Part V · Options · Chapter 5F

Put-call parity and the risks at expiry

Synthetics, pin risk and assignment

Where we are

Chapters 5A to 5E built an option's price from its payoff, its Greeks and its volatility, to which this chapter adds two related ideas. The first is put-call parity, a no-arbitrage identity that ties every call to every put with the same strike and expiration. It explains why many "different" strategies are economically identical. The second is a survey of the risks that do not show up in the Greeks but absolutely show up in your P&L.

What you need first

The four basic positions, European and American exercise, and the contract multiplier (Chapter 5A).
Intrinsic value and time value (Chapter 5B).
Gamma's spike at the strike as expiry nears (Chapters 5C and 5D).
Present value with continuous compounding, $e^{-rT}$ (Chapter 1A).
Margin, margin calls and forced liquidation (Chapter 1E).

1 · Put-call parity

Put-call parity links the prices of a European call, $C$ , and a European put, $P$ , on the same underlying with the same strike $K$ and time to expiry $T$ . With spot $S$ and the risk-free rate $r$ , it says

C - P = S - K e^{-rT}

(5F.1)

Adding a dividend yield $q$ gives the more general form:

C - P = S e^{-qT} - K e^{-rT}

(5F.2)

The term $K e^{-rT}$ is the present value of the strike: the cash that, invested at the rate $r$ , grows to exactly $K$ by expiry (Chapter 1A).

The intuition is that a portfolio of a long call, a short put and cash equal to the present value of $K$ replicates exactly the payoff of holding the underlying at expiration. By no-arbitrage, two positions with the same payoff must cost the same, so the cost of the portfolio must equal the cost of the underlying. The table checks the payoff at expiry, where $S_T$ is the underlying's price then.

position	payoff if $S_T < K$	payoff if $S_T \ge K$
long call	0	$S_T - K$
short put	$-(K - S_T)$	0
cash grown to $K$	$K$	$K$
total	$S_T$	$S_T$

In both columns the three legs add up to one unit of the underlying. Their costs today, $C - P + Ke^{-rT}$ , must therefore equal $S$ , which rearranges to (5F.1).

Worked example: Parity on a three-month option

Step 1: the present value of the strike. Take $S = 100$ , $K = 100$ , $r = 4\%$ and $T = 0.25$ . The strike's present value is $100 \times e^{-0.01} = 99.00$ , so parity requires $C - P = 100 - 99.00 = 1.00$ .

Step 2: price the put from the call. If the call trades at $4.50, the put must trade at $4.50 - 1.00 = \$3.50$ , to the nearest cent.

Step 3: an arbitrage, if parity breaks. Suppose the put traded at $3.00 instead. Selling the call for $4.50, buying the put for $3.00, buying the underlying for $100 and borrowing $99.00 leaves $0.50 in hand today. At expiry the put and the short call together deliver the underlying at exactly $100 whatever happens, which repays the loan. The $0.50 is a riskless profit from a trade called a conversion (Chapter 12E), which traders compete away by pushing the prices back into line.

Synthetic positions

Rearranging parity gives the synthetic equivalents: every combination of options and stock has a synthetic position, a counterpart constructed differently with the same payoff. In each row, the parity form says which legs to hold. A plus sign means long and a minus sign short, with $Ke^{-rT}$ standing for cash lent (plus) or borrowed (minus).

synthetic position	construction	parity form	equivalent to
long stock	long call, short put (same $K$ ), cash	$S = C - P + Ke^{-rT}$	the underlying
long call	long stock, long put	$C = S + P - Ke^{-rT}$	the "protective put"
long put	short stock, long call	$P = C - S + Ke^{-rT}$	a synthetic protective short
short put	long stock, short call	$-P = S - C - Ke^{-rT}$	the "covered call"
short call	short stock, short put	$-C = Ke^{-rT} - S - P$	risky
short stock	short call, long put	$-S = P - C - Ke^{-rT}$	used when borrow is hard

The most operationally important equivalence is that a covered call is identical in payoff to a short cash-secured put at the same strike. Both positions:

Receive a credit up front.
Have capped upside: at the strike for the covered call, at the premium for the short put.
Have large downside as the underlying falls, all the way to a price of zero.
Have identical net Greeks.

If you think covered calls are "conservative" but selling puts is "reckless", you are being fooled by labels. They are the same payoff, since naming things differently does not change what you own. The choice between them is operational, not economic: a covered call requires you to own the stock, while a cash-secured put requires you to have cash collateral.

