Why Contract Matching Comes First
Put-call parity compares portfolios that have the same value at expiration. That comparison is valid only when the call and put refer to the same underlying asset, strike, expiration, and relevant settlement terms. A pair of prices can look close enough to compare and still fail this basic test. If the contracts do not match, a numerical difference from a parity equation is not, by itself, evidence of mispricing or arbitrage.
This tutorial focuses on that contract check. The calculations use the standard European, non-dividend-paying asset case unless stated otherwise. Known dividends or other cash flows require the corresponding adjustment, but they do not remove the need to match the contracts being compared.
The Baseline Parity Relation
Assume European options on the same non-dividend-paying asset, with one unit of the underlying represented by each option. Both options expire at time \(T\), have strike \(K\), and settle on compatible terms. Assume a frictionless market and a known effective annual risk-free rate \(i\) over the relevant term, so the discount factor to time 0 is \(v^T=(1+i)^{-T}\). Let \(S_0\) be the asset price today, \(C\) the call price, and \(P\) the put price.
This equation is a statement about matched contracts, not a general rule for any call and any put. The expiration date determines when the options pay, the strike determines the cash amount in the payoff comparison, and the underlying determines the asset delivered or referenced. Changing any one of these can change the terminal cash flows.
Why Matching Makes the Equation Work
At expiration, a European call pays \(\max(S_T-K,0)\), and a European put pays \(\max(K-S_T,0)\), where \(S_T\) is the underlying price at time \(T\). For every possible value of \(S_T\), their payoff difference is \(S_T-K\):
To see this, if \(S_T\ge K\), the call pays \(S_T-K\) and the put pays zero. If \(S_T<K\), the call pays zero and subtracting the put payoff gives \(-(K-S_T)=S_T-K\). Thus a long call and short put have the same expiration payoff as owning one unit of the asset and owing \(K\) at expiration. The present value of that future amount \(K\) is \(Kv^T\), which gives the parity relation.
The identity depends on the same \(S_T\), \(K\), and \(T\) appearing in both option payoffs. If the call and put use different strikes or expirations, this payoff identity no longer describes their combined position. If they refer to different underlyings, there is no single \(S_T\) for the identity. This is why checking the contract description must precede any arithmetic.
Verify that both contracts reference the same asset or precisely defined index, including any relevant currency or reference-price convention.
Confirm that the exercise price \(K\) is identical. A nearby strike is still a different contract.
Confirm the same expiration date and time. Options expiring at different dates do not have payoffs at a common date merely because their quoted terms are otherwise similar.
Check the unit quantity, multiplier, exercise style, and how and when settlement occurs. Convert quantities or cash flows explicitly before treating portfolios as equivalent.
Use the baseline relation only if its assumptions hold. For example, known dividends require an adjusted relation; a contract mismatch is not fixed by inserting a different number into the baseline formula.
Worked Examples
Worked Example: A Put Quote at the Wrong Strike
A non-dividend-paying asset has \(S_0=\$120\). The effective annual risk-free rate is \(i=3\%\). A one-year European call with strike \(K=\$120\) costs \(\$14\). A one-year European put with strike \(K=\$125\) is quoted at \(\$10\). Can these two option quotes be tested directly as a parity pair? What price does parity imply for the put that matches the call?
Method. Check the contract terms first. The underlying and expiration match, but the strikes do not: the call has \(K=\$120\), while the quoted put has \(K=\$125\). Therefore, the quoted call and put are not a matched pair, and their prices cannot be substituted together in one parity equation.
To find the parity-implied price of the put that would match the call, use the call's strike and expiration. The one-year discount factor is
For a matched pair, \(C-P=S_0-Kv^T\), so solve for \(P\):
Substitute \(C=14\), \(S_0=120\), \(K=120\), and \(T=1\):
Rounded to the nearest cent, the matched one-year \(K=\$120\) put has a parity-implied price of \(\$10.50\) per unit. The quoted \(K=\$125\) put is not the put whose price this calculation determines. The common mistake is to use its \(\$10\) quote as if it shared the call's strike; correcting the strike match removes that invalid comparison. As a check, \(14-10.504854\ldots=3.495146\ldots\), and \(120-120/1.03=3.495146\ldots\), so the matched-pair equation balances.
Worked Example: Same Strike, Different Expirations
A non-dividend-paying asset has \(S_0=\$50\), and the effective annual risk-free rate is \(i=10\%\). A European call with strike \(K=\$55\) expires in one year and costs \(\$7\). A European put with the same strike expires in two years and costs \(\$8\). Is the put a direct parity match for the call? Find the one-year put price implied by parity for the call's expiration.
Method. The underlying and strike match, but the expiration times differ: \(T=1\) for the call and \(T=2\) for the put. The options pay at different dates, so they are not a direct parity pair. Use the one-year term when calculating the put that matches the call.
The one-year discount factor is
The parity-implied price of the one-year put is
Thus the matched one-year put price is \(\$7.00\). The quoted two-year put at \(\$8\) does not contradict this result: it is a different contract with a different expiration and cannot be inserted as \(P\) in the one-year equation. The common mistake is to treat equal strikes as sufficient matching. The check is direct: \(C-P=7-7=0\), while \(S_0-Kv^1=50-55/1.10=0\).
Worked Example: Options on Different Underlyings
A European call on asset \(U\) has strike \(K=\$72\), expires in one year, and costs \(\$9\). Asset \(U\) currently costs \(\$70\). A quoted European put costs \(\$8\), has the same strike and expiration, but is written on a different asset \(V\), currently priced at \(\$68\). Assume both assets pay no dividends and the effective annual risk-free rate is \(i=5\%\). Can these quotes be tested directly as a parity pair? What put price is implied for a put on \(U\) that matches the call?
