Options Math

Option Payoff
Let KKK
 be the exercise price and STS_{T}ST​
 the price of the underlying stock at the maturity date.
Call Option Payoff
The payoff of a call option is: 
cT={0if ST≤KST−KifST>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*}cT​={0ST​−K​if ST​≤KifST​>K​}​
or
cT=max[ST−K,0]c_{T} = max [S_{T} - K, 0] cT​=max[ST​−K,0]
If STS_{T}ST​
 > 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}ST​
.
Put Option Payoff
The payoff of a put option is: 
pT={K−STif ST≤K0if 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*}
pT​={K−ST​0​if ST​≤Kif ST​>K​}​
or
pT=max[K−ST,0]p_{T} = max [K - S_{T}, 0] pT​=max[K−ST​,0]
If STS_{T}ST​
 < K, then the put is said to expire in-the-money and the option will be exercised. 
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Last modified 25d ago