Worked example: A covered call and a cash-secured put

Take a stock at $100 and options struck at 100, with no interest rate, so that parity makes the call and the put cost the same, say $4.00 each.

stock at expiry	covered call	cash-secured put
$80	$80 - 96 = -16$	$4 - 20 = -16$
$100	$100 - 96 = +4$	$4 - 0 = +4$
$120	$100 - 96 = +4$	$4 - 0 = +4$

Step 1: the covered call. You pay $100 for the stock and collect $4 for the call, a net outlay of $96. At expiry you hold stock worth $S_T$ , less anything above $100 that the short call takes away.

Step 2: the cash-secured put. Having collected $4 and set aside $100 of cash, at expiry you keep the $4 less whatever the put pays its holder, $\max(100 - S_T, 0)$ .

Step 3: compare. The two columns match at every price, as do the deltas: the covered call's is $1 - \Delta_{\text{call}}$ and the short put's is $-\Delta_{\text{put}}$ , which parity makes the same number.

2 · Put-call parity in practice

Put-call parity provides one of the cleanest sanity checks in options trading. If parity does not hold, an arbitrage is available. In liquid markets, parity holds tightly enough that no retail trader can exploit deviations. Monitoring it is still useful: comparing the bid-ask of a synthetic position with the bid-ask of its direct equivalent shows which is the cheaper execution route to the same exposure.

Suppose you want long stock exposure but the share borrow is expensive, as in a short squeeze in progress. The synthetic long, a long call and a short put, can then be cheaper. Parity shows why: a fee $b$ a year for borrowing shares enters (5F.2) the way a dividend yield does, lowering $C - P$ , so puts cost more relative to calls. Selling that expensive put is how the synthetic long captures the difference. At a hypothetical borrow fee of 20% a year, a three-month synthetic long on a $100 stock costs $100 \times (1 - e^{-0.05}) = \$4.88$ less than the shares.

Similarly, if you want to hedge with a protective put but the put's spread is wide, the equivalent structure of short stock and a long call sometimes has tighter execution. These are edge cases, but they exist.

3 · Early exercise of American options

European-style options exercise only at expiration, while American-style options can exercise at any time before it. Although this optionality has economic value, holders exercise early only when it is rational, which takes specific conditions.

Early exercise of American puts is rational when:

The put is deep in the money, so little time value remains.
Interest rates are non-trivial, so receiving the strike's cash earlier is worth more than holding the option.
The dividend yield is low, so the opportunity cost against the short-stock equivalent is low.

A rule of thumb says that early exercise is rational for a deep ITM put with very low time value (under $0.50) whose strike would earn more than 2% a year in interest on the released cash. A 0.30-delta put with $3 of time value never gets exercised early. Chapter 12H states the test as a comparison: exercise when the strike times the interest rate times the time to expiry exceeds the remaining time value.

Early exercise of American calls is rational only before large dividends. For non-dividend-paying stocks, early exercise of an American call is strictly suboptimal, because you give up time value and get nothing in return. The exception is the dividend: a call holder who exercises before the ex-dividend date receives the dividend, so early exercise can be rational when

\text{dividend} > \text{time value of the call}

(5F.3)

For deep ITM American calls with low time value and a large upcoming dividend, this condition fires.

Worked example: Two early exercises

Step 1: a deep ITM put. A put struck at $100 on a stock at $60 has three months left and $0.20 of time value, with rates at 4%. Exercising now releases $100 of cash, which earns about $100 \times 0.04 \times 0.25 = \$1.00$ of interest by expiry. That is more than the $0.20 of time value given up, so exercise is rational.

Step 2: a deep ITM call before a dividend. A call struck at $100 on a stock at $150 trades at $50.30, which is $0.30 of time value, on the day before a $1.00 dividend goes ex. Exercising today captures the $1.00 at the cost of the $0.30, so by (5F.3) the holder gains $0.70 by exercising.

For short-option positions, these conditions mean that a trader short a deep ITM American put can wake up to find it exercised early, which converts the short put into long stock at the strike. That is generally fine, since cash becomes stock at a known price, although it ends the option's continued role. A trader short a deep ITM American call on a dividend stock may face early assignment before the ex-dividend date. That leaves the trader short stock and owing the dividend: rare, but operationally important.

