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# Options Math

## Option Payoff

Let
$K$
be the exercise price and
$S_{T}$
the price of the underlying stock at the maturity date.

### Call Option Payoff

The payoff of a call option is:
$\begin{equation*} c_{T} = \{ \begin{array}{ll} 0 & \quad \text{if } S_{T} \leq K \\ S_{T}-K & \quad \text{if} S_{T} > K \end{array} \} \end{equation*}$
or
$c_{T} = max [S_{T} - K, 0]$
If
$S_{T}$
> K, then the call is said to expire in-the-money and the option holder can exercise the right to buy the underlying asset at price K rather than at the current market price
$S_{T}$
.

### Put Option Payoff

The payoff of a put option is:
$\begin{equation*} p_{T} = \{ \begin{array}{ll} K - S_{T} & \quad \text{if } S_{T} \leq K \\ 0 & \quad \text{if } S_{T} > K \\ \end{array} \} \end{equation*}$
or
$p_{T} = max [K - S_{T}, 0]$
If
$S_{T}$
< K, then the put is said to expire in-the-money and the option will be exercised.
Last modified 1mo ago