# 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.&#x20;


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