The SPX, XSP and RUT exemption: European-style index options (SPX, XSP, RUT, NDX) have no early exercise. This eliminates an entire class of operational risk relative to SPY, which is American-style, and relative to individual-stock options.

4 · The other risks: the ones the Greeks miss

The Greeks describe sensitivities to smooth changes in price, time and volatility. They do not describe discontinuous events, structural-market risks or path-dependent dynamics. Each of the risks below has destroyed more retail accounts than missed delta calls.

Pin risk

Pin risk is the chaos that descends on option sellers when the underlying closes near a strike at expiration. In a perfectly rational world, the holder exercises an option that is in the money by even one cent and lets expire one that is out of the money by any amount. In practice, the final-minute price fluctuates and settlement prints may differ from the close. You go home on Friday not knowing whether you are to be assigned. Come Monday morning, you might wake up with 100 shares per contract, long or short, that you did not plan for and that are fully exposed to the weekend gap.

Chapter 5D showed why the strike is so treacherous at the end: color compresses gamma into a spike there, so the smallest move flips the option between in and out of the money.

The defense against pin risk is to close any short option trading near its strike in the final hour before expiration. The small premium you give up by closing early is dwarfed by the cost of an unplanned weekend position. Structurally, you can instead trade cash-settled index options (SPX, XSP), which have no physical delivery and therefore no pin risk in the traditional sense. They have a related risk: the settlement print can differ from the previous close. For the standard monthly SPX options that print is the SOQ, set on the morning of expiry, and Chapter 12H gives the expiration timeline.

The figure sets up a short call or put at one strike and lets you watch expiry arrive, pricing with a rate of 4% and no dividends. The sliders set the short strike (400 to 700, starting at 580), the current spot, the hours to expiry (0.5 to 48, starting at 24) and the IV (5% to 80%, starting at 20%). Spot starts at 580 and stays within $20 of the strike. The curve is the option's gamma across spot from 3% below the strike to 3% above, scaled to its own peak. The readouts give the call's delta at the current spot, a "pin probability" and the distance from the strike. The table below moves spot by weekend gaps of 1% to 3% either way and says which short option then expires in the money and is assigned.

The "pin probability" is a rough gauge, $1 - 2\,|\Delta - 0.5|$ , where $\Delta$ is the call's delta. It reads 100% at a delta of 0.5 and zero as the delta reaches 0 or 1, which makes it a gauge of closeness rather than a probability of anything. Run the hours down from 24 to 0.5 with spot at the strike, then set spot $2 above the strike and run them down again.

short strike K $580

current spot $580.0

hours to expiry 24h

IV σ 20%

call delta at S = 0.506

pin probability = 98.7%

distance from K = 0.00%

Weekend gap	Spot	Call	Put
+1% gap	585.80	ITM · assigned	OTM
−1% gap	574.20	OTM	ITM · assigned
+2% gap	591.60	ITM · assigned	OTM
−2% gap	568.40	OTM	ITM · assigned
+3% gap	597.40	ITM · assigned	OTM
−3% gap	562.60	OTM	ITM · assigned

Gamma at the strike as expiry closes in. A short call or put at strike K with spot within $20 of it. The curve is the option's gamma across spot from 3% below the strike to 3% above, scaled to its own peak: run the hours to expiry down and it narrows to a spike at the strike, where a small move flips the option between in and out of the money. The table moves spot by each weekend gap and says which short option then expires in the money and is assigned. The pin probability is a rough gauge, 1 − 2 × |Δ − 0.5|: 100% at a call delta of 0.5, zero as the delta reaches 0 or 1.

At the starting settings, with spot on the strike and 24 hours left, the call's delta is 0.506 and the gauge reads 98.7%. As the hours run down, the gamma curve narrows to a spike at the strike. With spot $2 above the strike, the gauge reads 73% at 24 hours. At half an hour the delta has reached 0.99 and the gauge only 2.3%: the outcome is all but settled unless spot moves. With spot on the strike, every row of the gap table puts one of the two in the money and assigned: the call after an up-gap, the put after a down-gap.

Worked example: What a pin costs over a weekend

Step 1: the assignment. You are short one SPY put struck at 580, and SPY closes Friday just below 580. The put is assigned, leaving you 100 shares bought at $580, or $58,000 of stock you did not plan to hold.

Step 2: the gap. SPY opens Monday about 2% below the strike, at $580 \times 0.98 = 568.40$ . The shares have lost $(580 - 568.40) \times 100 = \$1{,}160$ before you can trade.

