What Is Put-Call Parity? Formula, Examples, and How It Prices Options

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Put-call parity is a no-arbitrage relationship between the prices of a European call and a European put that share the same underlying asset, strike price and expiration date. It states that a call plus the present value of the strike must cost the same as a put plus the underlying asset:

C + PV(K) = P + S

Parity fixes the price of one option relative to the other, the underlying and interest rates. If you know three of the inputs, parity gives you the fourth. If the market quotes all four and they do not balance, a trader can in principle lock in a riskless profit by buying the cheap side and selling the expensive side.

What Is Put-Call Parity?

Put-call parity describes how the prices of four instruments must line up when no arbitrage is possible:

• C: the price of a European call option
• P: the price of a European put option with the same strike and expiration
• S: the current spot price of the underlying asset
• PV(K): the present value of the strike price K, discounted from the expiration date at the risk-free rate

PV(K) is not the strike price itself. It is the amount you would need to invest today at the risk-free rate to have exactly K at expiration. With annual compounding over T years, PV(K) = K / (1 + r)^T. With continuous compounding it is K × e^(-rT). The difference between K and PV(K) matters: a common shorthand, C - P = S - K, is only correct when interest rates are zero or the option expires immediately.

The relationship was formalized by economist Hans R. Stoll in his 1969 paper “The Relationship Between Put and Call Option Prices,” published in the Journal of Finance. Dealers had used the idea long before, converting puts into calls and back with stock positions, but Stoll set out the equilibrium argument that options pricing theory now treats as a basic building block.

Why Put-Call Parity Holds

The logic rests on the law of one price: two portfolios that pay exactly the same amount in every possible future state must cost the same today. Otherwise, you could sell the expensive one, buy the cheap one, and keep the difference with no risk.

Compare two portfolios, each held until expiration:

• Portfolio A (fiduciary call): one European call with strike K, plus a risk-free bond that pays K at expiration (cost today: PV(K)).
• Portfolio B (protective put): one European put with strike K, plus one unit of the underlying asset.

At expiration, the underlying trades at some price S_T. The payoffs work out as follows:

Both portfolios pay the greater of K and S_T in every case, so their costs today must match:

C + PV(K) = P + S

This derivation needs no assumption about volatility, the probability distribution of prices, or investor risk preferences. That is why parity is model-free and holds under any consistent pricing model, including the Black-Scholes model. It also explains why the argument requires European options: the payoffs above are measured only at expiration, and an American option holder could exercise earlier and break the match.

The Formula and Its Rearrangements

Moving terms around produces several useful versions of the same relationship.

Call minus put:

C - P = S - PV(K)

The difference between the call and put premium equals the difference between the spot price and the discounted strike. When S is above PV(K), the call is worth more than the put; when S is below PV(K), the put is worth more.

Synthetic positions: each instrument can be replicated with the other three.

• Synthetic long underlying: S = C - P + PV(K) (long call, short put, lend the present value of the strike)
• Synthetic call: C = P + S - PV(K) (long put, long underlying, borrow the present value of the strike)
• Synthetic put: P = C - S + PV(K) (long call, short underlying, lend the present value of the strike)

This answers a frequent point of confusion. A protective put (stock plus put) is not equivalent to a call alone. It is equivalent to a call plus a bond paying K. To turn a protective put into a synthetic call, you have to finance part of it by borrowing PV(K).

Put-call-forward parity: a long call and a short put with the same strike behave like a forward contract to buy the underlying at K. If F is the forward price for the same expiration, then

C - P = PV(F - K)

When the strike equals the forward price, the call and put have the same value. This form is especially convenient for currency and commodity options, where forward and futures prices are quoted directly.

Worked Examples

All examples below are hypothetical and ignore commissions, spreads, taxes and margin requirements unless stated.

Checking a Put Price

Suppose a stock, XYZ, trades at $50. A six-month European call with a $55 strike costs $3.00, and the present value of the $55 strike at the risk-free rate is $54.46. What should the matching six-month European put cost?

Start from C + PV(K) = P + S:

• $3.00 + $54.46 = P + $50
• $57.46 = P + $50
• P = $7.46

If the market quotes the put at $7.46, parity holds and there is no arbitrage. Notice that the put is worth more than the call here. The stock is below the strike, so the put already carries $5 of intrinsic value, while the call needs the stock to rise more than 10% just to finish in the money.

