On a free boundary problem for an American put option under the CEV process

We consider an American put option under the CEV process. This corresponds to a free boundary problem for a PDE. We show that this free boundary satisfies a nonlinear integral equation, and analyze it in the limit of small ρ = 2 r / σ 2 , where r is the interest rate and σ is the volatility. We use perturbation methods to find that the free boundary behaves differently for five ranges of time to expiry.