Introduction
A European call and a European put on the same asset are linked even though their expiration payoffs differ. When the options have the same strike and expiration, and the asset pays no dividends before expiration, their current prices satisfy a precise relationship with the asset price and the present value of the strike. That relationship is European put-call parity.
The essential point is that the strike in the equation is not generally its cash amount today. It is the present value of the amount due at expiration. Keeping the contract terms and timing straight is as important as knowing the equation.
Contracts, Prices, and Assumptions
The standard setup assumes that the options are European, so exercise is permitted only at expiration; both options refer to the same underlying asset, strike, and expiration; the asset pays no dividends during the option term; and the market is frictionless, with no taxes or transaction costs. It also assumes that the risk-free borrowing and lending rate used to value the payment at expiration is known and consistent with the option term.
For example, if expiration is exactly \(n\) years from now and the effective annual risk-free rate is \(i\), then \(v=1/(1+i)\) and $$ PV(K)=K v^n=\frac{K}{(1+i)^n}. $$ This discounting formula uses an effective annual rate and an integer number of years. For other maturities or rate structures, the appropriate risk-free discount factor must be used; do not assume that the integer-year formula applies unchanged.
The relation does not apply as written if the options have different strikes or expiration dates, if one is American, or if the asset pays dividends during the term. Those differences change the assumptions behind the equality. This tutorial focuses on the stated non-dividend-paying European case.
Why the Equality Holds
At expiration, write the asset price as \(S_T\). A call’s payoff is \(\max(S_T-K,0)\), and a put’s payoff is \(\max(K-S_T,0)\). The following terminal-payoff identity holds for every possible value of \(S_T\): $$ \max(S_T-K,0)+K=\max(K-S_T,0)+S_T. $$
To verify it, consider the two regions separated by the strike. If \(S_T\geq K\), the left side is \((S_T-K)+K=S_T\), while the right side is \(0+S_T=S_T\). If \(S_T<K\), the left side is \(0+K=K\), while the right side is \((K-S_T)+S_T=K\). The two sides therefore have identical values at expiration in either region, including at \(S_T=K\).
The strike amount \(K\) in that identity is paid at expiration, not today. Its current value is \(PV(K)\). Since the two sides produce the same expiration value in every outcome, no-arbitrage requires their current values to agree. The current value of the call payoff together with the discounted strike is \(C+PV(K)\); the current value of the put payoff together with the asset is \(P+S\). Equating them gives European put-call parity.
This argument explains why the equation compares current prices rather than just comparing option payoffs. The strike contribution has to be brought back from its payment date before it is added to a current option price.
Interpreting the Two Sides
In \(C+PV(K)=P+S\), each side combines current values of positions whose expiration values match. The left side contains the call and the present value of the strike payment. The right side contains the put and the asset. The equation says these two combinations must have the same value now under the stated assumptions.
Parity is not a claim that a call price equals a put price. It also does not say that the call and put have equal payoffs at expiration. Instead, it relates the prices of the two different options after accounting for both the asset price and the discounted strike. A call can be more expensive than its matching put while parity still holds.
A useful calculation order is:
Confirm that the call and put refer to the same asset, strike, and expiration, and that both are European.
Confirm that the asset pays no dividends before expiration and identify the applicable risk-free rate convention.
Use a discount factor consistent with the rate and time stated in the problem.
Calculate \(C+PV(K)\) and \(P+S\) separately, then check whether they agree.
Worked Examples
Worked Example: Checking One-Year Quotes
A non-dividend-paying asset is priced at \(S=\$104\). A European call and put have strike \(K=\$110\) and expire in one year. The call costs \(\$13\), the put costs \(\$9\), and the effective annual risk-free rate is \(10\%\). Determine whether the quotes satisfy put-call parity.
Method. Check the matching terms, discount the strike for one year, and compare both sides of the parity equation.
Both options are European, have the same strike and expiration, and refer to the same asset. The asset pays no dividends, so the stated parity relation applies. With \(i=0.10\) and \(n=1\), $$ PV(K)=\frac{110}{1.10}=\$100. $$
The left side is $$ C+PV(K)=13+100=\$113. $$
The right side is $$ P+S=9+104=\$113. $$
The two sides agree, so the quotes satisfy parity. As a check, the discounted strike of \(\$100\) is less than the \(\$110\) due at expiration, as it should be at a positive interest rate. The common mistake this calculation avoids is adding the full \(\$110\) strike to the current call price instead of discounting it.
Worked Example: Keeping Expiration Payoffs Separate From Current Prices
A non-dividend-paying asset currently costs \(\$95\). A European call and put both have strike \(K=\$108\), expire in one year, and have prices \(C=\$4.20\) and \(P=\$9.20\). The effective annual risk-free rate is \(8\%\). First check parity, then verify the matching expiration values of the two sides for \(S_T=\$80\), \(\$108\), and \(\$140\).
Method. Discount the strike to compare current values; separately use the option payoff formulas to check the expiration identity.