Step 3: a stress weekend. A 5% gap, at the top of the occasional 2% to 5% range for SPY, costs $580 \times 0.05 \times 100 = \$2{,}900$ per contract, on a put that was worth a few cents in Friday's final minutes.

Familiar ground

Pin risk is the equivocal result at a diagnostic cut-off. A biomarker measured far from its threshold classifies a patient reliably. A value sitting on the cut-off can fall either side on a repeat assay, because small measurement noise decides the call. An option at its strike in the last minutes is in that position. The price is the assay and the strike the cut-off, while the noise of the final prints decides whether you are assigned.

Early exercise

American-style options can be exercised at any time, not just at expiration. The holder of an ITM call has an economic incentive to exercise early just before a dividend, to capture the dividend at the cost of the time value forgone. The holder of a deep ITM put has an incentive to exercise early to receive the strike's cash and reinvest it at the risk-free rate. Short positions can be assigned without warning when this happens.

For European-style, cash-settled index options (SPX, XSP, NDX, VIX, RUT), early exercise is impossible: the contract structure precludes it. This is one of several structural advantages of trading SPX and XSP for systematic income strategies.

Gap risk

Gap risk is the risk that the underlying opens at a price meaningfully different from the previous close, leaving a gap on the chart. Short positions held overnight or over weekends are exposed to the full gap, while long positions are insulated by their bounded loss. Gap risk is large for single stocks, where earnings releases produce gaps of 10% to 20% regularly. It is modest for SPY, whose typical Monday gap is 0.2% to 0.5%, occasionally 2% to 5% on stress weekends. It is zero for cash-settled SPX options at expiry, where the settlement reference is a single print (the SOQ, for the monthly options) rather than a continuous price. No shares are delivered to carry through a weekend.

For physically settled SPY positions held into expiry, gap risk includes the weekend between Friday's settlement and Monday's open. You receive shares on Friday and absorb Monday's gap in full, as in the worked example above.

Liquidity risk

Liquidity risk is the risk that bid-ask spreads widen dramatically, or that depth disappears, at exactly the moment you need to exit. It is small for SPY ATM options on a typical day and large for deep OTM single-name options in stress. In the March 2020 drawdown, bid-ask spreads on liquid index options expanded to 10 times their typical width during the most volatile sessions. Traders attempting to roll positions during those sessions faced execution costs that wiped out months of accumulated premium.

Worked example: A roll at ten times the spread

Step 1: normal conditions. Suppose an option's spread is normally $0.20 wide, so each trade at the far side of the quote costs $0.10 against the midpoint, or $10 per contract on a multiplier of 100.

Step 2: stress. At 10 times the width the spread is $2.00, so each trade costs $1.00 against the midpoint, or $100 per contract.

Step 3: the roll. A roll closes one option and opens another, so it crosses the spread twice. On 10 contracts it costs $2 \times 100 \times 10 = \$2{,}000$ in stress, against $200 normally.

The defense is to avoid trading positions you cannot afford to hold to expiration. Avoid OTM strikes with low open interest (OI). Plan to close positions during normal liquidity conditions, such as mid-day with the VIX low, not during stress.

Borrow cost

For strategies involving short stock, the cost to borrow shares can be substantial, particularly for stocks under short-squeeze pressure. Retail traders rarely short stock directly, although short stock underlies certain synthetic positions. GameStop in early 2021 had borrow rates exceeding 100% annualized at peak. The borrow cost, paid by the short side, is invisible in standard option pricing models but very real in execution. Parity shows where it goes: it lowers $C - P$ , as the synthetic-long example above showed.

Margin expansion risk

The discussion of portfolio margin in stress, in Chapters 1E and 2E, applies with full force to options. Position sizing that is comfortable in calm conditions can become forced-liquidation territory in stress, as margin requirements expand. The retail discipline is never to run margin-leveraged positions whose stress-scenario losses would force liquidation. Use cash-secured strategies, or pre-committed kill rules, to avoid it.

Counterparty risk

Counterparty risk is the risk that the other side of a contract fails to pay. For exchange-listed options cleared through the OCC, this is effectively zero: the OCC has never failed to deliver in its history. For OTC options the risk is real, which is the primary reason most retail traders should never trade OTC products (Chapter 4A).

