Put–Call Parity and Synthetics

One identity, four instruments

Parity is easier to use than to prove. The identity is that a call minus a put, plus the strike in cash, is worth the stock: owning the call and writing the put gives you the same outcomes as owning the shares, because whichever way the stock finishes, one of the two contracts is in the money and the other is not. Buy the call at $4.20, sell the put at $3.60, and hold $100 of cash earning the risk-free rate; if the stock finishes at $130 the call is worth $30 and the put is worthless, so you own $130 of value having paid $100.60; if it finishes at $70 you are assigned on the put, buy the stock at $100, and own $70 of stock having parted with $100.60. Either way you have behaved exactly like a shareholder who bought at $100.60 rather than $100.00. That single identity generates the whole vocabulary of synthetics, which is worth building once rather than memorising. Long stock = long call + short put. Long call = long stock + long put. Long put = short stock + long call. And the mirrors: short stock = short call + long put, and so on. What the synthetic costs relative to the real thing is not an opportunity but the financing of the position, which is why the fully worked comparison matters: a $0.60 gap on 90 days at a 4% rate is *less* than the $0.99 of interest, so the market is trading inside the carry-implied band rather than outside it. Reading parity as a law that quotes must obey is the first mistake; reading it as a band whose width is set by financing, spreads and borrow is what makes it useful. The identity does real work in three places. It audits your own arithmetic: if you compute a call’s fair value from a put and the answer is wildly different from the quoted call, one of your four inputs is wrong, and the relationship localises which. It explains the quotes you see — a call and a put at the same strike look like different instruments with different prices, and parity shows they are the same instrument with the financing and the expected dividend taken out. And it underwrites the strategies later in this rung: a conversion, a reversal and a box are all parity rearranged, and the reason a market maker can quote a call at all is that they can hedge it with the stock, the put and cash rather than with an opinion about direction. The same position, written four ways — Synthetic long stock: call − put + strike: 4.20 − 3.60 + 100 = $100.60 against a $100.00 stock ← · Synthetic long call: stock + put: Buy the shares and the protection instead of the call · Synthetic long put: short stock + call: The protective-put position read backwards · Carrying cost of the real stock: $100 × 4.0% × 90 ÷ 365 = $0.99 — which is why the $0.60 gap is inside the band ← Read the last row carefully: the apparent mispricing is smaller than the cost of the position needed to exploit it. That is the ordinary condition, not a market failure — which is the difference between a textbook identity and a trade.

What widens the band

Four things. The first is the spread on both legs: with a call quoted 4.15/4.25 and a put 3.55/3.65, the “free” $0.60 becomes a range, and only the extreme edges of the range are candidates. The second is the borrow rate on the stock, which is the cost of being short it — a synthetic position frequently requires shorting one leg or the stock itself, and in a hard-to-borrow name that cost can dwarf the basis. The third is the dividend, which is the one that surprises newcomers: a stock that pays $2.00 inside the 90 days makes the call relatively less valuable, because the holder of the call does not receive the dividend while the holder of the shares does. Parity with dividends says the call should trade lower than the no-dividend value by the present value of the dividend, and a trader who tests parity without accounting for the ex-date will find a “mispricing” every quarter that is exactly the dividend. The fourth is execution and capital. Even a genuine basis gap has to be harvested with a position that ties up capital, requires margin, and can be held for months, which is why the annualised return on a basis trade is usually a small number on a large, leveraged balance sheet — the business of a market maker rather than a retail strategy. That is the last honest point of the lesson: parity is not a weapon, it is a map. It tells you what the quotes mean relative to each other and to the cost of money, and it tells you when your own arithmetic is wrong. Treat it as the check that runs before any structure in the rest of the subject, because every structure is a combination of these four instruments and therefore obeys the same relationship somewhere inside it. • Leg spreads: a basis range rather than a price. • Borrow cost: the price of being short the stock required by the synthetic. • Dividends: the call should trade lower by the present value of any dividend paid before expiry. • Capital and margin: the return on a basis trade is small on a large balance sheet. • Use it as the arithmetic check before a structure, not as a strategy. The classic error is testing parity at an ex-dividend date. A call that looks $1.80 cheap in the week before a $1.80 dividend has priced the dividend correctly and is not cheap at all — and the “arbitrage” that a trader constructs will be exactly the dividend they did not count. Before reading any parity gap as a mispricing, check three things in order: the ex-date inside the expiry, the borrow rate on the stock, and the spread on each leg.

