Put-Call Parity: Formula, Dividends, and Synthetic Positions

Editorial responsibility: Oliver Sinz · Last verified: October 27, 2026

Put-call parity is the no-arbitrage relationship between a European call, a European put with the same strike and expiration, the underlying asset, and the present value of the strike price. For a stock that pays no dividends before expiration, the call price minus the put price must equal the stock price minus the present value of the strike: C − P = S − PV(K).

Parity needs no pricing model, no volatility estimate, and no forecast. It follows from one observation: two portfolios that pay the same amount in every possible outcome must cost the same today. Its practical meaning is about financing. A long call plus a short put at the same strike is a forward purchase of the stock, so it behaves like shares bought with borrowed money, and a price gap that looks like arbitrage often disappears once dividends, financing, short-sale costs, and bid-ask spreads are counted.

This guide derives the formula, adds dividends, builds the synthetic positions it implies, turns it into a trade through conversions and reversals, and shows how the equality becomes a range for American-style options.

All prices, quotes, rates, and fees in this article are hypothetical, not market data. Nothing here is a recommendation to trade.

The put-call parity formula

For European options on a stock that pays no dividends before expiration:

C + PV(K) = P + S, or equivalently C − P = S − PV(K)

• C and P are the prices of a European call and put on the same underlying, with the same strike K and expiration.
• S is the current stock price.
• PV(K) is the amount that, invested at the risk-free rate, grows to K by expiration. With a continuously compounded rate r and time T in years, PV(K) = K × e^(−rT).

The Options Industry Council gives the same relationship as c = S + p − Xe^(−r(T−t)) (OIC). Hans Stoll is widely credited with the first detailed academic treatment, tested on the dealer market for puts and calls that preceded listed options (Stoll, 1969), but traders knew the relationship much earlier: a 1904 trading manual described it at length (Haug and Taleb, 2011). Solved for the call, the put, the stock, or the bond, the equation gives the synthetic positions covered below.

Checking parity on an option chain

Hypothetical chain: stock XYZ at $100.00, 45 calendar days to expiration (T = 45 ÷ 365 = 0.1233), a 4.0% continuously compounded risk-free rate, and no dividends. Premiums are Black–Scholes–Merton values at 25% volatility, rounded to the cent.

The differences are rounding: model prices satisfy parity by construction, so the real test is whether a chain’s quotes do.

The table also shows the structure parity imposes. C − P falls by $4.98, the present value of $5, for every $5 increase in the strike. Plotted against the strike, the call-minus-put difference is a straight line with a slope of −e^(−rT), crossing zero where the strike equals the forward price, about $100.49. Volatility, skew, and the choice of model decide how value is divided between the call and the put at each strike. For European options with known rates and dividends, they cannot change the difference.

Why it holds: two portfolios with the same payoff

Take the 105 strike. Portfolio A holds the call plus a Treasury bill that pays $105 at expiration and costs PV(K) today. Portfolio B holds the put plus one share.

Both portfolios pay the greater of the stock price and $105, whatever happens. If A cost less than B, a trader could buy A, sell B, keep the difference today, and owe nothing at expiration. Competition for that trade forces the prices together. This is the law of one price.

Three features separate parity from a pricing model:

• It is static. The portfolio is built once and held to expiration; the Black–Scholes argument requires continuous rebalancing.
• It assumes nothing about volatility or the distribution of returns. The table holds for any price path, so parity holds under every arbitrage-free model.
• It implies one implied volatility per strike. Black–Scholes–Merton prices satisfy parity, so a European call and put whose prices satisfy parity imply the same volatility, as long as both are inverted with the same rate and dividend assumptions. A persistent gap between call and put implied volatilities at one strike is either a parity deviation or a sign that one of those assumptions is wrong.

The payoff table relies on three hidden conditions: neither option can be exercised early, portfolio B receives no dividend before expiration, and money can be borrowed and lent at the same rate. The rest of this guide relaxes each one.

Put-call parity with dividends

A shareholder receives dividends; an option holder does not. If XYZ pays a dividend before expiration, portfolio B collects it and becomes worth more than portfolio A by the dividend’s present value, PV(D). Parity becomes:

C − P = S − PV(D) − PV(K) for known cash dividends

C − P = S × e^(−qT) − K × e^(−rT) for a continuous dividend yield q, the usual description of a broad stock index

Both are special cases of one form written with the forward price F, the price agreed today for delivery of the stock at expiration:

C − P = e^(−rT) × (F − K), where F = (S − PV(D)) × e^(rT), or F = S × e^((r − q)T) with a yield

A long call and a short put at the same strike amount to a forward purchase of the stock at K, valued today. Dividends enter only by lowering the forward price; for options on futures, F is the futures price.

