Put-call parity is the no-arbitrage relationship linking a call, a put, the underlying asset and the present value of the strike. For European options on a non-dividend-paying asset with the same strike and expiration, the relationship is C + PV(K) = P + S, or, equivalently, C − P = S − PV(K). If the two sides produce the same future cash flows but have different current prices, the cheaper side can theoretically be bought and the more expensive side sold. In practice, spreads, funding, dividends, stock borrow, exercise rules, taxes and execution risk determine whether a difference is actually tradable.
What put-call parity says
Put-call parity applies to a call and a put written on the same underlying, with the same strike price and the same expiration date. The Options Industry Council describes this as the pricing relationship between those matched contracts. The cleanest version assumes European exercise, meaning the options can be exercised only at expiration, and an underlying that pays no dividends during the contract’s life.
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Use these symbols:
- C: current premium of the call.
- P: current premium of the put.
- S: current price of the underlying.
- K: strike price payable or received at exercise.
- PV(K): present value today of the strike payment due at expiration.
The parity equation is:
C + PV(K) = P + S
Rearranging it gives:
C − P = S − PV(K)
The equation does not predict whether the stock will rise or fall. It says that two portfolios with identical expiration cash flows should have the same value today after financing is included. The strike is not treated as a current cash payment at its full face value; it is discounted because the payment occurs later.
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How the equation works economically
Consider the portfolio consisting of a long call, a short put and the present value of the strike invested in a risk-free instrument. At expiration, the option combination has the same directional payoff as owning the underlying, while the financing position supplies the strike amount when required.
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For a call, the expiration payoff is max(ST − K, 0). A short put has payoff −max(K − ST, 0). Adding them produces:
max(ST − K, 0) − max(K − ST, 0) = ST − K
If the underlying finishes above the strike, the call is worth ST − K and the put expires worthless. If it finishes below the strike, the short put loses K − ST and the call expires worthless; the combined result is still ST − K. Adding the strike-financing asset, which grows to K, leaves ST, the same terminal value as owning the underlying.
This is why the financing leg matters. The two option legs alone create the underlying’s payoff minus the strike obligation. Put-call parity accounts for that obligation through PV(K).
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Why strike and expiration must match
Matching the strike
The call and put must use the same K because exercise transfers the same amount of cash in either direction. A 100-strike call and a 110-strike put do not offset one another’s exercise obligations. Their payoff kinks occur at different prices, so the portfolios do not have identical terminal cash flows.
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Matching the expiration
The options must expire on the same date. A position that settles in three months cannot be directly replicated by one that settles in six months: the later contract has additional time value, a different discount period and more exposure to dividends, volatility and financing. Even options described as expiring in the same month can differ if their exact expiration date or settlement convention differs.
Matching the underlying and contract terms
Parity also requires the same underlying reference, contract multiplier, currency, settlement method and other material specifications. An option on an index, an exchange-traded fund and a futures contract may have related economics but are not interchangeable without adjusting for their distinct dividends, settlement and financing.
Synthetic positions created by parity
A synthetic position uses options and, when necessary, the underlying or a financing instrument to reproduce another position’s payoff. The table shows the familiar option combinations and the financing qualification needed for an exact replication.
| Target exposure | Option combination | What the combination delivers |
|---|---|---|
| Synthetic long stock | Long call + short put, same K and expiration | Expiration payoff of ST − K; add the present value of K to reproduce the full stock payoff. |
| Synthetic short stock | Long put + short call, same K and expiration | Expiration payoff of K − ST; the financing leg completes the short-stock cash-flow pattern. |
| Synthetic long call | Long stock + long put | A protective-put combination. Its payoff equals a call payoff plus the strike amount, so borrowing or otherwise financing K is required when comparing it with a standalone call. |
| Synthetic long put | Long call + short stock | The reverse construction; a cash or financing adjustment for K is required for an exact standalone-put comparison. |
Market participants often use the shorter labels even when the cash leg is implicit. When calculating a trade, write every cash flow explicitly rather than assuming that the financing adjustment is free.
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A numerical illustration
Suppose a non-dividend-paying stock is trading at S = 100. A call and put both have K = 100 and the same expiration. Assume the present value of the strike is PV(K) = 98 and the call premium is C = 7.
Substitute those values into parity:
7 + 98 = P + 100
Therefore, the parity-implied put premium is P = 5. This is an algebraic illustration, not a live quote or a promise of executable profit. If an observed put were materially above 5 after using executable bid and ask prices and deducting all costs, a trader would investigate the reverse combination: sell the expensive portfolio and buy the cheaper one. The trade would be attractive only if the resulting spread remained positive after commissions, financing, stock borrow, margin and execution costs.
Dividends, interest and forward prices
Interest-rate discounting
PV(K) depends on the time to expiration and the relevant funding or risk-free discount curve. Under continuous compounding, a common representation is PV(K) = K e−rT, where r is the applicable annualized rate and T is the time in years. Actual desks may use a currency-specific curve, day-count convention and compounding method. A higher rate lowers the present value of a future strike and changes the call-minus-put relationship.
