> [!tldr] > For a fixed strike $K$, constant risk-free rate $r$, and time-to-expiry $\tau$, the price of an European put $P_{t}$ and a call $C_{t}$ satisfy the **put-call parity**: $C_{t}-P_{t}=F_{t}=S_{t}-Ke^{-r\tau}.$where $F_{t}$ is the price of a [[Futures Contract|futures]] with the same strike and expiry. Without further explanation, any futures contract is assumed to have the same strike and expiry. ### Parity by No-Arb Portfolio Consider the portfolio of longing the call, shorting the put, shorting the underlying, and a saving of $K e^{-r\tau}$: $\underbrace{C_{t}-P_{t}}_{A} + \underbrace{Ke^{-r\tau}-S_{t}}_{B}$ At expiry, by considering the cases $S_{T}> K$ and $S_{T} < K$, we see that in both cases the portfolio has a flat position and $0$ PnL: - If $S_{T} > K$, you will exercise the call to buy the stock (closing the short position) for $K$, paid for by the savings. - If $S_{T} < K$, the put will be exercised by its holder, making you buying the stock for $K$, with the same result. Therefore $A$ and $-B$ must equal at expiry; however, since there are no changes or cashflow before the expiry (the options are European), they must equal at time $t$ too: $A_{t}=-B_{t}=S_{t}-Ke^{-r\tau},$and this is exactly the put-call-parity. ### Parity as Futures and Synthetics The position $C_{t}-P_{t}$ (longing the call, shorting the put) is called a **synthetic**. Note that at expiry, *it gives the same position and PnL as a futures*: - If $S_{T} > K$, then you will exercise the call to buy the underlying at $K$ -- same as longing a futures. - If $S_{T} < K$, then the holder of the put will exercise to sell you the underlying at $K$, again the same as longing the futures. Therefore, because there can be no change before the expiry (the options are European), the two should worth the same at present too, giving $C_{t}-P_{t}=F_{t}=S_{t}-K e^{-r\tau}.$ > [!info] No-Arb Justification for Equality at Present > Consider the portfolio of longing a synthetic and shorting a futures, and borrow or save the extra money $F_{t}-(C_{t}-P_{t})$ with the bank. > > At expiry, the portfolio has no position and a PnL of: $e^{r\tau}(F_{t}-(C_{t}-P_{t})).$By no-arb, this should equal $0$. ### Parity for American Options If $C,P$ are American instead, the European no-arb arguments fail because the counterparty can early-exercise the put. Instead, the parity becomes two inequalities: > [!theorem|*] Put-Call-Parity (American) > Assuming the underlying pays no dividends, $S-K \le C-P \le S-K e^{-r\tau}.$ For the first inequality, consider the portfolio $C-P-S+K$, where $K$ is either held as cash (no interest rate) or in the bank. - If $-P$ is exercised early, then we pay $K$ and close the short on $S$, leaving a position of $C$. - If both $C$ and $-P$ are held till expiry, exactly one of them will be exercised to give a position of $S-K$, canceling out the $-S+K$ part of the portfolio. We are left with no position if $K$ is held as cash or $K(e^{r\tau}-1)$ if left in bank. Therefore either case, we end with a positive position, so the portfolio must have positive worth, i.e. $C-P \ge S-K$. The second inequality follows from the portfolio $S-Ke^{-r\tau}-C+P$, - If $-C$ is exercised early, we are left with $P+K(1-e^{-r\tau})$, which is worth positive and can be held till expiry for a gain. - Otherwise, the whole portfolio is held till expiry, giving the same $0$ position as the European portfolio.