Method. The strike, expiration, and stated rate match, but the underlying assets do not. The put on \(V\) has a payoff determined by \(V_T\); the call on \(U\) has a payoff determined by \(U_T\). They do not form the call-put payoff difference used in the parity derivation, so the \(\$8\) quote cannot be used as the matched put price.
For the put on \(U\), the one-year discount factor is
Using the spot price of \(U\), the parity-implied price is
Rounded to the nearest cent, a matched put on \(U\) is implied to cost \(\$7.57\) per unit. For a check, \(9-7.571428\ldots=1.428571\ldots\), and \(70-72/1.05=1.428571\ldots\). Comparing the call on \(U\) directly with the put on \(V\) would instead compare different risk exposures; any resulting difference is not a parity test.
Worked Example: Settlement Terms Need an Explicit Comparison
Consider two hypothetical European calls with the same reference share, strike \(K=\$75\), and expiration. One is cash-settled using the expiration share price; the other is physically settled by delivering one share for \(\$75\). At expiration, the reference share price is \(S_T=\$78\). Describe each settlement and explain what must be checked before treating the contracts as interchangeable in a parity comparison.
Method. First calculate what each contract specifies at expiration. The cash-settled call pays the positive difference between the reference price and strike:
The physically settled call gives the holder one share and requires a \(\$75\) payment. At the stated reference price, that share is worth \(\$78\); after paying \(\$75\), the net value is \(\$3\), if the share can be valued or sold at that price without additional costs.
The two outcomes can be economically comparable under those stated conditions, but their settlement instructions are not identical: one delivers cash, and the other delivers a share in exchange for cash. Before applying parity, verify the exact reference price, delivery timing, quantity, and any conversion or trading costs needed to compare the resulting positions. The common mistake is to assume that a similar-looking payoff description guarantees identical settlement. A parity comparison is justified only after the actual terminal cash flows and assets have been matched.
What a Mismatch Does—and Does Not—Show
A mismatch means that the standard parity equation cannot be applied directly to the quoted pair. It does not prove that either option is incorrectly priced, and it does not establish an arbitrage opportunity. The proper conclusion is narrower: the quotes do not supply the matched call and put prices needed for that direct test.
| Feature that differs | Why direct parity fails | What to verify |
|---|---|---|
| Strike | The call and put payoff difference uses one common \(K\); two strikes create different payoff functions. | Compare options at the same exercise price or calculate the parity value for the call's strike. |
| Expiration | The options pay at different times, so their terminal payoffs are not simultaneous. | Match expiration dates and times before using one discount factor and one parity equation. |
| Underlying | The payoffs depend on different terminal asset values. | Confirm the exact asset or index, including reference and currency conventions. |
| Settlement or contract unit | The terminal deliverables, cash flows, or quantities may differ. | Read the terms and explicitly convert to comparable amounts and settlement outcomes. |
This discipline matters especially when quotes are assembled from a screen or from several sources. Similar symbols, nearby dates, or the same displayed strike can conceal a different underlying, expiration time, or settlement convention. Record the contract features beside each price before computing a residual such as \(C-P-(S_0-Kv^T)\). That residual is meaningful as a parity check only after the contracts and model assumptions match.
Common Exam Trap
A frequent error is to begin with the parity equation and select the nearest available put quote, even though its strike or expiration differs. For example, if a call has \(K=120\) and the put quote has \(K=125\), substituting both option prices into the same equation silently assumes a shared strike that the contracts do not have. The correction is not to adjust the parity result by guesswork: identify the mismatched term, state that the quoted pair is not directly testable, and calculate an implied value only for the correctly matched contract.
A second error is to use the right strike but the wrong term in the discount factor. In \(Kv^T\), \(T\) belongs to the expiration of the matched options. A one-year call cannot be paired in the equation with a two-year put simply by using either option's expiration at convenience.
Check Your Understanding
For each question, check the contracts before applying the equation. Unless stated otherwise, assume European options, a non-dividend-paying underlying, and no transaction costs.
- A call on an asset with \(S_0=\$90\) has \(K=\$95\), \(T=1\), and price \(\$11\). The effective annual rate is \(5\%\). A one-year put at \(K=\$95\) is quoted. What price is implied for a matched put? (Answer: \(\$11.48\) per unit.)
- A call with \(K=\$60\) expires in one year and costs \(\$8\). A put with \(K=\$60\) expires in two years and costs \(\$7\). The asset has \(S_0=\$60\), and \(i=10\%\). Can the quoted options be tested directly as a parity pair? What is the parity-implied one-year put price? (Answer: No; the expirations differ. The matched one-year put is \(8-60+60/1.10=\$2.55\) per unit.)
- A call and put have the same stated strike and expiration, but the call references a share priced at \(\$40\) and the put references a different share priced at \(\$42\). Is their price difference a direct parity residual? (Answer: No; the underlying differs, so they are not a matched pair.)
- A one-year call has \(K=\$80\), \(S_0=\$76\), and price \(\$9.50\). The effective annual rate is \(5\%\). A put quote at \(K=\$80\) expires in six months. Is that quote a direct match? What one-year put price is implied? (Answer: No; the expirations differ. The matched one-year put is \(9.50-76+80/1.05=\$9.69\) per unit.)
- A cash-settled put and a physically settled put have the same stated underlying, strike, and expiration. Name two settlement details to verify before using them as interchangeable parity contracts. (Answer: Verify the precise reference or settlement price and timing, and verify the delivered quantity or cash amount and any conversion needed to compare outcomes.)