Example. Same 45-day 105 options, but XYZ pays a $1.00 dividend with an ex-dividend date in 20 days. PV(D) = $1.00 × e^(−0.04 × 20/365) = $0.9978, so C − P = $100 − $0.9978 − $104.4835 = −$5.4813. Without the dividend it was −$4.4835: the call-minus-put difference falls by exactly the dividend’s present value.

How that change is split between the call and the put depends on how the dividend is modeled. The difference does not:

The two cash-dividend models disagree by about 1.3 cents on each option and agree on C − P to four decimals. Parity needs the dividend’s amount and timing, not a model of how the stock moves.

What weakens parity in practice is uncertainty about the dividend itself. Option prices reflect the dividend the market expects, and a cut, an increase, or a special dividend changes the true value of C − P. Ordinary cash dividends do not adjust listed option terms; OCC generally adjusts only for non-ordinary cash dividends of $0.125 per share or more (SEC Release 34-99641).

Running parity in reverse

Because parity has no unobservable input, traders often solve it for whichever quantity is least certain:

• Implied forward. F = K + e^(rT) × (C − P). From the 100-strike quotes, F = $100 + 1.00494 × $0.49 = $100.49. Cboe’s methodology for the VIX index sets its forward index level from SPX call and put prices with the same expression (Cboe).
• Implied dividend. From the rounded escrowed-model quotes ($1.47 call, $6.95 put), PV(D) = S − PV(K) − (C − P) = $100 − $104.4835 + $5.48 = $1.00.
• Implied rate. The 100-strike quotes imply about 3.98%, close to 4% given penny rounding. Each cent of C − P moves the implied 45-day rate by about 0.08 percentage points.

Synthetic positions: each leg from the other three

Rearranging C − P = S − PV(K) produces six synthetic positions. Each includes a bond leg, lending or borrowing PV(K), that the common shorthand “stock = call − put” omits; the OIC’s own table of synthetics assumes zero interest rates and dividends (OIC).

With known cash dividends, S in each line becomes S − PV(D). That replicates a prepaid forward, the shares delivered at expiration without the dividends paid before then, which is all the options deliver. A shareholder also receives those dividends, so replicating the full stock position needs one more lending leg: S = C − P + PV(K) + PV(D). Two familiar strategies are synthetics under other names: stock plus a protective put has the same expiration payoff as a call plus a T-bill paying K, and a covered call has the same payoff as a short put plus that T-bill.

Synthetic long stock: at the money, the debit is interest

Buying the 100 call at $3.74 and selling the 100 put at $3.25 costs a net $0.49 per share, or $49 per contract. That is not a discount on the stock. Parity says the debit equals S − PV(K) = $100.00 − $99.51: the present value of 45 days of interest on $100 at 4%. (At other strikes the debit also contains the difference between the stock price and the strike; at the 90 strike it is $10.44.) The synthetic buyer pays the financing cost up front and keeps $99.51 that can earn interest. With that money in T-bills, the position pays the same as the shares at expiration; without it, the synthetic is like buying shares with money borrowed at the rate embedded in the options.

The Black–Scholes Greeks of the European pair say the same thing:

• Delta: the call’s delta minus the put’s delta is exactly 1, the same as one share, or 100 shares per contract.
• Gamma and vega: identical for a call and a put with the same strike and expiration, so they cancel.
• Theta: the call loses about $4.42 per contract per day and the short put earns about $3.33, a net −$1.09 per day. That is the daily interest on $9,951 at 4% (r × PV(K) ÷ 365, per contract). The negative theta is not lost option value; it is interest accruing on the implied loan.

Unlike shares, the synthetic carries no dividends or voting rights, expires, and exposes the short put to early assignment, generally when it is deep in the money (OIC). It also requires margin: under Cboe’s minimum rules, an uncovered equity put requires the premium plus 20% of the underlying’s value, less any out-of-the-money amount, about $2,325 per contract here, and firms may require more (Cboe). The cash outlay is smaller than buying shares; the exposure is the same.

Parity as a trade: conversions and reversals

A conversion buys the stock, buys the put, and sells the call at the same strike and expiration. At expiration, whichever option is in the money delivers the stock at K, so the position pays exactly K whatever the stock does: it is a synthetic T-bill. A reversal, or reverse conversion, shorts the stock, sells the put, and buys the call. It receives cash today and pays K at expiration: a synthetic loan. The OIC calls these conversion arbitrage and reversal arbitrage (OIC). A conversion at one strike combined with a reversal at another is a box spread; the stock legs cancel.