In practice

Before expiry week, sort each short option you hold by how it settles. A European, cash-settled index option such as SPX or XSP cannot be exercised early and delivers no shares. Its remaining expiry risk is the settlement print, which for the monthly SPX options is the SOQ. An American, physically settled option on SPY or a single stock carries all of the risks above: early assignment, pin risk at the close and the weekend gap on any shares delivered.

For the second kind, decide in advance what to do in the final hour. Close shorts near the strike rather than wait for the settlement print. Multiply every assignment by 100 shares a contract before you trade (Chapter 5A), so that you know how much stock could arrive on Monday.

When it fails

Parity is exact for European options with no frictions. For American options, early exercise loosens it from an equation to a range. Dividends and borrow fees shift it, as (5F.2) shows, while the bid-ask spread on each leg blurs it further. Closing a short option in the final hour assumes a market to close into. In stress, liquidity risk and pin risk can arrive together, so the exit costs more than the calm-day spread.

Decision rule

Close short options near the strike in the final hour before expiry, unless they are cash-settled and you accept the settlement print.
For systematic income strategies, prefer European, cash-settled index options (SPX, XSP), which remove early exercise and delivery.
Before an ex-dividend date, expect a short deep ITM American call to be assigned whenever the dividend exceeds its time value. Close or roll it first if you do not want to be short the stock.
Size every short-option position for the stressed margin requirement, not the calm one, or hold it cash-secured.

Common mistake: "covered calls are conservative, selling puts is reckless"

A covered call and a cash-secured put at the same strike have the same payoff, the same downside and the same Greeks. If one is too risky for your account, so is the other. Choose between them on operational grounds, such as whether you already own the stock or would rather post cash (Chapter 2E).

Common mistake: "cash-settled means no expiry risk"

Cash settlement removes delivery and the weekend gap on delivered shares without removing the settlement print. For the monthly SPX options the print is the SOQ, which can differ from the previous close. A short option that was out of the money on Thursday's close can settle in the money on Friday morning.

What to carry forward

Put-call parity, $C - P = S - Ke^{-rT}$ , ties every European call to the put with the same strike and expiry. Every position has a synthetic twin with the same payoff.
A covered call and a cash-secured put at the same strike are the same position under two names.
American options can be exercised early: puts when deep in the money and rates are material, calls just before a dividend larger than their time value. SPX, XSP, NDX, VIX and RUT options are European and cannot be.
The Greeks miss pin risk, early exercise, gap risk, liquidity risk, borrow cost, margin expansion and counterparty risk. Cash-settled European index options remove several of them, though not the settlement print.

5 · Check yourself

Question 1

A stock is at $50, and European options struck at 50 have six months left, with a rate of 4% and no dividends. If the call costs $3.50, what must the put cost?

Answer

The strike's present value is $50 \times e^{-0.02} = 49.01$ , so parity requires $C - P = 50 - 49.01 = 0.99$ . The put must cost $3.50 - 0.99 = \$2.51$ .

Question 2

A friend who sells covered calls on a stock calls it conservative. The same friend refuses to sell cash-secured puts on that stock at the same strike, because that is "too risky". What is wrong with this reasoning?

Answer

By put-call parity the two positions have the same payoff: a credit up front, the same capped gain and the same loss if the stock falls. Their Greeks are the same too, leaving only operational differences, such as needing to own the shares for the covered call.

Question 3

You are short an American put struck at $100 with six months left. The stock is at $70, the put has $0.15 of time value and rates are 5%. Should you expect early assignment?

Answer

Yes, you should: exercising releases $100 of cash, which earns about $100 \times 0.05 \times 0.5 = \$2.50$ of interest by expiry, far more than the $0.15 of time value the holder gives up. It is also under the rule of thumb's $0.50 of time value.

Question 4

A deep ITM American call has $1.20 of time value, and a $0.80 dividend goes ex tomorrow. Should its holder exercise?

Answer

No: by (5F.3) exercise pays only when the dividend exceeds the time value, whereas here the holder would give up $1.20 to collect $0.80.

Question 5

You are short five SPY puts struck at 580 when SPY closes Friday at 579 and then opens 3% lower on Monday. What has the weekend cost you?

Answer

The puts are in the money at the close, so you are assigned 500 shares at $580. Monday's open is $579 \times 0.97 = 561.63$ , so the shares are worth $(580 - 561.63) \times 500 = \$9{,}185$ less than you paid, before any premium you collected. With cash-settled XSP options at the same strike, the loss would have been fixed by the settlement print, with no shares to carry through the weekend.