Part VIII · The whole market · Chapter 8E

Pinning

Why prices stick to strikes at expiry, and when they are pushed away

On expiration dates, the closing prices of stocks with listed options cluster at strike prices more than on other days (Ni, Pearson and Poteshman 2005). The pull of a heavily held strike on the price near expiry is called pinning. It is the market-wide face of the pin risk of Chapter 5F, and it has a mechanism, published models and measured sizes. It also has a mirror image that vendors rarely mention: a strike can push the price away.

1 · The mechanism: two forces at one strike

Take dealers who hold a large net position in options at one strike $K$, all expiring today, and who hedge it through the day. Calls and puts at the same strike and expiry have the same gamma, and with no dividends the same charm, so only the net position matters. Two forces act on the price, and both flip with the sign of the position.

The charm force. Hold the price still a few points above $K$. As the minutes pass, a call's delta rises toward 1, and the put's rises toward 0, by the same amount. A dealer long those options gains delta through them without any move, and sells the underlying to stay flat. A few points below $K$ the deltas fall, and the dealer buys. Selling above the strike and buying below it pushes the price toward the strike from both sides. For a dealer short the options, every trade reverses and the push is away from the strike.

The gamma force. A dealer long gamma at $K$ trades against every move near the strike, and a dealer short gamma trades with it. This is Chapter 8D's feedback, confined to the prices near one strike. For a long position it divides each move near the strike by a factor that grows as expiry approaches, because gamma at the money grows like $1/\sqrt{T}$. For a short position it multiplies them.

The numbers in step 4 show why the last hour matters. The same 20,000 contracts carry about $11.7 billion of gamma per 1% move at the open, $30 billion with an hour left and $42 billion with 30 minutes left. With ten minutes left they carry $73 billion, above most of the Dealer Mechanics Lab's range for the whole market's GEX. That is Chapter 5D's color at work: gamma compressing into the strike as expiry nears.

The equation behind the figure

The two forces fit in one equation, which the chapter's figure solves minute by minute.

One strike, one equation

Step 1: the hedge. Dealers hold $N$ contracts at strike $K$, positive when they are long, each on 100 units. To stay flat they hold $H = -100\,N\,\Delta(S, t)$ units of the underlying.

Step 2: what moves the hedge. Over a minute $dt$, with implied volatility fixed, the delta changes by $\Gamma\,dS + \text{charm}\,dt$, the gamma and charm terms of (7B.1). So the hedge changes by $dH = -100\,N\,(\Gamma\,dS + \text{charm}\,dt)$ units.

Step 3: the impact. That trade is worth $S\,dH$ dollars, and each $1 billion moves the index $\lambda$ points. The dealers' trade therefore moves the index by $\lambda\,S\,dH/10^9$.

Step 4: solve for the move. The minute's move is the news plus the dealers' impact:

$$\begin{aligned} dS ={}& \text{news} \\ &- \frac{\lambda\,S\,100\,N\,(\Gamma\,dS + \text{charm}\,dt)}{10^9} \end{aligned}$$ (8E.1)

Collecting the $dS$ terms on the left gives (8E.3), with

$$\begin{aligned} w &= \lambda N\,100\,\Gamma S/10^9 \\ c &= \lambda N\,100\,\text{charm}\,dt\,S/10^9 \end{aligned}$$ (8E.2)

Step 5: read it. $w$ is the strike's gamma exposure in billions per 1% move times $\lambda/(0.01S)$, which is Chapter 8D's $\kappa G$ with the sign reversed. For $N > 0$ it divides every move, and $-c/(1 + w)$ is a drift toward the strike, because a call's charm is positive above the strike and negative below it near expiry. For $N < 0$ both reverse, and where $w$ reaches −1 the equation is singular: Frey's singular point of Chapter 8D, at a single strike.

In (8E.3), $w$ measures the gamma force and $c$ the charm force. The drift $c$ is weak for most of the day. At 6,005 with 30 minutes left it is about 0.04 index points a minute, while news moves have a standard deviation of about 1.2 points a minute. The damping $w$, by contrast, is already 0.2 at the open and grows all afternoon. The pull on any single path is gentle. What the two forces change is where the price tends to spend the last hour.

2 · The theory in the literature

The equation is not the book's invention. It is the construction of Avellaneda and Lipkin (2003), in Quantitative Finance, written in the form that Jeannin, Iori and Samuel (2008) analyze.