The useful way to read a conversion is as an interest rate. Hypothetical quotes for the 45-day 100 strike:

At mid-quotes, the options are priced consistently with a 4% rate. At executable prices, a conversion lends at 3.25% and a reversal borrows at 4.72%. Neither beats a 4% Treasury bill, so these quotes offer no arbitrage even though parity holds only at the midpoint. The band between the two rates is the cost of crossing three bid-ask spreads; commissions, fees, stock-borrow costs, and the interest a broker actually pays on short-sale proceeds widen it further.

Now suppose the call is quoted at $3.86 bid, $3.94 ask. At the midpoint it looks $0.16 per share too expensive, about $16 per contract. A conversion at executable prices costs $100.01 + $3.29 − $3.86 = $99.44 and returns $100.00, an implied rate of 4.55%: about $6.84 per contract more than lending at 4%. Hypothetical commissions of $0.65 per option contract leave about $5.54, before any charge on the stock leg, on roughly $9,944 of capital for 45 days. Market makers compete for gaps like this, so quotes rarely stay out of line, and with American-style options, as standard U.S. stock options are, the trade carries risks the identity leaves out:

• Early assignment of the short call. A conversion is priced assuming its holder keeps any dividend paid before expiration. Assignment before the ex-dividend date delivers the shares early and forfeits the dividend.
• Early assignment of the short put. In a reversal, assignment forces the trader to buy back the short stock at K early, giving up interest on the short-sale proceeds.
• Expiration risk. If XYZ closes near the strike, the trader may not learn whether the short option was assigned until after expiration (OIC) and can be left with an unhedged stock position.
• Execution and borrowing. Three legs must fill together, and borrowed shares can be recalled or repriced.

American options: parity becomes a range

Early exercise breaks the payoff equivalence. Robert Merton pointed out that Stoll’s equality therefore holds exactly only for European options; for American puts, which may be exercised early, it does not pin down a unique price (Merton, 1973a; Merton, 1973b). With a nonnegative interest rate and no dividends, an American call should never be exercised early and is worth the same as the European call, while the American put can be worth more. The result is a pair of bounds:

S − K ≤ C − P ≤ S − PV(K)

The upper bound is European parity. The lower bound reflects the put holder’s right to exercise immediately. Using a binomial tree for the American puts on the same 45-day chain (C − P from unrounded prices):

The band’s width, K − PV(K), is the discount on the strike: the interest that PV(K) earns by expiration. Over 45 days at 4% it is about half a dollar, and American C − P sits in its upper part, pulled down by the put’s early-exercise value, which grows as the put moves into the money. For a one-year option at the 100 strike, the band widens to $3.92. At a zero interest rate it collapses, and the European equality holds for American options too. With negative rates the inequalities reverse.

With known cash dividends, the bounds become:

S − PV(D) − K ≤ C − P ≤ S − PV(K)

Dividends lower only the lower bound. The upper bound stays where it is because an American call holder can exercise just before an ex-dividend date to collect the dividend, which can make the American call worth more than the European one. A version that subtracts PV(D) from both bounds is too tight. Take a 95 strike with the $1.00 dividend in 20 days: the binomial tree gives American C − P = $4.65, above the $4.47 that version would allow and well inside the correct band of $4.00 to $5.47.

Why parity gaps are rarely arbitrage

Parity is an identity about payoffs. An executable arbitrage also needs prices that can be traded at the same moment, in size, at known costs, by someone who can finance and borrow on the assumed terms. Each friction opens a band within which quotes can sit without offering a riskless profit:

Short-sale costs are a major source of persistent apparent parity violations. A shareholder who lends out stock earns a fee that a short seller pays, so the fee works like an extra dividend yield; a Montréal Exchange guide adds the lending fee to the dividend yield in pricing (Kazemi, 2019). The example below follows that convention: European-style Black–Scholes–Merton values, with a constant borrow fee b modeled as an additional continuous yield, so that C − P = S × e^(−bT) − PV(K). For the 45-day 100 strike at 25% volatility (C − P and the gap from unrounded prices):

The puts look expensive relative to the calls, by $244 per contract at a 20% fee. The reversal that would capture the gap requires shorting the stock and paying the very fee that produces it, if shares can be located at all. For actual American-style equity options, early-exercise premiums can add further differences, and real borrow fees vary by lender and over time.

