Put-Call Parity and Arbitrage: Conversions, Reversals, Box Spreads, and Early Exercise

Put-call parity is the rule that, for European options with the same strike and expiry, a call minus a put must equal a forward position in the underlying: the stock price minus the present value of the strike (and of any dividends). It holds because a call plus cash and a put plus stock pay off identically at expiry, so any gap between them can be locked in without taking a view on the market.

In short: parity says C - P = S - PV(dividends) - PV(K). Professionals trade it in three ways. A conversion (long stock, long put, short call) and a reversal (the opposite) are synthetic loans whose interest rate is set by option prices. A box spread combines two parity trades at different strikes into a pure zero-coupon bond, and on European-style SPX options it is a real lending and borrowing market: Cboe reported box yields averaging about 32 basis points over SOFR, rising to 69 basis points in early August 2026. For American options parity becomes a range rather than an equation, because early exercise and dividends give one side extra value.

This deep dive assumes you know how calls and puts pay off and what the Black-Scholes formula does (see Black-Scholes explained). It describes mechanisms used by market makers and institutions. It is not a recommendation, and the “arbitrages” described here are, in practice, thin margins competed away by firms with lower costs than any individual.

A common misconception: “a put is just the opposite of a call”

A put is not the mirror image of a call in the sense that one is cheap when the other is dear for unrelated reasons. Calls and puts on the same strike and expiry are tied together by arithmetic. If you know the stock price, the interest rate, the dividends, and the call price, you know the European put price. That is why traders say there is only one implied volatility per strike: if parity holds, the call and put at a strike must imply the same volatility, and a quote that implies different volatilities for the two is usually stale, or is pricing something the simple formula leaves out, such as borrow costs, dividends, or early exercise.

The derivation

Compare two portfolios held to the expiry date T on a stock that pays no dividends:

Portfolio A: one European call with strike K, plus cash equal to the present value of K, invested at the risk-free rate.
Portfolio B: one European put with strike K, plus one share.

At expiry:

Stock at expiry	Portfolio A (call + cash)	Portfolio B (put + share)
Below K	0 + K = K	(K - S) + S = K
Above K	(S - K) + K = S	0 + S = S

Both are worth the larger of S and K in every state. Two portfolios with identical payoffs and no cash flows in between must cost the same today, or someone buys the cheap one, sells the dear one, and pockets the difference. So:

C + K x e^(-rT) = P + S
C - P = S - K x e^(-rT)

With known cash dividends D paid before expiry, the share in portfolio B collects the dividend and the option holders do not, so the stock’s price is reduced by the dividends’ present value. With a forward price F for the same date the relation becomes compact:

C - P = S - PV(D) - K x e^(-rT)  =  e^(-rT) x (F - K)

The derivation used no model of how the stock moves: no volatility, no probability, no Black-Scholes. That is what makes parity a no-arbitrage condition rather than a pricing model.

Synthetic positions

Rearranging parity gives every position as a combination of the other three instruments:

Position	Built from
Synthetic long stock	Long call, short put, plus lending PV(K)
Synthetic short stock	Short call, long put, minus PV(K)
Synthetic long call	Long put, long stock, borrowing PV(K)
Synthetic long put	Long call, short stock, lending PV(K)

Market makers use these equivalences constantly. A desk that buys calls from customers can hedge by selling puts and buying stock, and whether it prefers that to buying the call back depends on which side of parity is cheaper after costs.

Worked example: checking parity

Hypothetical Maple Widgets Inc. trades at $50 with no dividends. Three-month options (T = 0.25), strike $50, 4% rate. At 30% volatility, Black-Scholes gives a call of $3.2297 and a put of $2.7322.

C - P = 3.2297 - 2.7322 = $0.4975
S - K x e^(-rT) = 50 - 50 x e^(-0.01) = 50 - 49.5025 = $0.4975

They match to the fourth decimal, as they must: Black-Scholes builds parity in. The more useful point is the reverse. If the market quoted a call at $3.2297 and a put at any price other than $2.7322, the put would imply a different volatility, and in principle, before costs, the difference could be locked in.