A description first. Krishnan and Nelken (2001), in the trade magazine Risk, modeled a pinned price as a Brownian bridge. That is a random path made to end at a given value, here the strike, at a given time. Their empirical illustration was Microsoft, whose shares "clustered near integer multiples of $5 more regularly on expiration dates than on any other days", in a search summary's words. A Brownian bridge describes a pin. It does not derive one from anyone's trades, so it says nothing about the sign of dealer gamma. It also shows that a pinned path can be modeled with no impact at all. So when the figure below shows zero impact changing nothing, that is a statement about the hedging mechanism, not about pinning in general.

The mechanism. Avellaneda and Lipkin gave a model of how hedging makes the pin. Floor market makers hold unusually large open interest at one strike and hedge it, and each trade moves the price in proportion to its size through a "price elasticity" constant $E$. In their abstract, if the open interest on a particular contract is "unusually large, delta-hedging in aggregate by floor market-makers can impact the stock price and drive it to the strike price of the option". The stock price then "has a finite probability of pinning at a strike".

As summaries of the authors' work put it, the mechanism needs hedgers "who are net long options, hence long Gamma". Its strength is "proportional to the open interest n and the price elasticity constant, E". The probability of a pin rises with open interest and impact, and falls with volatility and with time left. The pin comes from "a subtle interaction between the singular drift ... which drives the price to the strike, and independent diffusive shocks".

Two effects, and robustness. Jeannin, Iori and Samuel (2008), also in Quantitative Finance, put the Avellaneda-Lipkin idea into the Frey and Stremme feedback model of Chapter 8D. In their abstract, "pinning is driven by two effects". One is "a hedging-dependent drift term that pushes the stock price toward the strike price". The other is "a hedging-dependent volatility term that constrains the stock price near the strike as it approaches it". Those are the figure's $c$ and $w$. They also simulated trading in a double auction, with hedging done in steps and impact that varies at random. Pinning appears there too, "consistent with the Frey and Stremme model when both discrete hedging and stochastic impact are taken into account".

Curved impact, and the sign. Avellaneda, Kasyan and Lipkin (2012), in Communications on Pure and Applied Mathematics, extend the model to "non-linear, power-law, price impact functions", and pinning persists. That matters, because Chapter 8D showed that measured impact is concave, not linear. Their summary of the evidence states the sign condition: "When market-makers are net long options, the frequency of pinning at option expiration dates is much higher than if market-makers are net short."

The mirror image. The same equations with the sign reversed predict repulsion: a drift away from the strike and amplified volatility near it. The published models concentrate on the long case, and the evidence for repulsion is indirect, as the next section shows.

Away from expiry. Charm grows like $1/T$ as expiry approaches (Chapter 7B), so days before expiry the drift toward a strike is tiny. What remains is the volatility term: in the model, the price moves slowly where hedgers are long gamma, so it spends longer there, and quickly through zones where they are short. That is lingering, not a pull, and no test of it for SPX strikes was found.

3 · Watch a strike pin, or repel

The figure runs (8E.3) for one heavily held strike on a made-up expiration day. Dealers hold a net position in same-day options struck at 6,000, from −30,000 to +30,000 contracts, starting at +20,000: long, and so long gamma. They hedge every minute, and each hedge trade moves the index by the impact, from 0 to 3 points per $1 billion, starting at 1. Spot at the open runs from 5,970 to 6,030, starting at 6,012, and implied volatility from 10% to 30%, starting at 15%.

The upper panel draws ten sessions beside twins that get the same news with no dealers. The lower panel shows where 400 sessions close. Readouts give the share of closes within 5 points of 6,000 and the median distance from 6,000, each beside the no-dealer twins, and the largest feedback $w$ on any path. Start at the defaults. Then set the dealers to −20,000, then set the impact to zero. Finally compare +10,000 contracts at 1 point with +20,000 at half a point.

dealers at 6,000

+20,000

impact

1.0 pt per $1B

spot at the open

6,012

IV

15%

new paths

within 5 of 6,000: 36% (none: 20%)

median miss: 12.3 (none: 18.7)

largest feedback w = +3.86

At the defaults, 36% of sessions close within 5 points of 6,000, against 20% of their twins, and the median miss falls from 18.7 points to 12.3. The histogram grows a spike at the strike. The largest feedback reaches $w = +3.86$, in the last minutes, where same-day gamma at the money explodes. At the money it is about 0.20 at the open, 0.50 with an hour left and 0.70 with 30 minutes left, and it passes 1 with about 15 minutes left.

