Put-Call Parity

Put-call parity is the sturdiest relationship in options, it holds by pure arbitrage, with no model and no distributional assumption about the underlying. It says a call and a put of the same strike and expiry are tied together through the stock and a bond, so knowing three of the four prices pins the fourth. Because it is model-free, it is the sanity check every options desk runs and the first tool for spotting mispricings and building synthetics.

The statement

For European options on a non-dividend-paying stock, with strike $K$, expiry $T$, and continuously compounded rate $r$:

$\boxed{\,C - P = S_0 - K e^{-rT}.\,}$

The right side is exactly the value of a forward struck at $K$: $C - P$ behaves like being long the stock and short a bond worth $Ke^{-rT}$.

Derivation by replication

Consider two portfolios held to expiry $T$:

• Portfolio A: one call plus cash $Ke^{-rT}$ invested at the risk-free rate (growing to $K$ at $T$).
• Portfolio B: one put plus one share of stock.

Tabulate the payoff at $T$ in the two states:

	$S_T \le K$	$S_T > K$
A: call + $K$ cash	$0 + K = K$	$(S_T - K) + K = S_T$
B: put + stock	$(K - S_T) + S_T = K$	$0 + S_T = S_T$

Both portfolios pay $\max(S_T, K)$ in every state, at every possible terminal price. Two portfolios with identical payoffs in all states must have identical value today, or a riskless arbitrage exists. Therefore

$C + Ke^{-rT} = P + S_0 \;\Longrightarrow\; C - P = S_0 - Ke^{-rT}.$

The only inputs were the payoff definitions and the ability to discount cash at $r$, no volatility, no lognormality, no Black-Scholes.

Why "model-free" matters

The Black-Scholes formula assumes geometric Brownian motion; put-call parity assumes essentially nothing. This has a sharp consequence: whatever the true distribution of $S_T$, whatever the volatility smile looks like, the call and put at a given strike must satisfy this identity. It follows that a European call and put at the same strike must carry the same implied volatility, if you plug both market prices into Black-Scholes and back out $\sigma$, you get the same number, because parity holds in both the market and the model (Black-Scholes itself satisfies parity exactly). A quoted call-IV and put-IV that disagree at the same strike signal a data error or a stale price, not a real trading opportunity.

Dividends and the general form

If the stock pays a continuous dividend yield $q$, holding the stock earns dividends that the option does not, so replace $S_0$ by $S_0 e^{-qT}$:

$C - P = S_0 e^{-qT} - K e^{-rT}.$

For discrete dividends with present value $D$, the form is $C - P = (S_0 - D) - Ke^{-rT}$. And in terms of the forward $F_0 = S_0 e^{(r-q)T}$, parity takes its cleanest shape:

$C - P = e^{-rT}(F_0 - K),$

which shows directly that call and put values are equal precisely at the forward-at-the-money strike $K = F_0$, the reason desks quote "ATM" relative to the forward, not spot.

Synthetic positions and arbitrage

Rearranging parity manufactures any leg from the others:

• Synthetic stock: $S_0 = C - P + Ke^{-rT}$, long call, short put, plus a bond replicates the share (a combo or conversion).
• Synthetic call: $C = P + S_0 - Ke^{-rT}$.
• Box spread: combining conversions at two strikes creates a pure interest-rate instrument (a well-known way traders, and once, famously, a Reddit trader, get burned when "riskless" boxes carry early-exercise or funding risk).

If the market violates parity, say $C - P > S_0 - Ke^{-rT}$, you sell the call, buy the put, buy the stock, and borrow $Ke^{-rT}$: a zero-cost package today that pays a certain profit at expiry. Market makers enforce parity to within transaction costs precisely by standing ready to do this.

Worked example

$S_0 = 100$, $K = 100$, $r = 5\%$, $T = 1$, and the market call trades at $C = 10$. Parity gives

$P = C - S_0 + Ke^{-rT} = 10 - 100 + 100\,e^{-0.05} = 10 - 100 + 95.12 = 5.12.$

If the put actually traded at $6.00$, it is $0.88$ rich: buy the call, sell the put, short the stock, lend $Ke^{-rT}$, locking $0.88$ per unit with no market risk (ignoring frictions and borrow costs).

What breaks in practice

• American options. Parity is an equality only for European options. American options can be exercised early, so the clean identity becomes a pair of inequalities; early-exercise premium on the put (and on the call if there are dividends) breaks equality.
• Hard-to-borrow / dividend uncertainty. The synthetic-stock relation assumes you can short the stock at rate $r$. Borrow costs, locate fees, and uncertain dividends shift the parity line, apparent violations are usually just an unmodeled borrow rate. Traders back out the implied borrow/dividend from parity rather than assuming a violation.
• Discrete vs continuous rates, bid-ask. Real quotes have spreads; parity holds to within the cost of executing all four legs, not to the penny.

In interviews

The classic question is "derive put-call parity", set up the two portfolios, show both pay $\max(S_T,K)$, invoke no-arbitrage. Emphasize that it is model-free: no volatility appears, which is why it holds regardless of the smile and why call-IV must equal put-IV at a strike. Common follow-ups: "does it hold for American options?" (no, only inequalities, because of early exercise), "how do dividends change it?" (subtract PV of dividends or use $S e^{-qT}$), and "if put-call parity is violated, what's the trade?" (the conversion/reversal arbitrage). A subtle one: "at which strike are the call and put equally priced?", the forward, $K = F_0 = S_0 e^{(r-q)T}$, not spot.