Contingent claim payoff ($X$)

The monetary amount delivered by a contingent claim at its payment time, represented by a random variable measurable with respect to the information then available. For a European call option it is $(S_T-K{)}^{+}$, and for a European put option it is $(K-S_T{)}^{+}$. A payoff is the delivered amount, whereas the claim is the contract and its current price is the cost of acquiring or replicating that contract.

The financial payoff obtained by exercising a financial option at a permitted stopping time $\tau$. An American option compares this immediate payoff with the value of keeping the exercise right, whereas a European option restricts exercise to expiry.