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 discrete dividend with present value $D$ before expiry, parity becomes:

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

For underlyings with a continuous dividend yield $q$ (the standard model for equity indices, ETFs, and FX), the dividend stream is paid as a constant rate. Holding the stock from $t = 0$ to $T$ accumulates a yield of $q T$ (in log terms), so the "effective" stock you can spend at expiry is $S e^{-qT}$. Parity becomes:

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

The intuition is the same as the discrete case (the dividend stream lowers the cost of the synthetic forward), but with the dividend yield $q$ playing the role of a continuous coupon. Foreign exchange is the canonical use case: a USD/EUR option treats the foreign rate $r_{ ext{EUR}}$ as the "dividend yield" $q$.

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.

American Options: Parity Becomes an Inequality

For American options on a non-dividend-paying stock, the strict parity identity is replaced by a two-sided inequality:

$S - K \;\le\; C_A - P_A \;\le\; S - Ke^{-rT}.$

Upper bound ($C_A - P_A \le S - Ke^{-rT}$). On a non-dividend stock, $C_A = C_E$ (never optimal to exercise an American call early — see the American Option Pricing lesson), while $P_A \ge P_E$ because early exercise of the put adds value. Subtracting gives $C_A - P_A \le C_E - P_E = S - Ke^{-rT}$.

Lower bound ($C_A - P_A \ge S - K$). Suppose for contradiction $C_A - P_A < S - K$. Set up the portfolio: buy the call, short the put, short the stock, lend $K$ in cash at rate $r$. Net initial inflow $= S + P_A - C_A - K > 0$ by hypothesis. If the put is exercised early at some $t < T$, the put holder forces us to buy stock at $K$; we pay $K$ from the lent cash (which has grown to $Ke^{rt}$, so we still retain $K(e^{rt} - 1) > 0$ in carry profit), and we use the stock to close our short. The call we still own has non-negative value. If the put is held to expiry, the call-plus-short-stock-plus-cash position pays a non-negative amount in every state. So we lock in a riskless profit — contradiction.

The gap between the two bounds, $(S - Ke^{-rT}) - (S - K) = K(1 - e^{-rT})$, is precisely the maximum early-exercise premium of the American put (the interest you could earn by exercising and banking $K$ now). At $r = 0$ the bounds collapse to a single equality $C_A - P_A = S - K$, reflecting that early exercise carries no premium without interest.

With a continuous dividend yield $q$, the upper bound's $S$ becomes $Se^{-qT}$ (call early exercise may now be optimal, so the bound is no longer tight), but the lower bound's structure is unchanged: $Se^{-qT} - K \le C_A - P_A \le Se^{-qT} - Ke^{-rT}$.

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 a two-sided inequality $S - K \le C_A - P_A \le S - Ke^{-rT}$, with the gap equal to the maximum early-exercise premium $K(1 - e^{-rT})$.
• Violations imply arbitrage opportunities — a fact that keeps markets efficient.