In this chapter we compute numerically the prices of American put options, and find the optimal exercise boundary, which is a free boundary not given explicitly in the problem. First, as a motivation we introduce a free boundary problem for ice melting, called the Stefan problem, and extend the idea to the main subject of pricing American put options. Since the Black–Scholes–Merton equation is essentially a heat equation, one may build helpful intuition for option pricing by studying the ice melting problem.