Put-Call Parity

Put-Call Parity for European Options.  Fix $t>0$ and let $T>t$ be a fixed future time. Denote the continuously compounded risk-free interest rate of tenor $T-t$ at time $t$ by $r_{t}(t,T)$, and let $K$ be the strike price on some asset $S$ negotiated at time $t=0$, whose price at time $t\geq0$ is denoted by $S_{t}$.  Then if $c_{t}(K,T)$ and $p_{t}(K,T)$ are the respective prices of European call and put options on $S$ with strike $K$ and expiry $T$, then we have the following result: $$c_{t}(K,T)-p_{t}(K,T)=e^{-r_{t}(t,T)(T-t)}\left(F_{t}(T)-K\right),$$ where $F_{t}(T)$ is the price of a forward contract on $S$ expiring at time $T$, calculated at time $t$. Theoretically, $$F_{t}(T)=S_{t}e^{-r_{t}(t,T)(T-t)}.$$ We are assuming throughout that the asset $S$ pays no income and has no further funding/carrying cost over the period $[t,T]$ beyond the risk-free rate $r_{t}(t,T)$.  The usual adjustments apply if there are dividends, storage costs, etc., with income benefiting the short party and funding/carrying costs benefiting the long party.

Proof.  The proof consists of a simple replication argument.  Let $V_{t}=c_{t}-p_{t}$ be the value of the portfolio at time $t$ consisting of a long position on $c$ and a short position on $p$.  At time $T$ the payoff from our portfolio is
$$V_{T}=c_{T}-p_{T}=\max(S_{T}-K,0)-\max(K-S_{T},0)=S_{T}-K.$$
Therefore, the payoff from our contract is equal to a long forward position on $S$.  If we took opposite positions, then we would have a short forward position.

Thus our portfolio replicates the payoff of a forward contract on $S$.  It follows that

If the value of a derivative is known at time $T$ with certainty, then the value at any previous time $t$ is equal to the value at time $T$ discounted back to time $t$.  In particular, since $V_{T}$ is known with certainty,
$$V_{t}=e^{-r_{t}(T-t)}(S_{T}-K)\;\;\;\;0\leq t\leq T.$$
This assertion follows from the fact that forward contracts have a certain known payoff at their time of expiration, the fact that our portfolio is equal to this payoff, and the principle of Rational Pricing.  It follows that
$$V_{t}=e^{-r_{t}(T-t)}(F_{t}(T)-K)\;\;\;\;0\leq t\leq T.$$