Mathematical formula

A call option has positive monetary value at expiration when the underlying has a spot price (S) above the strike price (K). Since the option will not be exercised unless it is in-the-money, the payoff for a call option is

${\displaystyle \max \left[(S-K);0\right]}$

also written as

${\displaystyle (S-K)^{+}\ }$

where

${\displaystyle (x)^{+}={\begin{cases}x&{\text{if }}x\geq 0,\\0&{\text{if }}x<0.\end{cases}}}$

A put option has positive monetary value at expiration when the underlying has a spot price below the strike price; it is "out-the-money" otherwise, and will not be exercised. The payoff is therefore:

${\displaystyle \max \left[(K-S);0\right]}$

or

${\displaystyle (K-S)^{+}\ }$

For a digital option payoff is ${\displaystyle 1_{S\geq K}}$, where ${\displaystyle 1_{\{\}}}$ is the indicator function:

${\displaystyle 1_{S\geq K}={\begin{cases}1&{\text{if }}S\geq K,\\0&{\text{otherwise.}}\end{cases}}}$