curve. Based on their results we prove that the optimal exercise boundary is infinitely differentiable using PDE method.