We consider an American call option and let C(S, T0) be the price of an option corresponding
to asset price S at some time T0 prior to the expiration time TF . We analyze C(S, T0) in various
asymptotic limits. These include situations where the interest and dividend rates are large or
small, compared to the volatility of the asset. We also analyze the optimal exercise boundary
for the option. We use perturbation methods to analyze either the PDE that C(S, T0) satisfies,
or a nonlinear integral equation that is satisfied by the optimal exercise boundary.