Put-Call Parity

In one sentence: Put-call parity says that a call and a put on the same stock, with the same strike and expiry, are priced so that their gap equals the stock price minus the discounted strike.

$C - P = S_0 - K e^{-r T}$

Read it out loud: “C minus P equals S 0 minus K times e to the power minus r T. C is the call price, P is the put price, S 0 is the stock price today, K is the strike price, r is the risk-free interest rate, and T is the time to expiry in years.”

Before you start

The ideas and signs this page uses, explained from zero. For powers of ten, Greek letters and other notation in general, see Math basics.

Call option: A call option is a contract that gives its owner the right, but not the duty, to buy a stock at a fixed price, the strike price K, on the expiry date. The owner pays a price today for this right, and that price is the call price C.
Put option: A put option gives its owner the right, but not the duty, to sell a stock at the strike price K on the expiry date. A put gains value when the stock falls below the strike. The put price, written P, is what the right costs today.
European option: A European option can be used only on its expiry date. The rule on this page holds for European options. American options can be used before expiry, so they need a different treatment.
Present value and $e^{-rT}$: A payment due in the future is worth less today. The factor $e^{-rT}$ brings a payment due in T years back to today at a continuous rate r. For example, a 100 dollar strike due in one year at 5% a year is worth $100 \times e^{-0.05} = 95.12$ dollars today. The letter e is Euler’s number, about 2.718, and a negative power means one divided by the positive power.
Dividend: A dividend is a cash payment that a company makes to its shareholders. The rule on this page assumes the stock pays no dividend, because a dividend changes what holding the share is worth and the exact rule then changes.

What each symbol means

Symbol	Meaning
$C$	The call price, the value today of the option to buy the stock at the strike price K on the expiry date, in dollars per share.
$P$	The put price, the value today of the option to sell the stock at the strike price K on the expiry date, in dollars per share.
$S_0$	The stock price today, in dollars per share. The 0 is a label meaning ‘at the start’.
$K$	The strike price, the fixed price written into both the call and the put, in dollars per share. Both options must share this strike.
$e$	Euler’s number, about 2.718. In $e^{-rT}$ it discounts the strike, which is paid at expiry, back to today.
$r$	The risk-free interest rate per year, as a decimal. It is the return on a very safe asset. A value of 0.05 means 5% a year.
$T$	The time to expiry in years, the time left until both options end. T = 1 means one year.

The formula piece by piece

$K e^{-r T}$: The present value of the strike. The strike is paid at expiry, so it is brought back to today by multiplying by $e^{-rT}$ , a number smaller than one.
$C - P$: The call price minus the put price. The call and the put are opposite sides of the same stock, and the rule fixes the gap between their values.
$S_0 - K e^{-r T}$: The stock price minus the present value of the strike. This is the value of holding one share while owing the strike at expiry.
$C - P = S_0 - K e^{-r T}$: The call price minus the put price equals the stock price minus the present value of the strike, for European options on a stock that pays no dividend.

What the formula is doing, and why

Put-call parity links the price of a call and the price of a put on the same stock, with the same strike and the same expiry date. If you know the call price, the stock price, the strike and the interest rate, the put price is fixed. The link does not come from a forecast or a model of how the stock moves. It comes from a fact about how the two options must relate to each other.

Two portfolios that match at expiry. Portfolio A holds one call plus cash equal to the present value of the strike, $K e^{-rT}$ . That cash grows to exactly $K$ by expiry. Portfolio B holds one put plus one share of stock. At expiry, portfolio A is worth the larger of the stock price and $K$ . Portfolio B is also worth the larger of the stock price and $K$ . Two portfolios that always end up worth the same must cost the same today.

Why the rule must hold. Suppose the two portfolios cost different amounts today. A trader could buy the cheaper one, sell the dearer one, and lock in a profit with no risk, because the two always end up equal. Such free profits are quickly competed away, so prices adjust until the equality holds. Finance calls this idea no-arbitrage.

Why the strike is discounted. The strike is paid at expiry, so only its value today counts, and that value is $K e^{-rT}$ . The rule says that the call minus the put equals the stock price minus the present value of the strike. The stock is worth its price today, and the strike is an amount to be paid later, so it is brought back to today before the subtraction.

Where the rule applies. It holds for European options and for a stock that pays no dividend. American options can be used before expiry, and dividends change the value of the stock, so for those cases the exact equality changes. Always check the assumptions before you use the rule on real prices.

