Asymptotic approximations for American options

I shall discuss a number of problems to do with approximating the value function of an American Put option in the Black-Scholes model. This is essentially a variant of the oxygen-consumption problem, a parabolic free boundary (obstacle) problem which is closely related to the Stefan problem. Having reviewed the short-time behaviour from the perspective of ray theory, I shall focus on constructing approximations in the case when there is a discretely paid dividend yield.