The empirical evidence fits this picture:

• Testing the first year of listed puts on the CBOE, 1977–1978, Klemkosky and Resnick (1979) concluded that prices were generally consistent with parity and market efficiency.
• Using European-style index options, which rule out early exercise, Kamara and Miller (1995) found violations much less frequent and smaller than in earlier studies of American-style options, and attributed those that remained to a premium for liquidity risk.
• In stock options, Ofek, Richardson, and Whitelaw (2004) found violations skewed in the direction short-sale constraints predict, strongly related to the cost of borrowing shares, and predictive of future stock returns. Muravyev, Pearson, and Pollet (2022) show that option prices reflect the present value of future borrowing fees.
• Deviations can carry information rather than free money. Measuring them as the implied-volatility spread between calls and puts, Cremers and Weinbaum (2010) found that stocks with relatively expensive calls outperformed those with relatively expensive puts by about 50 basis points a week, with predictability declining over their sample. Muravyev, Pearson, and Pollet (2025) find that such predictability falls by at least two-thirds once high-fee stocks are excluded.

Key takeaways

• For European options on a stock without dividends, C − P = S − PV(K), because a call plus a T-bill paying K has the same payoff as a put plus the stock.
• Parity needs no volatility input and holds under any arbitrage-free model. Models and skews split value between a same-strike call and put but cannot change their difference, and the two should imply the same volatility under consistent assumptions.
• With dividends, C − P = S − PV(D) − PV(K), or S e^(−qT) − K e^(−rT) with a yield. In general, C − P = e^(−rT)(F − K): a call minus a put is a forward purchase at the strike.
• Every synthetic position has a financing leg. A long call and short put at the same strike is stock bought with borrowed money; at the money, its net debit is the interest.
• A conversion is a synthetic T-bill and a reversal a synthetic loan. At executable prices they reveal implied lending and borrowing rates, which usually leave no arbitrage.
• For American options with nonnegative rates, parity becomes S − PV(D) − K ≤ C − P ≤ S − PV(K). Without dividends, the band’s width is the discount on the strike; dividends widen it by PV(D).
• Apparent violations usually reflect borrow fees, financing differences, dividends, early exercise, or stale quotes. A mathematical identity is not an executable arbitrage.

Primary sources

• Stoll, H. R. (1969). The Relationship Between Put and Call Option Prices. Journal of Finance, 24(5), 801–824.
• Merton, R. C. (1973a). The Relationship Between Put and Call Option Prices: Comment. Journal of Finance, 28(1), 183–184.
• Merton, R. C. (1973b). Theory of Rational Option Pricing. Bell Journal of Economics and Management Science, 4(1), 141–183.
• Klemkosky, R. C., and Resnick, B. G. (1979). Put-Call Parity and Market Efficiency. Journal of Finance, 34(5), 1141–1155.
• Kamara, A., and Miller, T. W. (1995). Daily and Intradaily Tests of European Put-Call Parity. Journal of Financial and Quantitative Analysis, 30(4), 519–539.
• Ofek, E., Richardson, M., and Whitelaw, R. F. (2004). Limited Arbitrage and Short Sales Restrictions: Evidence from the Options Markets. Journal of Financial Economics, 74(2), 305–342.
• Cremers, M., and Weinbaum, D. (2010). Deviations from Put-Call Parity and Stock Return Predictability. Journal of Financial and Quantitative Analysis, 45(2), 335–367.
• Haug, E. G., and Taleb, N. N. (2011). Option Traders Use (Very) Sophisticated Heuristics, Never the Black–Scholes–Merton Formula. Journal of Economic Behavior & Organization, 77(2), 97–106.
• van Binsbergen, J. H., Diamond, W. F., and Grotteria, M. (2022). Risk-Free Interest Rates. Journal of Financial Economics, 143(1), 1–29.
• Muravyev, D., Pearson, N. D., and Pollet, J. M. (2022). Is There a Risk Premium in the Stock Lending Market? Evidence from Equity Options. Journal of Finance, 77(3), 1787–1828.
• Muravyev, D., Pearson, N. D., and Pollet, J. M. (2025). Why Does Options Market Information Predict Stock Returns? Journal of Financial Economics, 172, 104153.
• Options Industry Council: Put/Call Parity · Synthetic Long Stock
• Cboe: Strategy-Based Margin · Cboe Volatility Index Methodology
• Montréal Exchange: Kazemi, K. (2019). Stock Financing and Option Prices
• SEC: Key Points About Regulation SHO · Release 34-99641, OCC interpretive guidance on cash dividend adjustments