An everyday picture

Think of two sealed boxes that must hold the same amount of money on a fixed date. Box one holds a ticket that gives the right to buy at a set price, plus a set amount of cash. Box two holds a ticket that gives the right to sell at that same price, plus one share. Whatever the market does, both boxes end up worth the same on that date, so the two boxes must cost the same today. That equality of the two boxes is the whole rule.

Worked example: Finding the put price from a 10.45 dollar call

The call price of 10.45 dollars comes from the Black-Scholes example: a made-up stock at 100 dollars, a strike of 100 dollars, a risk-free rate of 5% a year and 1 year to expiry. We want the price of the put with the same strike and expiry. In the formula’s symbols, $C = 10.45$ , $S_0 = 100$ , $K = 100$ , $r = 0.05$ and $T = 1$ .

Find the present value of the strike. The strike of 100 dollars is paid at expiry, one year away, so we bring it back to today by multiplying by $e^{-0.05 \times 1}$ : $K e^{-rT} = 100 \times e^{-0.05} = 100 \times 0.9512 = 95.12$ dollars.
Rearrange the formula to find the put. Start from $C - P = S_0 - K e^{-rT}$ . Add $P$ to both sides, then take $S_0$ away and add $K e^{-rT}$ to both sides. This leaves $P = C - S_0 + K e^{-rT}$ , a formula where P stands alone on the left.
Put in the numbers. The call is 10.45 dollars, the stock is 100 dollars, and the present value of the strike is 95.12 dollars. So $P = 10.45 - 100 + 95.12 = 5.57$ dollars, which is the put price.
Check the result with the Black-Scholes put formula. The put value from that model is $K e^{-rT} N(-d_2) - S_0 N(-d_1)$ . Using $N(-0.15) = 1 - 0.5596 = 0.4404$ and $N(-0.35) = 1 - 0.6368 = 0.3632$ gives $95.12 \times 0.4404 - 100 \times 0.3632 = 41.89 - 36.32 = 5.57$ dollars. The two routes agree.
Check that both sides of the rule match. The left side, call minus put, is $10.45 - 5.57 = 4.88$ dollars. The right side, stock price minus the present value of the strike, is $100 - 95.12 = 4.88$ dollars. Both sides are equal, as the rule requires.

Answer: With a call price of 10.45 dollars, the put with the same strike and expiry must cost about 5.57 dollars. Both sides of the rule come to 4.88 dollars, so the numbers are consistent. In this example the stock price is above the discounted strike, which is why the call costs more than the put.

The numbers in this example are recomputed by script every time the site is built.

Common mistakes

Mixing up the call and the put. The call is the right to buy, the put is the right to sell, and the letter C always means the call in this formula.
Forgetting to discount the strike. Writing $S_0 - K$ instead of $S_0 - K e^{-rT}$ gives a wrong answer, because the strike is paid at expiry, not today.
Using the rule for American options or for a stock that pays dividends. In those cases the exact equality changes, so the rule can give a misleading answer.
Treating the rule as a forecast of the stock price. It links the prices of the two options and the stock at one moment, and it says nothing about which way the stock will move.
Comparing a call and a put with different strikes or different expiry dates. Parity only holds when both options share the same strike and the same expiry.

Where you will meet it

Checking whether quoted call and put prices on the same stock are consistent with each other.
Building equivalent positions: a call plus cash can imitate a share plus a put, which is how option traders describe the rule.
Pricing and hedging on option trading desks, where the rule is one of the first consistency checks.
Teaching no-arbitrage pricing in finance courses and textbooks.

Questions people ask

Why is it called parity?

Parity means equality. The rule says that two combinations of the same assets must have equal value today, so the call and the put are linked by a fixed equality.

Does the rule tell me which option is more expensive?

Not by itself. It ties the call and the put together, so if one price is known, the other follows. Which one is larger depends on the stock price compared with the discounted strike. In this example the stock at 100 dollars is above the discounted strike of 95.12 dollars, so the call costs more than the put.

What changes if the options are American?

The exact equality no longer holds. American options can be used before expiry, so the rule becomes a pair of inequalities that limit the two prices, rather than fixing one from the other.

Black-Scholes Formula for a Call Option: The Black-Scholes formula works out what the right to buy a stock later is worth today, using the stock price, the agreed price, the interest rate, the time left and the stock’s ups and downs.
Present Value: Present value says how much money you must put aside today, at a steady interest rate, so that it grows to a known amount by a later date.
Continuous Compounding: Continuous compounding says money grows as if interest were added every instant, and the number e, about 2.718, sets the size of that growth.