Non-convexity of the optimal exercise boundary for an American put option on a dividend-paying asset

In this thesis, we prove that the optimal exercise boundary of the American put option is not convex when the dividend rate of the underlying assetwhich follows a geometric Brownian motion, is slightly larger than the risk-free interest rate. We show that the non-convex region occurs very near the expiry time. Numerical evidence is also provided which suggests that the convexity of the optimal exercise boundary is restored when the dividend rate is sufficiently larger than the interest rate. In addition we provide the near-expiry and far-from-expiry behavior of the boundary. To complete the rigorous proofs, we also show that the optimal exercise boundary has $C^infty$ regularity