Set the dealers short 20,000 and only 1% of sessions close within 5 points, a hole in the histogram where the spike was, and the median miss widens to 26.4. The readout shows the feedback past −0.9: the model has crossed Frey's singular point in the final minutes, and the figure caps $1 + w$ at 0.1 to keep the paths finite. Set the impact to zero, and the dealers change nothing: 20% and 18.7, the twins' numbers exactly.

Now compare the last two settings. Long 10,000 contracts at 1 point per $1 billion gives 28% within 5 points and a median miss of 15.4, and so does long 20,000 at half a point. Only the product of position and impact matters, as in Avellaneda and Lipkin's pinning strength, $n$ times $E$. A small position does little: long 2,000 contracts leaves 23% within 5 points, against the twins' 20%.

The default position is deliberately large, so that the mechanism shows. Cboe's 2023 note measured market makers' net gamma across all same-day SPX options at $170 million to $670 million through the day, hedging flows of 0.04% to 0.17% of daily S&P 500 futures liquidity (Chapter 8G). The figure's single strike carries $11.7 billion of gamma per 1% move at the open, with spot at the strike. The figure also leaves out several things the theory and evidence add:

• Impact that is concave and partly transient, not constant and permanent (Chapter 8D). Pinning survives power-law impact (Avellaneda, Kasyan and Lipkin 2012).
• Hedging in steps with noisy impact, which pinning also survives (Jeannin, Iori and Samuel 2008).
• Changes in the dealers' position during the day, which new trades bring (Egebjerg and Kokholm 2024, Chapter 8D).
• Other strikes, other expiries and other flows, all of which the real market has at once.
• The manipulation channel below, by which traders short options at a strike can pull the price there without any hedging.

The pinning figure shows where closes land. The figure below shows the clock behind them: the feedback $w$ at one strike from the open to the close, with spot held at a distance from the strike that you set. Sliders set the dealers' position, from short 30,000 to long 30,000 contracts and starting long 20,000. They also set spot minus the strike, from −40 to +40 points and starting at 0, the impact, starting at 1 point per $1 billion, and implied volatility, starting at 15%. The lower panel turns $w$ into the factor $1/(1 + w)$ that multiplies each news move, on a logarithmic scale.

dealers at the strike

long 20,000

spot − strike

at the strike

impact

1.0 pt per $1B

IV

15%

w at the open: +0.20

with 30 min left: +0.70

w passes 1 at 3:46, 14 minutes left

Damping. News moves are multiplied by 0.84 at the open and 0.59 with 30 minutes left. From 3:46 the dealers' own trades offset more than half of every news move.

At the start, $w$ is +0.20 at the open and +0.70 with 30 minutes left, and it passes 1 at 3:46, with 14 minutes left. From then on the dealers' trades offset more than half of every news move. Set the dealers short 20,000: the factor starts at 1.24, reaches 3.38 with 30 minutes left, and at 3:46 the feedback reaches $w = -1$, the singular point. Now move spot 10 points from the strike. The short position's $w$ is −0.18 at the open and −0.24 with 30 minutes left, and it never nears the singular point. At 25 points it has faded to zero by the last half hour. A same-day strike takes over late, and only if the price sits at it.

4 · The evidence

Single stocks: clustering at strikes

Ni, Pearson and Poteshman (2005), in the Journal of Financial Economics, compared US stocks with listed options against stocks without them, from 1996 to 2002. They used end-of-day data on every listed stock option, and Cboe data that split open interest and volume by type of trader. The Cboe data let them see the positions of market makers and of firm proprietary traders. On expiration dates, the closing prices of optionable stocks cluster at strike prices, and stocks without options show no such change. The size:

Clustering is much more frequent when the market makers who hedge are net long options than when they are net short, as the model requires. And the authors name two channels: "hedge re-balancing by option market-makers and stock price manipulation by firm proprietary traders contribute to the clustering".

Read the 16.5 basis points for what it is: a lower bound on the average distortion per stock per expiration date, for single stocks, from 1996 to 2002. It is not a directional return, and it is not an index number. The paper does not study the index, so its figure does not convert into SPX points.

