Contingent claim

A contingent claim is a future payoff whose value depends on uncertain market outcomes. A replicable claim has the unique no-arbitrage price of its replicating portfolio.

A replicating strategy is a self-financing portfolio whose terminal wealth equals the prescribed payoff of a contingent claim. Admissibility specifies the permitted wealth bounds and integrability of its holdings.

A European contingent claim pays a specified function of market variables at one fixed maturity. Under an equivalent martingale measure, an attainable claim is priced by the discounted conditional expectation of its payoff.

The holder chooses at a specified time between a European call option and a European put option with the same later maturity and strike. Put-call parity makes its value equal to a call of the original maturity plus a put of the choice-time maturity with appropriately discounted strike.

A power option is a European contingent claim whose payoff is a fixed power of the terminal underlying price. In the Black-Scholes model, its value is $S_t^p\exp ((p-1)r(T-t)+\frac{1}{2}p(p-1){\sigma }^2(T-t))$ when the required moments are finite.

A binary option pays one fixed amount when a specified event occurs at maturity and zero otherwise.

A digital call option with strike $K$ pays one unit if $S_T\ge K$ and zero otherwise.

A digital put option with strike $K$ pays one unit if $S_T<K$ and zero otherwise. With these complementary boundary conventions, one digital call plus one digital put equals a zero-coupon bond paying one unit at maturity.