ABSTRACT
This chapter presents the theory for pricing early-exercise (American) options in continuous time. The key difference between European-style and American-style options is that a holder of an American option can exercise her rights any time before the expiration date. This additional early exercise privilege should not be worthless. Thus, an American option is expected to be worth more than its European analogue. An American option cannot be worth less than its corresponding intrinsic value, which is the payoff associated with immediate exercise. The American option should be exercised whenever the asset price is in the stopping domain. A perpetual option is considered with infinite time to expiration. That is, a perpetual option has no expiration date and can be exercised at any future time. Pricing an American option can be formulated as a boundary initial value problem for a partial differential equation with a time-dependent free boundary.