Now add a $0.50 dividend paid in one month. Its present value is 0.50 x e^(-0.04/12) = $0.4983, so parity now requires C - P = 50 - 0.4983 - 49.5025 = minus $0.0008. A dividend almost exactly the size of the strike’s interest cost erases the call’s premium over the put at the money.

Conversions and reversals: synthetic loans

A conversion buys the stock, buys the put, and sells the call at the same strike and expiry. At expiry the position is worth exactly K whatever happens: if the stock is above K the call is assigned and the stock is delivered at K; if below, the put is exercised and the stock is sold at K. Paying S + P - C today to receive K at T is lending money. A reversal (or reverse conversion) is the mirror image: short stock, sell the put, buy the call, receiving cash today and paying K at expiry, which is borrowing.

The rate on that loan is set by option prices. Suppose the Maple Widgets quotes are:

Instrument	Bid	Ask
Stock	49.99	50.01
50 call (3 months)	3.30	3.40
50 put (3 months)	2.55	2.65

Conversion (buy stock at 50.01, buy put at 2.65, sell call at 3.30):

Cost today: 50.01 + 2.65 - 3.30 = $49.36
Receive at expiry: $50.00
Implied lending rate: ln(50 / 49.36) / 0.25 = 5.15% continuously compounded
Against a 4% rate, the present value of $50 is $49.5025, so the position locks in about $0.1425 per share, or $14.25 per contract before commissions, borrow, and exchange fees

Reversal (short stock at 49.99, sell put at 2.55, buy call at 3.40):

Received today: 49.99 + 2.55 - 3.40 = $49.14
Pay at expiry: $50.00
Implied borrowing rate: ln(50 / 49.14) / 0.25 = 6.94%, which loses about $36.25 per contract against a 4% cost of money

These quotes are deliberately mispriced to show the mechanics; the call bid sits above fair value and the put ask below it. In a liquid market, conversion and reversal desks compete such gaps down to the level of their own costs, which is why individuals with retail commissions and borrow costs almost never see a true arbitrage. What remains visible is information: the rate implied by conversions and reversals tells you what the options market thinks the cost of carrying the stock is.

When parity looks broken: hard-to-borrow stocks

A reversal needs a short sale. When a stock is expensive or impossible to borrow, the short seller earns less than the full interest rate on the proceeds (the rebate is lower), and puts become dear relative to calls. Eli Ofek, Matthew Richardson and Robert Whitelaw studied this in “Limited Arbitrage and Short Sales Restrictions: Evidence from the Options Markets” (Journal of Financial Economics, 2004, volume 74, pages 305-342). Their abstract reports that apparent parity violations “are asymmetric in the direction of short sales constraints,” that their size is “strongly related to the rebate rate spread,” and that they persist after transaction costs in both the options and stock-lending markets. They also found the violations predicted future stock returns. A wide put premium on a hard-to-borrow stock is the market pricing the borrow, not a free trade.

Box spreads: parity as a bond

A box spread combines a conversion at one strike with a reversal at another, which cancels the stock:

Long the K1 call and short the K1 put (a synthetic long at K1)
Short the K2 call and long the K2 put (a synthetic short at K2), with K2 above K1

Equivalently, it is a bull call spread plus a bear put spread on the same strikes. At expiry it pays K2 - K1 in every state. That is a zero-coupon bond. Its price today must be (K2 - K1) x e^(-rT), whatever volatility each strike trades at. We checked this on a hypothetical one-year box on an index at 6,500 (4.2% rate, 1.3% dividend yield) using a 22% volatility at the 6,000 strike and 15% at 7,000 (a steep skew): the 6,000 call and put priced at 928.19 and 265.36 index points, the 7,000 call and put at 261.99 and 558.03, and the box came to 928.19 - 265.36 - 261.99 + 558.03 = 958.87, exactly 1,000 x e^(-0.042). Skew cancels out.

Why SPX boxes in particular

The Options Industry Council paper “Box Spreads: Exchange-listed Options Strategies for Borrowing or Lending Cash” (published by the Options Clearing Corporation) explains why the market uses S&P 500 index (SPX) options: they are European-style, which “ensure[s] that the box spread cannot be exercised early which would result in the cancellation of the effective loan before the term date.” Cboe’s SPX specifications confirm the details that matter for a box:

Exercise: European; SPX options “generally may be exercised only on the expiration date.”
Settlement: cash, on the business day after expiration. Standard monthly SPX settles to the SET, calculated from the opening prices of the component stocks on the expiration date.
Size: multiplier of 100, so a 1,000-point box pays $100,000 at expiry.
Position limits: Cboe states no position and exercise limits apply to SPX.
Tenor: near-term monthly expirations plus long-dated LEAPS expiring several years out, which allows multi-year loans.

