Put-Call Parity & Payoffs

The Most Important Equation in Options

Put-call parity is the foundational no-arbitrage relationship linking European calls, puts, the underlying asset, and a risk-free bond. If you remember only one formula from options theory, make it this one:

$C - P = S - Ke^{-rT}$

where $C$ is the European call price, $P$ is the European put price, $S$ is the current stock price, $K$ is the strike price, $r$ is the continuously compounded risk-free rate, and $T$ is the time to expiration in years.

Derivation from Replicating Portfolios

The derivation proceeds by constructing two portfolios that produce identical payoffs at expiration and invoking the law of one price.

Portfolio A: Buy one European call and sell one European put, both with strike $K$ and expiry $T$. The payoff at expiration is:

$\max(S_T - K, 0) - \max(K - S_T, 0) = S_T - K$

regardless of whether $S_T$ finishes above or below $K$. You can verify this by checking both cases: if $S_T > K$, the call pays $S_T - K$ and the put expires worthless; if $S_T \leq K$, the call expires worthless and you owe $K - S_T$ on the short put.

Portfolio B: Buy one share of stock and borrow $Ke^{-rT}$ at the risk-free rate. At expiration the loan grows to $K$, so the net payoff is $S_T - K$.

Since both portfolios pay $S_T - K$ at time $T$ under all scenarios, they must have the same price today:

$C - P = S - Ke^{-rT}$

Any deviation would create a riskless arbitrage opportunity.

Worked Numerical Example

Suppose a stock trades at $S = 100$. A European call with strike $K = 105$ expiring in 6 months ($T = 0.5$) costs $C = 8.50$. The risk-free rate is $r = 4\%$. What should the put cost?

Step 1 — Compute the present value of the strike:

$Ke^{-rT} = 105 \cdot e^{-0.04 \times 0.5} = 105 \cdot e^{-0.02} \approx 105 \times 0.9802 = 102.92$

Step 2 — Rearrange parity for $P$:

$P = C - S + Ke^{-rT} = 8.50 - 100 + 102.92 = 11.42$

The put should trade at approximately $11.42.

Arbitrage from Violations

Suppose the put in the example above were quoted at $P = 10.00$ instead of $11.42$. The put is too cheap relative to parity. An arbitrageur would:

1. Buy the underpriced put for $10.00
2. Buy the stock for $100.00
3. Sell the call for $8.50
4. Borrow $Ke^{-rT}$ = $102.92 at the risk-free rate

Net initial cash flow: $-10.00 - 100.00 + 8.50 + 102.92 = +1.42$. At expiration the combined position nets to zero regardless of $S_T$, so the $1.42 is pure profit. In practice, transaction costs narrow this window, but the logic is airtight.

Synthetic Positions

Parity lets you replicate any single instrument from the other three:

• Synthetic long call: Buy stock + buy put + borrow $Ke^{-rT}$
• Synthetic long put: Buy call + short stock + lend $Ke^{-rT}$
• Synthetic long stock: Buy call + sell put + lend $Ke^{-rT}$

These synthetic constructions are central to options market-making. When a particular contract is illiquid, a trader can manufacture it from liquid components.

Adjustments for Dividends

When the underlying pays a known dividend with present value $D$, parity becomes:

$C - P = S - D - Ke^{-rT}$

Forgetting to adjust for dividends is one of the most common mistakes in interview settings.

Interview Tip: When asked to price a call or put, always check whether put-call parity gives you a shortcut. Interviewers frequently provide three of the four quantities and ask for the fourth. State the parity equation, plug in, and solve — it demonstrates you understand the no-arbitrage framework before reaching for Black-Scholes.

Key Takeaways

• Put-call parity is model-free: it relies only on no-arbitrage, not on any distributional assumption about the stock.
• It holds exactly for European options. American options satisfy an inequality instead.
• Violations imply arbitrage opportunities — a fact that keeps markets efficient.