The index: pinning, and pushing away

Golez and Jackwerth (2012), in the Journal of Financial Economics, looked in the deepest market there is, S&P 500 futures, around two kinds of expiration. Their abstract reports both directions. Futures "are pulled toward the at-the-money strike price on days when serial options on the S&P 500 futures expire (pinning)". They "are pushed away from the cost-of-carry adjusted at-the-money strike price right before the expiration of options on the S&P 500 index", which the authors call anti-cross-pinning. The effects, in their words, "are driven by the interplay of market makers' rebalancing of delta hedges due to the time decay of those hedges". Market makers also rebalance "in response to reselling (and early exercise) of in-the-money options by individual investors". The size: "The associated shift in notional futures value is at least $115 million per expiration day".

A summary of the paper's results goes further on how the two channels combine. The time decay of hedges on its own produces anti-pinning. At the serial expirations, investors' reselling and early exercise of in-the-money options outweigh it, so the net effect there is pinning. In the model's terms, decay that pushes away is the short-gamma case. Read that way, charm repelled in this sample, and the pull came from investor behavior that market makers absorbed.

Three limits bound what the paper says about the index today. Its sample predates same-day options. Its pinning channel depends on American-style futures options, which can be exercised early, while SPX options are European. And its $115 million per expiration day is small beside the S&P 500 futures market, with no figure in basis points to set beside the single-stock 16.5.

As expirations multiply

Pinning is not uniform. Zhang and co-authors (2013), in a small journal, studied 96 stocks with weekly options from 2010 to 2012. Clustering still appeared on monthly expiration days, but it was much weaker for the newly listed weeklies, and close to three quarters of the firms showed no clustering at all. A 2017 working paper from the same group found significant clustering only for weeklies with high volume and a wide strike grid. It found evidence for both the time decay of delta hedges and manipulation. The sample is small, but it points one way: more expirations did not bring more pins.

Since 2022, SPX options expire every trading day (Chapter 9B). No study found tests strike pinning in SPX in this era: the 0DTE papers study volatility, not pins. What they report bears on the model's conditions. Dim, Eraker and Vilkov, in a working paper first posted in 2023, find market makers' gamma in same-day SPX options "on average positive". Amaya, Garcia-Ares, Pearson and Vasquez (2025) find it "typically positive" but "often negative". Adams, Fontaine and Ornthanalai (2024) find that market makers "match most 0DTE order flows, but absorb end-users' trades in long-dated options that eventually become 0DTE positions". Long gamma at the money on most days is the condition the model needs for a pull, and Adams and co-authors trace it mostly to positions opened days earlier. Whether SPX closes cluster at those strikes has not been tested.

Other ways a strike can win

Not every expiration effect is hedging. Four other channels appear in the research:

• Manipulation. Ni, Pearson and Poteshman's second channel: traders who hold options can profit from moving the close relative to a strike. Jarrow (1994) showed in theory that "the introduction of the derivative security generates market manipulation trading strategies that would otherwise not exist". Later models make the same point for option issuers with price impact (Kraft and Kühn 2011) and for option holders who trade the underlying to raise their payoff (Horst and Naujokat 2011). In NASDAQ order data, Figueiredo and co-authors (2023) find more fleeting orders, placed and quickly canceled, in optionable stocks on expiration days. Their patterns "suggest spoofing and price manipulation rather than a simple search for latent liquidity", and they "increase likelihood of stock prices crossing option strike prices on option expiration days".
• A retail level that worked, for another reason. Max pain is the strike at which the total payoff owed to option holders at expiry is smallest. Filippou, Garcia-Ares and Zapatero, in an SSRN working paper first posted in 2022, tested it on US stocks from 1996 to 2021. A long-short spread built on it earned about 0.4% a week, by a search summary's account, strongest in small and illiquid stocks. The authors attribute most of it to reversal in stocks that had fallen substantially, and cannot rule out manipulation as the reason its timing matches expiration.
• Exercise and sale. Chiang (2014) finds that stocks with a large amount of deep in-the-money call options "experience a significant return drop of 0.8 percentage point on option expiration dates", followed by a short-term reversal. The cause is holders who exercise and sell the delivered stock at once. SPX options settle in cash, so this channel does not apply to them.
• Another market, the same sign. In bitcoin, Weiss and co-authors (2026) find a return reversal around daily option expirations that "is strongest when cumulative gamma exposure is negative". That fits market makers hedging net short positions. They put the annual transfer between option writers and holders at about 50 million US dollars.

