Put-call parity: why the strike must be discounted
A one-year European options question gives a call price of 8, a stock price of 102, a strike of 100, and a risk-free rate of 5%. A candidate who treats the strike as cash owed today calculates the put at 6.00, because the future payment has not been discounted. The correct value is 1.24. The gap is timing, not option mathematics.
One payoff, two portfolios
Put-call parity compares two portfolios that produce the same cash flow at expiry. On one side sits the stock with a European put, the familiar protective put. On the other sits a European call with a risk-free bond whose value at maturity equals the strike. The options have the same strike and the same expiry, so neither side gains an advantage from different contract terms.
S₀ + p₀ = c₀ + X ÷ (1 + r)^T
The result is easier to see at expiry than it is to memorize today. If the stock finishes above the strike, the stock and put combination is worth the stock, while the call and bond combination reaches the same value through exercise of the call. If the stock finishes below the strike, the put protects the first portfolio at the strike and the bond does the same job for the second. Equal terminal cash flows require equal prices today.
The strike belongs to maturity
The strike, X, is exchanged when the options expire. The matching bond therefore needs to grow to X by that date; it does not need to cost X at the valuation date. Its current value is X divided by the risk-free growth factor for T, which is why the formula contains X ÷ (1 + r)^T rather than the undiscounted strike.
Using X directly overfunds the bond side of the comparison. There is a separate algebra risk as well: after starting from the parity relationship, moving the stock, call, or put across the equals sign changes its sign. A remembered rearrangement can conceal that change, whereas the original two-portfolio statement keeps each asset attached to the side where its payoff belongs.
Working the 8, 100, 102 case
Here the risk-free rate is 5% for one year, so the 100 due at expiry has a current value of 95.24. With the call worth 8 and the stock worth 102, the put is the amount required to restore equality between the portfolios.
p₀ = 8 + 100 ÷ 1.05 − 102 = 8 + 95.24 − 102 = 1.24
The 6.00 distractor comes from substituting 100 for 95.24. Every other input is handled correctly, which makes the wrong choice feel credible and explains why this is primarily a timing trap.
Quick reference
| Symbol | Meaning | Timing |
|---|---|---|
| S₀ | Price of the underlying stock | Today |
| p₀ | Price of the European put | Today |
| c₀ | Price of the European call | Today |
| X | Common exercise price | Paid at expiry |
| r | Risk-free rate | Applies through the option term |
| T | Time to expiry | Sets the discounting period |
The equality as written has a narrower scope than its compact formula suggests. It applies to European options with the same strike and maturity on an asset that pays no income during the period, or after that income has been adjusted for. An unadjusted dividend changes the cash flows from holding the stock, while an American option introduces possible early exercise. The missing adjustment can be buried in a short phrase about dividends, and the exercise style may appear only in the contract description. A question can test either condition in prose without asking for any calculation at all.
The same habit rescues the cost-of-carry signs. Ask what each cash flow does and when it happens, and the sign follows.
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