Critical price near maturity for an American option on a dividend-paying stock in a local volatility model

We consider an American put option on a dividend-paying stock whose volatility is a function of the stock value. Near the maturity of this option, an expansion of the critical stock price is given. If the stock dividend rate is greater than the market interest rate, the payoff function is smooth near the limit of the critical price. We deduce an expansion of the critical price near maturity from an expansion of the value function of an optimal stopping problem. It turns out that the behavior of the critical price is parabolic. In the other case, we are in a less regular situation and an extra logarithmic factor appears. To prove this result, we show that the American and European critical prices have the same first-order behavior near maturity. Finally, in order to get an expansion of the European critical price, we use a parity formula for exchanging the strike price and the spot price in the value functions of European puts.