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Derive option payoffs by terminal-price region

A vanilla option payoff is formed by a positive difference in one region and zero in the other.

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Partition a call at its strike

For strike K, a holder receives S−K when terminal price S exceeds K, and zero when S is at or below K. The partition describes expiry settlement. It does not estimate the price of the option before that date.

Reverse the difference for a put

A put holder receives K−S below the strike and zero at or above it. Nonnegative terminal underlying prices are assumed in these examples. A writer’s signed settlement is the negative of the holder’s amount.

Price regions
RegionCall holderPut holder
S < K0K−S
S = K00
S > KS−K0

Add the position’s premium after the payoff

Subtract a paid premium for a long position and add a received premium for a short one in the simple unfinanced profit calculation. The payoff kink remains at K, while the positive-premium profit boundary generally lies elsewhere.

Use regions to inspect a combination

For several same-expiry legs, break the price axis at every strike. Add the correct leg expression in each region. This method exposes sign errors that can remain hidden in a single numerical scenario.

Further references