The contract terms and assumptions match the European, non-dividend-paying setup. The one-year present value of the strike is $$ PV(K)=\frac{108}{1.08}=\$100. $$
The current-value comparison is $$ C+PV(K)=4.20+100=\$104.20, $$ and $$ P+S=9.20+95=\$104.20. $$
Thus the quotes satisfy parity. At expiration, the call-plus-strike value is \(\max(S_T-108,0)+108\), and the put-plus-asset value is \(\max(108-S_T,0)+S_T\). Evaluate each pair:
If \(S_T=\$80\), the call-plus-strike value is \(0+108=\$108\), and the put-plus-asset value is \(28+80=\$108\).
If \(S_T=\$108\), the call-plus-strike value is \(0+108=\$108\), and the put-plus-asset value is \(0+108=\$108\).
If \(S_T=\$140\), the call-plus-strike value is \(32+108=\$140\), and the put-plus-asset value is \(0+140=\$140\).
Each expiration comparison agrees, as does the current-price comparison after discounting the strike. The trap is to treat the expiration amount \(\$108\) as though it were already its present value; at an \(8\%\) effective annual rate, its present value is \(\$100\).
Worked Example: Matching the Maturity Before Comparing Quotes
A non-dividend-paying asset is priced at \(\$104\). A European call with strike \(\$112.36\) expires in two years and costs \(\$11.25\). A European put with the same strike and expiration costs \(\$7.25\). The effective annual risk-free rate is \(6\%\). A separate one-year put on the asset, also with strike \(\$112.36\), is quoted at \(\$5.60\). Check parity for the two-year options and explain whether the one-year quote belongs in the check.
Method. Identify the matching option pair, discount the strike for the full two-year term, and compare the corresponding sides. Do not substitute a quote with a different expiration.
The two-year call and put match in underlying, strike, and expiration, and the asset pays no dividends. With \(i=0.06\) and \(n=2\), $$ (1.06)^2=1.1236, \qquad PV(K)=\frac{112.36}{1.1236}=\$100. $$
For the two-year options, $$ C+PV(K)=11.25+100=\$111.25, $$ and $$ P+S=7.25+104=\$111.25. $$
The two-year quotes satisfy parity. The one-year put at \(\$5.60\) cannot be used in this comparison: despite having the same asset and strike, it expires at a different time from the call. Parity requires the call and put to have the same expiration. The common mistake avoided here is selecting an option based only on its strike and ignoring its maturity.
Common Mistakes and Exam Traps
The most direct error is to use the undiscounted strike. In the first worked example, the incorrect calculation would be $$ C+K=13+110=\$123, \qquad P+S=9+104=\$113. $$
This apparent mismatch does not show that parity fails. It shows that the calculation used \(K\), payable at expiration, where the equation requires \(PV(K)=\$100\), payable in current-value terms. The corrected comparison is \(13+100=9+104=\$113\).
- Mixing dates. Option premiums and the asset price are current values; \(K\) is due at expiration. Discount \(K\) before comparing current values.
- Mixing contracts. The call and put must have the same strike and expiration. Matching only one of those terms is insufficient.
- Ignoring the product type. The equality is for European options, not options that may be exercised early.
- Ignoring dividends. The stated equation assumes no dividends during the option term. If that condition is not met, do not apply this version unchanged.
- Equating the option prices alone. Parity relates \(C+PV(K)\) to \(P+S\), not \(C\) directly to \(P\).
Why Put-Call Parity Matters
Put-call parity is a consistency check across prices of options, the underlying asset, and a risk-free payment. It helps organize market information: the call and put cannot be considered independently once their strike, expiration, and underlying are fixed. If a problem gives several option quotes, first determine which contracts form a valid pair and then check that the strike is valued at the correct date.
The relation is aligned with the SOA Exam FM (Financial Mathematics) syllabus. For exam calculations, write down the assumptions, compute the discounting explicitly, and show both sides of the equality. That process makes errors in timing or contract matching visible before they affect the conclusion.
Check Your Understanding
For each question, identify the relevant assumptions and compare the required quantities.
- A non-dividend-paying asset has current price \(\$101\). Matching European options have strike \(\$105\), expire in one year, and cost \(\$8\) for the call and \(\$7\) for the put. The effective annual rate is \(5\%\). Do the quotes satisfy parity? (Answer: Yes; both sides equal \(\$108\).)
- A non-dividend-paying asset costs \(\$97\). A European call and put have strike \(\$108.16\), expire in two years, and cost \(\$12.40\) and \(\$15.40\), respectively. The effective annual rate is \(4\%\). Check parity. (Answer: Yes; \(PV(K)=\$100\), and both sides equal \(\$112.40\).)
- A non-dividend-paying asset costs \(\$103\). Matching European options have strike \(\$115.7625\), expire in three years, and cost \(\$14\) for the call and \(\$11\) for the put. The effective annual rate is \(5\%\). Check parity. (Answer: Yes; \(PV(K)=\$100\), and both sides equal \(\$114\).)
- A European call on a non-dividend-paying asset has strike \(\$90\) and expires in one year. A European put on the same asset has strike \(\$90\) but expires in two years. Can these two options be used as the matching pair in the stated parity equation? (Answer: No; their expiration dates differ.)
- A non-dividend-paying asset costs \(\$86\). Matching European options have strike \(\$100\), expire in two years, and cost \(\$5\) for the call and \(\$19\) for the put. The effective annual risk-free rate is \(0\%\). Check parity and state what happens to the present value of the strike. (Answer: Yes; \(PV(K)=\$100\), and both sides equal \(\$105\).)