A box is a bond: two identities, one loan

Parity links four instruments, so it can be applied twice at the same time to build something that contains no stock and no directional exposure at all. Buy a call and sell a put at a lower strike, then buy a put and sell a call at a higher one, and the four legs cancel every path the stock can take. What is left is a position that pays a fixed amount at expiry no matter where the underlying finishes — which makes it a bond, and makes its price an interest rate. The construction is easier to see as two spreads. A bullish call spread and a bearish put spread at the same two strikes, on the same expiry, is called a **box**. At expiry the stock is either above the upper strike, below the lower one, or somewhere between, and in all three cases the two spreads together deliver exactly the distance between the strikes. Nothing is left to chance, which is the point: a guaranteed future payment bought today is a loan, and what you pay for it is the repayment discounted at whatever rate the market is using. That rate is the interesting number. If a one-hundred-wide box trades for 98.80 with a year to run, the implied yield is about 1.2% — the rate at which the market will lend and borrow, expressed in options rather than in a money market. Compare that with what a Treasury bill of the same maturity yields and the gap is a cost: it is the financing and capital charge embedded in the option market, and it is usually a little wider than the risk-free rate, because the counterparty taking the other side is not a government. That is also why the box became famous as a retail financing tool. Borrowing against a portfolio at a broker’s margin rate can cost many percentage points; borrowing the same money through a box in an index option, where the settlement is cash and the legs are European, can cost less than a percent above the bill. The trade is real, and it is what turned a textbook arbitrage into a conversation about brokers closing positions they did not like. Its risks are operational rather than directional: a box should be held to expiry, and closing it early means paying the spread twice on four legs, which can exceed the interest saved. The distinction that decides whether a box is genuinely safe is **European versus American** settlement, and it is the same distinction that bounds parity itself. Parity in its clean form is a statement about European options, where early exercise does not exist, so the identity must hold exactly and any deviation is a financing rate. American options can be exercised at any moment, which gives the holder a privilege the European formula does not price, so an American option may trade slightly *above* its parity value and the small excess is not free money — it is the market charging for the right to act early. An index box in a European, cash-settled contract is the clean version; the same shape built from American equity options carries assignment risk on every leg. For a learner the payoff is a way of thinking rather than a trade to place. Once the four instruments are understood as one identity, every puzzle in the option market becomes an arithmetic question with a rate attached: why a deep position costs what it costs, what the market is charging to hold a synthetic, and whether the borrow is expensive enough to explain a strange quote. The box is simply the identity pushed until all the risk cancels — which leaves only money and time, the two things an interest rate is made of. • A bullish call spread plus a bearish put spread at the same strikes is a box: a guaranteed payment. • Its price is that payment discounted at the market’s financing rate, comparable with a bill. • The trade is real, and its risks are operational — an early close pays the spread on four legs. • Parity holds exactly for European options; American early exercise can push a contract above it, and that excess is not free money. The same identity read the other way explains a puzzle learners often hit: an American put trading above its parity value on a hard-to-borrow or high-dividend name. The excess is the right to exercise now, and the market prices it.

Sources

• Put–call parity, conversions and reversalsHull, “Options, Futures and Other Derivatives”
• Synthetic positions and the arbitrage band around parityNatenberg, “Option Volatility and Pricing”
• Dividends, borrow and the limits of arbitrageShleifer & Vishny, “The Limits of Arbitrage” (Journal of Finance)