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We study the behavior of the critical price of an American put option nearmaturity in the exponential L\'evy model when the underlying stock paysdividends at a continuous rate. In particular, we prove that, in situationswhere the limit of the critical price is equal to the stock price, the rate ofconvergence to the limit is linear if and only if the underlying L\'evy processhas finite variation. In the case of infinite variation, a variety of rates ofconvergence can be observed: we prove that, when the negative part of theL\'evy measure exhibits an $\alpha$-stable density near the origin, with$1<\alpha<2$, the convergence rate is ruled by $\theta^{1/\alpha}|\ln\theta|^{1-\frac{1}{\alpha}}$, where $\theta$ is time until maturity.

Exercise Boundary of the American Put Near Maturity in an Exponential L\'evy Model