Cboe’s Quoted Spread Book lets market makers quote designated SPX box spreads electronically on 1,000-point strikes across the first four serial, first three quarterly, and first three December expirations. Effective February 2, 2026, Cboe moved those designated strikes from 5,000/6,000 to 6,000/7,000.

Worked example: the implied rate

The OIC paper gives a historical example: a September 1,000/2,000 SPX box with 46 days to expiry quoted at 999.35 bid, 999.40 offered. Buying at 999.40 to receive 1,000 is a simple annual rate of (1,000 - 999.40) / 999.40 x 365 / 46 = 0.48%; selling at 999.35 is 0.52%. Those are rates from a near-zero-rate era.

At current rate levels the arithmetic is the same. Hypothetical box prices on a 1,000-point SPX box:

Days to expiry	Box price	Pays at expiry	Simple annual rate	Continuously compounded
91	989.50	1,000	4.26%	4.23%
365	958.00	1,000	4.38%	4.29%
730	915.00	1,000	4.64%	4.44%

A borrower who sells ten one-year boxes at 958.00 receives 10 x 958.00 x 100 = $958,000 and owes $1,000,000 in a year: $42,000 of interest, a 4.38% rate. The position has four legs per box. Assuming, purely for illustration, $0.50 of all-in fees per contract per leg, ten boxes cost $20, which moves the one-year rate by about 0.2 basis points. On a single three-month box the same $2 of fees adds about 0.8 basis points (4.256% to 4.264%). Costs matter far more for small, short boxes.

What box rates tell you

Box rates sit slightly above Treasury bill yields, and the gap has been measured:

Cboe’s white paper page “SPX Box Spreads: Trading and Market Microstructure” reports $929 million in average daily notional volume in 2024, rates “typically 10-55 basis points above three-month U.S. Treasury yields,” and standardized 1,000-point strikes creating a $100,000 notional order size.
In its Macro Volatility Digest for the week of August 3, 2026, Cboe’s Mandy Xu wrote that daily notional box loan volume had more than tripled in three years to a record $1.7 billion in 2026, and that the box yield, usually about 32 basis points over three-month SOFR on average, had more than doubled to a multi-year high of 69 basis points “as dealer balance sheets have gotten more constrained even as demand for borrowing has increased.”
Academically, Jules van Binsbergen, William Diamond and Marco Grotteria built a benchmark risk-free rate from SPX put-call parity (“Risk-Free Interest Rates,” Journal of Financial Economics, 2022, volume 143, pages 1-29). Over 2004-2018 they estimate the Treasury convenience yield (how much less Treasuries pay than a truly risk-free claim because they are money-like) at about 40 basis points on average, 65 basis points below three months, and much larger in 2008.

So the box-Treasury spread is not an arbitrage. It is the price of holding a claim that is safe (OCC-guaranteed) but less useful as collateral than a Treasury, plus whatever premium constrained dealer balance sheets demand.

The OIC paper also notes potential 60/40 long-term/short-term treatment under Section 1256 of the US tax code for SPX options. The interest on a box shows up as a capital gain or loss rather than interest. That is a reason some participants prefer boxes, and the rules are covered in how options are taxed in the US. This is not tax advice.