5 · What is not known

The evidence leaves gaps that matter for anyone trading SPX:

1. Pinning in SPX in the 0DTE era. No study found tests it. The single-stock evidence ends before weekly options, and the index evidence predates daily expirations.
2. Repulsion. The same equations predict it with the sign reversed, but it is tested only indirectly: less clustering when market makers are net short, and anti-cross-pinning in futures.
3. Pulls away from expiry. The theory predicts only lingering near long-gamma strikes and fast passage through short-gamma zones. No test of strikes attracting or repelling the SPX outside expiration days was found, and no test of walls as support and resistance.
4. The size of the impact. No published estimate gives the impact of dealer hedge flow in SPX in points per $1 billion (Chapter 8D), and the pin's strength is proportional to it.
5. The sign at a strike. The model needs to know whether hedgers are long or short there, and public data do not show it (Chapter 8B).

6 · What this means for "the King Node is where market makers settle price"

Chapter 8C describes Skylit's Heatseeker and its vocabulary. Its King Node is, in Skylit's words, "the strike carrying the largest absolute exposure value on the board. Not the largest positive, not the largest negative. The largest absolute". Skylit calls it the node "where Market Makers typically prefer to settle price by the end of day (EOD) or end of week (EOW) for swing trades". Its advice: "Treat it as your primary destination target for the session". One of its own pages defines the King differently, as "The strongest +GEX node across the heatmap".

Set against the theory and the evidence, the claim has a supported core and several parts that are not supported:

• Supported, weakly, at expiry. A large position that hedgers hold long at one strike pulls the price toward the strike and damps moves around it as expiry nears. The measured sizes are small: basis points in single stocks and at least $115 million of futures notional per expiration in the index.
• Not the sign-blind version. "The largest absolute" includes nodes where dealers are short. By every model here, those push the price away and make it more volatile near the strike. The sign decides attraction against repulsion, and size matters only relative to liquidity, through the product of position and impact. This is where Skylit's "size determines influence, regardless of sign" and the mechanics part ways.
• Not away from expiry. For an end-of-week anchor several days out, the charm drift that does the pulling is small, because charm grows like $1/T$ only as expiry nears. The theory leaves only lingering, which is untested.
• Not "prefer". In the hedging mechanism nobody chooses the pin. A dealer long the options at a strike, and hedged, earns from moves and pays time decay (Chapter 6D), so a pinned close is a poor outcome for it. The pin is the market's response to its hedging, not its preference. The traders who do prefer a close at a strike are those short the options there, and that is the manipulation channel, not the hedging one.
• Not tested as a target. No study was found that tests the largest absolute node as a daily or weekly target, and Skylit publishes no hit rate with a method (Chapter 8G).

One strike shows the sign problem plainly. The JPM collar is one of the few large index-option positions whose holder is public (Chapter 8B). The fund is long the upper put of its put spread, so whoever sold it, typically dealers, is short it. Vendors describe the market as holding "near the JP Morgan put strike through expiration", a claim with no published test. By one practitioner's account of the first quarter of 2026, SPX closed a Friday in the quarter's last week more than 100 points below that strike. By the model, a strike where dealers are short gamma is an accelerator on a break, not a magnet. While spot sits above it, the decay of the same short put makes dealers buy back hedges, which supports the price. That support still pushes away from the strike, not toward it.

SpotGamma makes a parallel claim about its "Absolute Gamma", "the strike with the largest total gamma", which it calls "a strong support/resistance level and a magnet". The same reading applies: size says how much hedging a strike can generate, and the sign says which way it pushes.

7 · Pin risk for the option holder

Chapter 5F described pin risk from the seat of a trader short an option near its strike at expiry. You do not know until after the close whether you are to be assigned, and an unplanned position can gap over the weekend. Pinning adds one thing: where hedgers are long gamma at a strike, closes near that strike are more likely than chance, so pin risk concentrates where the pull is. Where they are short, the price tends to leave the strike's neighborhood, which is the opposite risk for a short straddle.

What pin risk means depends on the contract:

• SPY and single-stock options are American-style and physically settled. A close near the strike leaves you unsure whether shares are coming, and the shares carry the weekend gap (Chapter 5F).
• SPX and XSP options are European-style and cash-settled, so there is no delivery and no early assignment. The risk is the settlement print. Monthly SPX options settle on the Friday-morning special opening quotation, and weekly SPXW options on the official 4:00 PM close (Chapters 2A and 12H).

A pin does not make a short option safe. A short straddle at a strike that pins keeps its premium. One at a strike that repels, or on a day with a news shock, loses on the move, and its short gamma is largest exactly in the final hour. A long straddle at a pinned strike loses its time value. Neither the pin nor its absence is predictable enough to size a trade on.

8 · Check yourself