What is put call parity?
Put call parity says that a long call plus a short put with the same strike and expiration behaves exactly like owning the stock with borrowed money. At expiration, the call and the short put together are worth S minus K no matter where the stock finishes. If the stock is above the strike, the call pays S minus K and the put expires worthless. If it is below, the call expires worthless and the short put costs K minus S, which again nets to S minus K.
Two portfolios with identical payoffs must cost the same today, or someone could buy the cheap one, sell the expensive one and keep the difference without risk. That no arbitrage argument is all you need to derive the formula. It does not depend on volatility, a pricing model or anyone's view of the stock, which is why parity holds so tightly in liquid markets. Hans Stoll described the relationship formally in 1969.
What is the put call parity formula?
In the formulas below, C and P are the European call and put prices, S is the stock price, K is the shared strike, r is the continuously compounded risk free rate, q is the continuous dividend yield and T is the time to expiration in years. K × e^(minus r × T) is the present value of the strike. Some people search for it as the call put parity formula; it is the same equation.
The formula and its common rearrangements are:
- C minus P = S minus K × e^(minus r × T), for a stock with no dividend
- C = P + S minus K × e^(minus r × T), which prices a call from a put
- P = C minus S + K × e^(minus r × T), which prices a put from a call
- C minus P = S × e^(minus q × T) minus K × e^(minus r × T), for a stock with a continuous dividend yield q
How does put call parity work in a worked example?
Use illustrative inputs: S = 100, K = 100, six months to expiration (T = 0.5), a risk free rate of 4 percent and no dividend. The present value of the strike is 100 × e^(minus 0.04 × 0.5) = 98.02, so parity says C minus P = 100 minus 98.02 = 1.98. With 25 percent volatility, the Black Scholes formula gives a call of 8.01 and a put of 6.03, and 8.01 minus 6.03 = 1.98, as expected.
Parity holds regardless of volatility. If implied volatility rose and the call climbed, the put would climb by the same amount, keeping the difference at 1.98. That is also why European calls and puts at the same strike and expiration should show the same implied volatility.
Now add a 2 percent continuous dividend yield. The stock term becomes 100 × e^(minus 0.02 × 0.5), about 99.005, so C minus P = 99.005 minus 98.020, about 0.985. Dividends lower call prices relative to put prices, which is exactly what the formula shows.
How do conversions and reversals use put call parity?
Conversions and reversals are the trades that enforce parity. Each combines stock with a synthetic position built from options, so its value at expiration is fixed in advance.
Suppose that, with the inputs above, the call traded at 8.50 while the put stayed at 6.03. Then C minus P = 2.47, above the parity value of 1.98. A conversion costs 100 + 6.03 minus 8.50 = 97.53 per share today and is worth exactly 100 at expiration, which has a present value of 98.02, so it locks in about 0.49 per share before costs. Market makers run these trades constantly, so gaps like this are rare in liquid names once commissions, borrowing costs and bid ask spreads are included.
The classic parity trades are:
- A conversion is long 100 shares, long one put and short one call at the same strike and expiration. It is always worth the strike at expiration, so it locks in a profit when the call is rich relative to the put.
- A reversal, also called a reverse conversion, is short 100 shares, short one put and long one call at the same strike and expiration. It locks in a profit when the put is rich relative to the call.
- A box spread combines a synthetic long at one strike with a synthetic short at another. It locks in the difference between the strikes and works like a loan or deposit at the interest rate implied by the option prices.
Why is put call parity an inequality for American options?
American options can be exercised early, so the two portfolios no longer have payoffs that are locked together. An American put can be worth more than its European twin because of early exercise, while an American call on a stock with no dividend is worth the same as the European call. The exact equation turns into a pair of bounds. For a stock that pays no dividend, S minus K ≤ C minus P ≤ S minus K × e^(minus r × T).
The upper bound holds because the American call equals the European call while the American put is worth at least the European put. The lower bound comes from comparing a call plus cash equal to the strike with a put plus the stock: the first is always worth at least as much, even if the put is exercised early.
In the example, the upper bound is 1.98 and the lower bound is 100 minus 100 = 0. A smoothed binomial tree values the American put at roughly 6.2, while the American call stays at 8.01, so C minus P is about 1.8, inside the range and below the European value. With dividends the bounds shift and widen, and calls also gain an early exercise premium before ex dividend dates. That is why CalcOptions prices every leg with an American model rather than deriving one option's price from the other.
How do traders use put call parity in practice?
Even if you never run a conversion, parity explains several everyday relationships between options and stock:
- Synthetic positions: a long call plus a short put at the same strike is synthetic long stock, and a short call plus a long put is synthetic short stock.
- Quote checks: if calls and puts at the same strike show very different implied volatilities, look for a dividend, a hard to borrow stock or an early exercise effect before assuming a mispricing.
- Market inputs: solving parity for r gives the interest rate implied by option prices, and solving for q gives an implied dividend or borrow cost.
- Strategy equivalents: a covered call has the same payoff shape as a short put at the same strike plus cash, and a protective put has the same shape as a long call plus cash.
Frequently asked questions
What is the put call parity formula?
The put call parity formula is C minus P = S minus K × e^(minus r × T) for European options on a stock with no dividends. With a continuous dividend yield q, the stock term becomes S × e^(minus q × T). C and P are the call and put prices at the same strike K and expiration T.
Does put call parity hold for American options?
Not as an exact equation, because early exercise can make American puts, and calls before dividends, worth more than their European versions. For a stock with no dividends, parity becomes the range S minus K ≤ C minus P ≤ S minus K × e^(minus r × T). Prices outside that range would allow arbitrage.
What happens if put call parity is violated?
If call and put prices break parity, a trader can lock in a riskless profit with a conversion when calls are rich or a reversal when puts are rich. In liquid markets these gaps are usually closed quickly by market makers. Apparent violations often reflect dividends, borrow costs, early exercise or stale quotes rather than free money.
Try it in a calculator
Conversion
A conversion is a market neutral arbitrage position that buys 100 shares, buys an at the money put, and sells an at the money call at the same strike, locking in a fixed value at expiration no matter where the stock goes.
Reversal
A reversal, also called a reverse conversion, is a market neutral arbitrage position that shorts 100 shares, sells an at the money put, and buys an at the money call at the same strike, locking in a fixed value at expiration.
Box Spread
A box spread is a four leg options position that combines a bull call spread and a bear put spread on the same two strikes, so it is worth exactly the distance between the strikes at expiration.
Educational tool, not financial advice. Calculations are theoretical and depend on your inputs; real fills, fees, early assignment and dividends can change results. Options involve risk and are not suitable for every investor.