Dividends and carry
Dividends paid before expiration reduce the value of holding the stock relative to an otherwise identical non-dividend-paying asset. For known cash dividends, a commonly used European-option form is:
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C − P = S − PV(K) − PV(dividends)
Equivalently, the stock leg is adjusted for the present value of dividends received during the option’s life. If dividends are uncertain, the parity calculation must use the market’s dividend assumptions rather than treating an estimate as a guaranteed cash flow.
Forwards and futures
When the underlying exposure is expressed through a forward or futures price, CME Group presents the compact notation c + k = f + p. In that notation, c and p are call and put prices, f is the forward or futures price, and k represents the financing-adjusted strike used in the lesson’s convention. Always check how a source defines k before comparing the formula with a spot-price equation.
When a parity difference may be an arbitrage opportunity
A theoretical arbitrage exists when two portfolios have exactly the same future cash flows but different current prices. The usual idea is to buy the cheaper portfolio, sell the dearer portfolio and hold the positions until the cash flows converge. A practical check should follow this order:
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- Use executable prices. A midpoint is not necessarily tradable. Test the price at which each leg can actually be bought or sold, including the bid-ask spread.
- Build both cash-flow schedules. Include option premiums, the strike-financing investment or borrowing, dividends, interest, stock borrow payments, margin transfers and settlement cash.
- Include every cost. Deduct commissions, exchange fees, taxes, funding charges and any cost of borrowing the stock or futures contract.
- Check exercise and assignment exposure. American options can be exercised before expiration, and a short option can be assigned before the planned closing date.
- Test liquidity and execution risk. A spread that exists for one contract may disappear before all legs fill. Partial fills can leave an unhedged position.
- Require a surplus, not merely equality. The expected locked-in difference must exceed a conservative allowance for costs, model error and operational risk.
Competitive trading normally pushes persistent, low-risk discrepancies toward equality. A screen showing a parity violation is therefore a prompt to investigate specifications and costs, not proof of free money.
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Synthetic exposure versus owning the underlying
| Comparison point | Direct underlying position | Synthetic option position |
|---|---|---|
| Expiration payoff | Direct exposure to the asset value at the measurement date. | Can match the target payoff only when strikes, expirations and financing are aligned. |
| Upfront cash | Usually requires the asset purchase price, possibly offset by borrowing. | May require a net premium, collateral and a separate strike-financing leg. |
| Dividends | Owner generally receives declared distributions, subject to the asset’s rules. | Dividend effects are embedded in option prices and parity; the synthetic holder may not receive the same distribution. |
| Borrow and financing | Long ownership has financing cost; short ownership requires borrow availability and a borrow fee. | Option premiums, margin and the financing leg create their own carry. |
| Exercise and assignment | No option exercise or assignment event. | American-style contracts can be exercised or assigned before expiration. |
| Liquidity and margin | Depends on the underlying market and account rules. | Several legs may have different spreads, margin requirements and liquidity. |
| Settlement | Follows the underlying’s delivery or cash-settlement rules. | May involve option exercise, assignment, cash settlement or delivery of shares. |
European versus American options
The simple equality is exact under the European, no-dividend assumptions stated above. American options may be exercised at any time before expiration, so early exercise creates an additional choice that a single European payoff does not contain. For that reason, practical American-option analysis often uses parity bounds or adjusted inequalities rather than treating the European equation as an exact identity.
Early exercise is especially relevant around ex-dividend dates for calls and puts whose time value is small. A short American option can also be assigned unexpectedly, changing the position before the intended parity trade reaches expiration. The possibility of assignment, together with stock-borrow availability and margin rules, must be included in any implementation.
Common mistakes and how to avoid them
- Comparing different strikes: Recalculate using a genuinely matched call and put.
- Comparing different expirations: Match the exact expiration date and settlement time, not just the calendar month.
- Ignoring the strike’s present value: Use PV(K), not automatically K, unless the notation explicitly incorporates financing.
- Forgetting dividends: Subtract the present value of expected distributions in a spot-based calculation.
- Using theoretical mids: Replace model or midpoint prices with executable bid and ask prices.
- Calling an American-option gap arbitrage: Check early exercise and assignment before assuming identical cash flows.
- Leaving out borrow or margin: A short-stock or short-option leg can consume collateral or become expensive to maintain.
- Assuming a synthetic is identical operationally: The terminal payoff may match while cash timing, voting rights, dividends, taxes and settlement differ.
A practical parity checklist
Before labeling a price difference as parity arbitrage, document the underlying, contract multiplier, strike, exact expiration, exercise style, settlement method, currency, dividend schedule, discount rate, borrow rate, bid and ask for every leg, commissions, taxes, margin and assignment treatment. Then map each portfolio’s cash flows at initiation, on every interim dividend or financing date, and at expiration. Only after those schedules match should the residual price difference be considered a candidate spread.
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