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Options Math

Option Payoff

Let
KK
be the exercise price and
STS_{T}
the price of the underlying stock at the maturity date.

Call Option Payoff

The payoff of a call option is:
cT={0if STKSTKifST>K}\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
cT=max[STK,0]c_{T} = max [S_{T} - K, 0]
If
STS_{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
STS_{T}
.

Put Option Payoff

The payoff of a put option is:
pT={KSTif STK0if ST>K}\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
pT=max[KST,0]p_{T} = max [K - S_{T}, 0]
If
STS_{T}
< K, then the put is said to expire in-the-money and the option will be exercised.