= European call option
{c}
{title2=$C_T=(S_T-K)^+$}
{wiki}
A <European call option> gives its holder the right, without an obligation, to buy one unit of an underlying <financial asset> at the <strike price> $K$ on the expiry date $T$. European describes exercise timing, not geography: exercise is allowed only at $T$. Here the underlying is one <stock> share, $S_T$ is its price at expiry, and the <financial payoff> is expressed as a cash amount. Physical exercise delivers a share against payment of $K$; immediate resale gives the same net amount under the assumed <frictionless market>.
The holder exercises only when buying at the <strike price> is advantageous. Writing $x^+=\max(x,0)$ for the <positive part>, the terminal <financial payoff> is
$$
C_T=(S_T-K)^+=\begin{cases}0,&S_T\leq K,\\S_T-K,&S_T>K.\end{cases}
$$
The <European call option> is a reusable building block for <claim replication>: its <financial payoff> is zero below the <strike price> and rises by one unit for each unit of <stock> price above it. The corner at $K$ lets combinations of <European call options> construct other payoff shapes, as in <static replication on a finite terminal support>.
For <strike price> $K=100$, the following are deterministic calculations conditional on each displayed terminal <stock> price. They assign no <probabilities> to the three scenarios. The <European put option> has the opposite exercise right and the <financial payoff> $(K-S_T)^+$.
|| Terminal <stock> price $S_T$
|| <European call option> $C_T$
|| <European put option> $P_T$
|| Difference $C_T-P_T$
| $80$
| $0$
| $20$
| $-20$
| $100$
| $0$
| $0$
| $0$
| $120$
| $20$
| $0$
| $20$
\Image[/european-option-payoffs.png]
{title=European call and put terminal payoffs at strike 100, with their linear difference}
A <financial payoff> excludes the initial contract price. If the illustrative <European call option> costs $C_0=10$, its <financial payoff> at $S_T=120$ is $20$, and subtracting its initial cost gives $10$. If that cost is instead borrowed for one year at the constant continuously compounded <interest rate> $r=0.05$, the terminal net amount is $20-10e^{0.05}\approx9.48728904$. At $S_T=80$ or $100$, the corresponding net amount is $-10e^{0.05}\approx-10.51271096$. Keeping all cash amounts at the same date avoids confusing a <financial payoff> with a financed gain.
Robert Shiller's https://oyc.yale.edu/economics/econ-252-11/lecture-17[Options Markets lecture, Yale ECON 252 (2011)] introduces these exercise rights and payoff shapes.
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