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.