 American option

An American option permits exercise at a chosen stopping time no later than maturity. Let $G_t$ be its exercise financial payoff and $B_t>0$ the bank account. Under a specified complete-market pricing model and the usual optimal-stopping integrability conditions, its value isThe Snell envelope of the discounted exercise process is $V_t/B_t$, not the unscaled currency price $V_t$. The distinction matters away from a date at which the bank account is normalized to one. At each exercise opportunity, one compares immediate payoff with the best conditional continuation value. A European option allows only the terminal stopping time, whereas an American call option or put can have a genuine early-exercise region depending on dividends, rates and payoff shape.