Optimal early exercise as a moving boundary

What is a free-boundary problem, and how does it determine the optimal time to exercise an American option?

It is a PDE problem in which part of the domain's boundary is unknown and found together with the solution. For an American put, the price satisfies the Black-Scholes PDE in the continuation region and equals the payoff in the exercise region; the boundary $S^*(t)$ is fixed by value matching and smooth pasting ($V$ and $\partial V/\partial S$ continuous). It is optimal to exercise the first time $S$ hits $S^*(t)$, which maximises the option's value.