When the Put Is Overpriced: The Arbitrage

Now suppose the same put is quoted at $8.00 instead of $7.46. The two sides of the equation no longer match:

• Fiduciary call side: C + PV(K) = $3.00 + $54.46 = $57.46
• Protective put side: P + S = $8.00 + $50 = $58.00

The protective put side is $0.54 too expensive. The arbitrage is to sell that side and buy the other:

1. Sell the put: receive $8.00
2. Short one share: receive $50.00
3. Buy the call: pay $3.00
4. Lend $54.46 at the risk-free rate (for example, buy a Treasury bill maturing at expiration)

Net cash today: $8.00 + $50.00 - $3.00 - $54.46 = +$0.54.

At expiration the bill pays $55, and the position closes out the same way whatever the stock does:

• If XYZ is above $55: exercise the call, buying the share for $55 with the bill proceeds to close the short. The put expires worthless.
• If XYZ is below $55: the put is assigned, so you buy the share for $55 and use it to close the short. The call expires worthless.
• If XYZ is exactly $55: both options expire worthless, and you buy the share at $55 to close the short.

The final cash flow is zero in every case, so the $0.54 collected up front is the profit. The trade combining a short put, long call and short stock is called a reversal; the opposite trade, used when the call side is too expensive (long stock, long put, short call), is a conversion.

A Second Example With Annual Compounding

Take a second hypothetical stock, TCKR, trading at $10. The risk-free rate is 4% a year, the stock pays no dividends, and both options have a $15 strike and expire in one year.

PV(K) = $15 / 1.04 = $14.42

Parity gives C + 14.42 = P + 10, so P - C = $4.42. With the stock at only about two-thirds of the strike, the put should trade $4.42 above the call, whatever their absolute prices. If the quoted gap differs, the same reversal or conversion shown above captures the difference.

How Dividends and Interest Rates Change the Equation

Dividends. The protective put portfolio holds the stock, but the call holder does not receive dividends. To keep the payoffs equal, the stock term is reduced by the present value of dividends paid before expiration:

C + PV(K) = P + S - PV(D)

With a continuous dividend yield q, the stock term becomes S × e^(-qT). Higher expected dividends lower call values and raise put values relative to each other, because the share price is expected to drop by the dividend amount on the ex-dividend date. The same adjustment applies to options on coupon-paying bonds, with the present value of coupons taking the place of dividends.

Interest rates. A higher risk-free rate lowers PV(K), so S - PV(K) becomes larger and the call gains value relative to the put. The intuition is a cost-of-carry one: buying a call instead of the asset defers paying the strike, and that deferral is worth more when rates are high. This is also why rho, the sensitivity to interest rates, is positive for calls and negative for puts.

Implications for the Greeks. Differentiating the parity equation shows that, for a non-dividend underlying, a call’s delta minus the put’s delta equals 1, and the call and put of the same strike and expiry share the same gamma and vega (see option Greeks).

Put-Call Parity in Forex Options

In a currency option, the underlying is one unit of a foreign currency, and holding that currency earns the foreign interest rate. The foreign rate therefore plays the same role as a dividend yield. With domestic rate r_d and foreign rate r_f over T years:

C - P = S × e^(-r_f × T) - K × e^(-r_d × T)

This is the parity relationship used in the Garman-Kohlhagen model for FX options. It ties directly to covered interest parity, because S × e^((r_d - r_f) × T) is the theoretical forward rate, so the forward form C - P = PV(F - K) applies naturally.

A hypothetical example with annual compounding: EUR/USD spot is 1.1000, the strike is 1.1000, the one-year USD rate is 4% and the one-year EUR rate is 2%.

• S / (1 + r_f) = 1.1000 / 1.02 = 1.0784
• K / (1 + r_d) = 1.1000 / 1.04 = 1.0577
• C - P = 1.0784 - 1.0577 = 0.0207

The euro call should be worth about 0.0207 USD per euro more than the euro put, even though the strike equals spot. The reason is that the higher USD rate puts the forward rate above spot, so a strike at spot is below the forward and favors the call.

