American Option

An American option is an option that may be exercised at any stopping time up to its expiration time . If is the exercise payoff, its time- value in a standard frictionless market is the optimal stopping value

where denotes the essential supremum, namely the smallest random variable that is almost surely at least every conditional expectation of the payoff in the family. The set consists of admissible stopping times with values in , is a valuation probability, and is a filtration. Thus, unlike a European option, an American option has an additional option on exercise timing.

For a Markov process with infinitesimal generator , the value formally satisfies the obstacle problem

The exercise region is where , while the continuation region is where . Their interface is an unknown boundary, so valuation is a free boundary problem or, equivalently, a variational inequality. The Snell envelope gives the corresponding probabilistic construction.

An early-exercise-premium decomposition writes the value as a term for a European option plus an integral over the exercise region. Imposing value matching on the unknown exercise boundary yields a nonlinear Volterra integral equation. In a two-factor stochastic-volatility model, this boundary becomes an early-exercise surface depending on both time and variance and need not be linear in variance (Andersen et al. 2026).