The Flow With No Opinion
A dealer who has sold an option holds a hedge that is a function of the price, which means the hedge cannot change until the price has. Its trading is a derivative, and a derivative can only follow. Priced on this course’s sixty closes, 10,000 contracts struck at 103 need a hedge that never exceeds a million shares, and holding it takes 3,079,024 shares of trading across the sixty bars — three times the largest position it ever carries, against a net change of 555,304. The rate is gamma, and gamma is a bell curve that grows a spike as expiry approaches: a dollar of movement at the strike moves the hedge by 57,861 shares with twenty bars left and by 258,898 with one, while six dollars away on that last bar it moves 91. What the dealer is betting on is not the size of the movement but where it lands. Sold at the series’ own realised 1.496 per cent a bar, the hedged short call finished 556,654 ahead, because it needed only 1.378 per cent.
Prerequisites: Lesson 59, for the square-root law and for the habit of asking what a large position actually trades rather than what it is worth, lesson 53, for the quoting business as inventory management rather than opinion, and lesson 44, for volatility as a number you can measure on a series instead of a mood.
The hedge is a position and the trading is its derivative
A dealer that sells a call to a customer has no view and does not want one. What it has is an obligation whose value moves with the underlying, so it holds shares against it: the number of contracts, times a hundred, times the option’s delta. That last term is the whole of the story, because delta is a function of the price. Write the hedge as a formula and it contains no clock, no opinion and no discretion. It contains the price.
Two consequences follow immediately and most of the folklore about dealers ignores both. The first is that the hedge cannot change until the price changes, so every share the dealer trades arrives after the move it is responding to. The flow is pro-cyclical rather than predictive, and there is nothing in it to position ahead of. The second is lesson 59’s point in a new costume: the size of the hedge is a position, and what trades is its derivative.
Put numbers on both. Take the sixty closes this course has carried since lesson 38, running from 100.7 to 106.5, and sell 10,000 calls struck at 103, expiring at the end of the series. Value them at the volatility the series actually has, 1.496 per cent a bar. Then rebalance the hedge once at every close and add up what it traded.
The hedge starts at 444,696 shares, drops to 313,203 at its lowest and finishes at a million, because the option finished in the money and the whole delta had to be there. So the largest position it ever holds is a million shares. The trading it took to hold that is 3,079,024 shares. The net change from the first bar to the last is 555,304. Three million shares moved to end up half a million away from where it started, and the position never got bigger than a million.
That ratio is the answer to every headline of the form: dealers have 11 billion dollars of hedges to unwind. Eleven billion is a position. The question is its derivative, and the derivative on any given day is a fraction of it. A hedge that does not move trades nothing at all.
Gamma is a bell curve that grows a spike
The rate at which the hedge changes per dollar of price movement is gamma, and gamma has a shape worth seeing rather than defining. It peaks at the strike, falls away on both sides, and gets taller and narrower as expiry approaches. Below is that shape on the same 10,000 contracts, in shares of rehedging per dollar the price moves, at three distances to expiry.
| Price relative to the strike | 20 bars left | 5 bars left | 1 bar left |
|---|---|---|---|
| -6 | 42,338 | 25,333 | 91 |
| -4 | 51,532 | 60,947 | 8,255 |
| -2 | 57,083 | 100,405 | 112,939 |
| 0 | 57,861 | 115,770 | 258,898 |
| +2 | 53,940 | 95,345 | 110,091 |
| +4 | 46,467 | 57,159 | 9,547 |
Read the first column and then the last. Twenty bars out the flow barely depends on where the price is: 57,861 shares a dollar at the strike against 42,338 six dollars away, a spread of 1.4 to one across the whole range. On the final bar the same span runs from 258,898 to 91, which is 2,845 to one. The concentration people describe as an expiration effect is not a property of options. It is a property of the last bar.
That is also where the amplification lives, and lesson 59 prices it. A dollar of movement on a 103 instrument is 0.97 per cent. On the final bar at the strike it obliges 258,898 shares of rehedging, and the square-root law turns that into an impact of the daily volatility times the square root of those shares over the day’s volume. In something trading 5 million shares a day that comes to 0.34 per cent, which is 35 per cent of the move that caused it. In something trading 70 million it is 0.09 per cent, or 9 per cent. Twenty bars out the same calculation gives 17 per cent and 4 per cent.
So the honest statement is a range with an input in it. Hedging flow adds something between a twenty-fifth and a third of a move, depending on the instrument’s volume, the distance to expiry and how far the price is from the strike. It does not create the move. It cannot: it arrives afterwards.
The sign decides whether a strike pulls or pushes
Everything above assumed the dealer sold the option. Reverse that and every conclusion inverts. A dealer who is long the call is long gamma, and rehedging a long gamma position means selling as the price rises and buying as it falls. The same 258,898 shares a dollar on the last bar now damp the move instead of amplifying it, and the strike holds price rather than pushing it away.
This is the one place where the popular account is not merely imprecise but exactly backwards half the time, and the reason is that the sign is not observable. Open interest counts contracts, not sides. A strike showing 200,000 contracts open tells you the gamma there is large; it does not tell you who holds it. Both configurations occur, they produce opposite behaviour from identical numbers, and no amount of staring at the chain resolves it.