American options: parity becomes a range

For American options, the derivation breaks because one side can be exercised early. Without dividends, an American call on a stock is never worth exercising early (exercising throws away the interest on the strike and the put-like protection), so it is worth the same as the European call. An American put can be worth exercising early, so it is worth more than the European put. The result is a pair of bounds:

S - K  <=  C(American) - P(American)  <=  S - K x e^(-rT)

We computed the Maple Widgets options with a 2,000-step Cox-Ross-Rubinstein binomial tree (the method is explained in pricing models beyond Black-Scholes), 30% volatility, 4% rate, three months:

Strike	European put	American put	Early-exercise premium	American C - P	Bounds: S - K to S - PV(K)
50	2.7319	2.7706	0.0387	0.4588	0 to 0.4975
55	5.8393	5.9517	0.1124	-4.5652	-5.0000 to -4.4527
60	9.9109	10.1658	0.2550	-9.6579	-10.0000 to -9.4030
65	14.5152	15.0000	0.4848	-14.8381	-15.0000 to -14.3532

The American C - P sits inside the bounds every time, and the deeper the put, the closer it moves to the lower bound S - K. At the 65 strike the American put is worth exactly its exercise value of $15.00: the tree says exercise now. A trader who applied European parity to these American quotes would conclude the put is overpriced by up to 48 cents and might sell it in a “conversion,” only to be assigned early.

Dividends and the American call

Dividends reverse the logic for calls. The stock drops by the dividend on the ex-date, and the call holder does not receive it, so a deep in-the-money call can be worth more exercised the day before. With dividends, the bounds widen:

S - PV(D) - K  <=  C(American) - P(American)  <=  S - K x e^(-rT)

The early-exercise test the day before the ex-date is: exercise if the dividend is larger than the remaining time value you would give up. Take Maple Widgets at $50 the day before a $1.00 ex-dividend date, with one month left after that date:

Exercise the 40-strike call now: worth $10.00
Hold it: after the dividend the stock is about $49, and a one-month 40 call at 30% volatility is worth $9.14
Exercise wins by about 86 cents. Equivalently, the $1.00 dividend exceeds the interest on the strike over the last month (40 x (1 - e^(-0.04/12)) = $0.13) plus the 40 put’s value ($0.01).

The same test with a $0.25 dividend on the at-the-money 50 call says hold ($1.68 of option value against $0 of exercise value), while on a 45 call with a $0.50 dividend exercise wins narrowly ($5.00 against $4.91). This is why market makers watch every in-the-money call ahead of ex-dates. A desk short such a call as part of a conversion should expect assignment before the ex-date: it delivers its stock, misses the dividend it had priced in, and is left holding only the put. A desk long such a call in a reversal that fails to exercise stays short stock over the ex-date and owes the dividend, giving up the difference. How assignment is allocated is covered in how exercise and assignment work.

What can go wrong

Early assignment on American legs. A box or conversion built from American-style options (for example, options on an ETF rather than on the SPX index) can be broken by early assignment of one leg, leaving an unhedged position or a stock delivery the account cannot carry. That is the main reason financing boxes use European SPX options.
Dividend surprises. Parity uses expected dividends. A special dividend, a cut, or a change in ex-date shifts the forward and with it the fair value of every conversion and reversal on the name.
Borrow recalls and rebate changes. A reversal’s short stock can be recalled or repriced. The Ofek, Richardson and Whitelaw evidence shows the largest apparent parity gaps are exactly where borrowing is hardest.
Settlement mismatch. Standard SPX settles to the SET, built from opening prices on expiration day, which can differ sharply from the prior close. A box is unaffected because every state pays the same, but a parity hedge against a futures or ETF position settling at a different time is not.
Legging risk. Executing four legs separately exposes you to the market moving between fills. Professionals trade boxes as single complex orders for this reason.
Funding, margin and clearing. The OIC paper notes that OCC applies a liquidity margin policy to concentrated expirations, beginning 30 days before expiry, because box sellers owe cash after expiration. Brokers also set their own margin treatment for short boxes, and an account that cannot meet a margin call can be forced out of a position whose payoff was never in doubt.

The short version

Put-call parity says a call minus a put equals the stock minus the present value of the strike and dividends. It is a no-arbitrage identity, not a model, and it ties every call to its put. Conversions and reversals turn it into synthetic lending and borrowing whose rate is set by option prices, and hard-to-borrow stocks show up as apparent parity violations that are really the cost of borrow. A box spread is two parity trades that cancel the stock, leaving a zero-coupon bond; on European SPX options it is a financing market, increasingly used by advisers, family offices and traders according to Cboe, priced a few tens of basis points above Treasuries, with the spread reflecting the Treasury convenience yield and dealer balance-sheet costs. For American options parity relaxes into a range, because early exercise of puts and of calls before dividends gives one side extra value.