= Put-call parity
{title2=$C_0-P_0=S_0-Ke^{-rT}$}
{wiki}
<Put-call parity> fixes the price difference between a <European call option> and a <European put option> on the same underlying <stock>, with the same <strike price> $K$ and expiry $T$. Assume no dividends or other payments from the <stock> before or at expiry, a known constant continuously compounded <interest rate> $r$, and a <frictionless market>. Borrowing and lending are available at that same <interest rate>; <short selling> of the <stock>, <European call option>, <European put option>, and <zero-coupon bond> is permitted. Settlement is certain, and the buy-and-hold <portfolios> below may be held until $T$ without interim funding restrictions or forced liquidation. These assumptions exclude transaction costs, taxes, borrowing fees, and execution or settlement risks from the <arbitrage> argument.
At expiry, the two <positive parts> cancel their corners:
$$
C_T-P_T=(S_T-K)^+-(K-S_T)^+=S_T-K.
$$
Thus a <European call option> and a short <European put option> together have the same terminal <financial payoff> as one <stock> share with a debt of $K$. A particularly readable <claim replication> uses two <portfolios> with nonnegative terminal amounts: portfolio A holds one <European call option> and a risk-free <zero-coupon bond> paying $K$ at $T$; portfolio B holds one <European put option> and one <stock> share. The <zero-coupon bond> costs $Ke^{-rT}$ today, since the <discount factor> is $e^{-rT}$. There are no intermediate cash flows. For every terminal <stock> price,
$$
A_T=C_T+K=\max(S_T,K)=P_T+S_T=B_T.
$$
For the <strike price> $K=100$, even the following very different outcomes leave the two <portfolios> equal:
|| Terminal <stock> price $S_T$
|| A: <European call option> + <zero-coupon bond>
|| B: <European put option> + <stock>
| $80$
| $0+100=100$
| $20+80=100$
| $100$
| $0+100=100$
| $0+100=100$
| $120$
| $20+100=120$
| $0+120=120$
The <law of one price> forces $A_0=B_0$ in this setting: otherwise buy the cheaper <portfolio> and sell the more expensive <portfolio> using <short selling>. Their terminal obligations cancel in every scenario, while the initial price difference can grow in the <bank account>. Hence absence of <arbitrage> gives
$$
C_0+Ke^{-rT}=P_0+S_0,\qquad C_0-P_0=S_0-Ke^{-rT}.
$$
With a constant per-period <interest rate> $i$ over $N$ discrete periods, the same reasoning gives $C_0-P_0=S_0-K(1+i)^{-N}$. This is a different compounding convention; $i$ is not the continuously compounded <interest rate> $r$.
For an illustrative <stock> price $S_0=100$, <strike price> $K=100$, expiry one year away, <interest rate> $r=0.05$ per year, and given <European call option> price $C_0=10$, <put-call parity> requires
$$
Ke^{-rT}=100e^{-0.05}\approx95.12294245,\qquad P_0=10+100e^{-0.05}-100\approx5.12294245.
$$
Both <portfolios> therefore cost approximately $105.12294245$. The inputs specify no <stock> price dynamics: $C_0=10$ is supplied for illustration, not calculated from the <Black-Scholes model>. <Put-call parity> determines the <European put option> price conditional on that <European call option> price; it does not determine both prices by itself. At zero <interest rate>, the same $S_0=K=100$ and $C_0=10$ give the exact result $P_0=10$.
\Image[/put-call-parity-replication.png]
{title=Two portfolios with identical terminal cash flows and equal initial costs under put-call parity}
To compute both directions of <arbitrage>, keep all the preceding inputs except the quoted <European put option> price. If it is $6$, portfolio B costs $106$ while portfolio A costs $105.12294245$. Buy A, sell B, and place the initial surplus $6-(10+100e^{-0.05}-100)\approx0.87705755$ in the <bank account>. At expiry, A and the obligations from selling B cancel exactly, leaving $96e^{0.05}-100\approx0.92202525$ in every scenario. If the <European put option> instead costs $4$, buy B for $104$ and sell A for $105.12294245$; the initial surplus is approximately $1.12294245$, growing to $100-94e^{0.05}\approx1.18051694$. Both constructions start with zero external capital. They are model calculations under the stated <frictionless market> assumptions.
The beauty of <put-call parity> is that uncertain individual <financial payoffs> produce an exact relation between today's prices. No forecast of the terminal <stock> price and no <probability distribution> are needed. It provides a consistency check and lets <claim replication> exchange one contract for a combination of others. Dividends introduce extra cash flows, and <American put options> can be exercised early, so the displayed European equality cannot simply be transferred to those contracts. K. Foster's https://kfoster.ccny.cuny.edu/classes/spring2010/eco275/lecturenotes5.html[CCNY lecture notes on option values (2010), Put-Call Parity section]{ref} explain the matching <portfolios>; Jun Pan's https://www.mit.edu/~junpan/Jun_Pan_Teaching_Notes.pdf[MIT Financial Markets teaching notes, Arbitrage Pricing and Dynamic Replication section (PDF p. 130)]{ref} discuss <put-call parity> and its dividend adjustment.
An original modern research source is Hans R. Stoll's https://doi.org/10.1111/j.1540-6261.1969.tb01694.x[The Relationship Between Put and Call Option Prices]{ref}, \i[The Journal of Finance] 24(5), 801-824 (December 1969). Stoll studied the relation between contract prices and tested it against market data; this was a relation between prices rather than a formula fixing each price individually. Robert C. Merton's https://doi.org/10.1111/j.1540-6261.1973.tb01357.x[1973 comment]{ref}, 28(1), 183-184, highlighted the importance of excluding premature exercise for an exact <arbitrage> equality. This distinction explains why the exercise assumption belongs in the statement of <put-call parity>.
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