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Exercise Boundary of the American Put Near Maturity in an Exponential Lévy Model

We study the behavior of the critical price of an American put option near maturity in the exponential Lévy model when the underlying stock pays dividends at a continuous rate. In particular, we prove that, in situations where the limit of the critical price is equal to the stock price, the rate of convergence to the limit is linear if and only if the underlying Lévy process has finite variation. In the case of infinite variation, a variety of rates of convergence can be observed: we prove that, when the negative part of the Lévy measure exhibits an $α$-stable density near the origin, with $1<α<2$, the convergence rate is ruled by $θ^{1/α}|\ln θ|^{1-\frac{1}α}$, where $θ$ is time until maturity.