= European put option
{title2=$P_T=(K-S_T)^+$}
A <European put option> gives its holder the right, without an obligation, to sell one unit of an underlying <financial asset> at the <strike price> $K$ only on the expiry date $T$. For one <stock> share, its terminal <financial payoff> is the <positive part>
$$
P_T=(K-S_T)^+=\begin{cases}K-S_T,&S_T<K,\\0,&S_T\geq K.\end{cases}
$$
For <strike price> $K=100$, terminal <stock> prices $80$, $100$, and $120$ give <financial payoffs> $20$, $0$, and $0$, respectively. At $S_T=80$, selling for $100$ rather than the market price $80$ supplies the difference $20$; at $S_T=120$, the holder can let the right expire. Equality at $S_T=100$ gives zero net exercise value. The <European call option> example plots both <financial payoffs> on the same axes.
Combining a <European put option> with the underlying <stock> share produces the terminal amount $S_T+(K-S_T)^+=\max(S_T,K)$. Its floor at the <strike price> is why a <European put option> can describe protection against a fall in terminal <stock> value. The protection has an initial cost and lasts to the specified expiry; the terminal floor alone says nothing about the <financial return>. <Put-call parity> replicates this same amount with a <European call option> and a <zero-coupon bond>.
Using the <put-call parity> example's price $P_0=10+100e^{-0.05}-100\approx5.12294245$, the cost financed for one year at <interest rate> $r=0.05$ grows to $P_0e^{0.05}=100-90e^{0.05}\approx5.38560133$. The terminal net amount is therefore approximately $14.61439867$ when $S_T=80$, and $-5.38560133$ when $S_T=100$ or $120$. These computations use the unrounded price and distinguish the <financial payoff> from the financed gain. Unlike an <American put option>, a <European put option> supplies no right to exercise before $T$.
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