European vs. American Options

Put-call parity as an equality applies only to European options, which can be exercised only at expiration. American options can be exercised at any time, and the right to exercise early has value, especially for puts.

For American options on a non-dividend-paying stock, the relationship becomes a pair of bounds:

S - K ≤ C - P ≤ S - PV(K)

The upper bound holds because an American call on a non-dividend stock should never be exercised early, so it is worth the same as the European call, while the American put is worth at least as much as its European counterpart. The lower bound comes from comparing portfolios that can be closed out the moment an early exercise occurs. Dividends can make early exercise of calls rational just before an ex-dividend date, which widens the bounds further.

Many listed single-stock options are American style, while many index options are European style. Traders who apply the exact equation to American options will see apparent “violations” that are really the value of the early exercise feature.

How Traders Use Put-Call Parity

Pricing consistency. Market makers use parity to keep calls and puts on the same strike in line. If a dealer adjusts a call quote, the matching put quote moves with it, otherwise the dealer’s own prices would offer arbitrage.

Conversions and reversals. Dealers and arbitrage desks run the conversion and reversal trades described above when quotes drift apart. Because these desks and automated systems watch the relationship continuously, meaningful gaps in liquid markets tend to close quickly, and the remaining edge is usually smaller than the cost of trading.

Extracting market information. Rearranging parity lets traders back out an implied forward price, an implied interest rate or an implied dividend from option quotes. If C, P, S and K are observed, solving for PV(K) reveals the financing rate the options market is pricing in.

Building synthetics. A trader who wants exposure that is hard to get directly can build it from the other legs. A long call and short put at the same strike create a synthetic long position in the underlying; a long put plus long stock replicates a call plus a bond. Combining a synthetic long at one strike with a synthetic short at another produces a box spread, whose value should equal the discounted difference between the strikes.

Checking implied volatility quotes. Since parity forces calls and puts at the same strike to share an implied volatility, a gap between call and put implied volatility at that strike usually points to a dividend, a borrowing cost, early exercise value or a stale quote rather than a genuine view on volatility.

Limitations of Put-Call Parity

The equation describes a frictionless market. Several real-world features cause small, persistent deviations:

• Bid-ask spreads and commissions. Arbitrage requires trading four legs, each at a worse price than the midpoint, so parity holds only within a band set by these costs.
• Borrowing and lending rates differ. The formula uses a single risk-free rate, but traders borrow at higher rates than they lend, which widens the band further.
• Short-sale constraints. Reversals require shorting the underlying. If the stock is hard to borrow or carries a borrowing fee, puts can stay expensive relative to calls without offering a usable arbitrage.
• Uncertain dividends. A surprise dividend change moves the true parity value after the trade is placed.
• Early exercise. Early assignment on American options can disrupt a hedged position before expiration.
• Margin, taxes and capital. Margin ties up capital, and tax treatment of the legs can differ. With profits often a few cents per contract, an arbitrage may need large size to be worthwhile.
• Timing and liquidity. Quotes can move between legs. Empirical research has found that parity generally holds in developed markets but can break down briefly during high volatility or thin trading, which is when executing all legs at once is hardest.

What Put-Call Parity Is Not

• Not a valuation model. Parity fixes the relative prices of calls and puts but cannot tell you whether both are expensive or cheap. A model such as Black-Scholes, or the market’s implied volatility, sets the absolute level.
• Not the put-call ratio. That ratio divides put volume or open interest by call volume or open interest and is a sentiment gauge with no pricing role.
• Not “trading at parity.” That phrase means a single option’s premium equals its intrinsic value.
• Not interest rate parity. Interest rate parity links spot and forward exchange rates to interest rate differentials. It feeds into FX put-call parity through the forward rate but is a separate relationship.
• Not a statement that puts and calls cost the same. Equal premiums occur only when the strike equals the forward price. An option struck at the current spot price normally has a call worth more than the put when interest rates are positive and the underlying pays little or no dividend.

The Bottom Line

Put-call parity matters to options traders, market makers and anyone pricing currency or equity options, because it keeps calls, puts and the underlying consistent without relying on any volatility model. Gaps between quotes can in principle be captured with conversions and reversals. In liquid markets, spreads, borrowing costs, short-sale limits and fast-moving arbitrage desks usually leave little usable profit, so parity works best as a pricing check.