The consequence is worth stating plainly because a great deal of writing avoids it. Pinning at a strike and repulsion from a strike are the same mechanism with the sign flipped, so the same table above predicts both. What a large open interest at a strike does tell you is that the last hour will be busy near it, and that is a reason to expect wide spreads and a poor fill rather than a reason to take a direction.
One more thing the formula settles. A hedge held against options that finish in the money is not sold at the close: the options are exercised and the shares are delivered against them, so nothing reaches the market. A hedge against options that finish out of the money does have to be unwound, but delta decays toward zero as the price walks away from the strike, which means that selling happens on the way, following the move. Neither case produces the scheduled liquidation at four o’clock that the folklore describes, and both are visible in the table: on the last bar, six dollars from the strike, the hedge changes by 91 shares a dollar. There is nothing left to unwind because there is nothing left to hedge.
What the dealer is actually betting on
The hedging is not the business. It is the cost of running the business, and the business is a bet on volatility. Every rebalance buys higher than the last sale or sells lower than the last purchase, so a short option pays a running cost as the price moves; against it, the option loses value with time, which the dealer collects. The whole position is the difference.
Run that on the sixty closes at five prices. The dealer sells 10,000 calls struck at 103 at a stated implied volatility, hedges once a bar, and the two columns are what time decay paid and what the rebalancing cost.
| Sold at, per bar | Time decay collected | Rebalancing cost | Net |
|---|---|---|---|
| 1.200% | 2,483,330 | 3,378,024 | -894,694 |
| 1.350% | 2,916,343 | 3,050,809 | -134,466 |
| 1.496% | 3,343,758 | 2,787,105 | +556,654 |
| 1.650% | 3,799,433 | 2,553,442 | +1,245,991 |
| 1.800% | 4,246,929 | 2,360,019 | +1,886,911 |
The third row is the interesting one, and it is not what the textbook leads you to expect. It sells the option at exactly the volatility the series turned out to have, 1.496 per cent a bar, and it makes 556,654 rather than nothing. Search for the price at which it breaks even and the answer is 1.378 per cent a bar, which annualises to 21.9 against the series’ own 23.7.
The gap is not an error and it is not an edge. It is the difference between how much a series moved and how much it moved where the gamma was. Squaring the two figures says this path delivered 85 per cent of its variance to the places that mattered to this option. A path with the same standard deviation that had spent more of its time near 103 would have cost more to hedge and could have lost money at the same price.
Which is the useful reframing of the whole subject. The dealer has no opinion about direction and is not trying to move anything. It has sold a number, it is paying to find out whether that number was too low, and the trading everybody watches is the payment rather than the position. When the flow looks like it is pushing price around, what is happening is that a cost is being incurred, in public, at a rate set by a bell curve.
What this does not settle
That one option is a book. A dealer holds thousands of lines across many strikes and expiries, and the aggregate gamma is a sum with cancellation in it, not the single bell curve drawn here. The published gamma-exposure figures are estimates of that sum built on an assumption about who is short what, and they inherit exactly the sign problem set out above. Everything on this page is right about one position and only illustrative about a book.
That rebalancing once a bar is what a dealer does. It is not. Real hedging is closer to continuous, triggered by bands in the delta rather than by clocks, and often left deliberately loose because trading costs money. Rebalancing more finely would not have changed the 2,787,105 by much on average, because the cost converges to the movement the path actually contained; what it would have done is narrow the spread of possible outcomes around it, and raise the turnover and the commissions in doing so. Rebalancing more loosely widens that spread and saves the trading. The 3,079,024 figure is therefore a property of the rule chosen here as much as of the option.
That the impact numbers stack the way they were used. Lesson 59’s law estimates the cost of one order worked over a session, and the amplification paragraph applies it to a single dollar of movement inside a bar. That is a stretch of the law past what it was calibrated on, and it is why the answer was given as a range with the volume as an input rather than as a figure. Read it as an order of magnitude, not as a measurement.
That the series is an equity index. It is sixty numbers with a 1.496 per cent standard deviation and no volume attached, which is why every share figure on this page is stated per 10,000 contracts and every impact figure needs a volume you supply. The shape of the results transfers. The levels do not.
That a zero interest rate and no dividend are innocent. The pricing here sets both to zero, which shifts delta at every point and therefore shifts the hedge. On a two-month option the effect is small next to everything else on the page, but it is not nothing, and anybody reproducing these figures against a real chain will find their deltas a little different for that reason before they find them different for any interesting one.
And the concession that costs this lesson most: it has explained the mechanism and given you almost nothing to do with it. The sign of the dealer’s gamma decides the behaviour and is not observable; the modelled versions of it are sold by firms with an interest in the answer; and the one thing you can see, open interest at a strike, is compatible with both outcomes. What is left is negative knowledge — do not trade an expiration hour on a story about forced selling, do not read a hedge position as a flow — and negative knowledge is worth having, but it is not a strategy, and this page is not going to pretend otherwise by dressing the range in the second section as an entry.
Problems
- Build the bell curve for one real strike. Pick a liquid name, one expiry, and one strike near the money. Get the delta at five prices around the strike from any option calculator, and difference them to get shares per dollar per 100 contracts. Then repeat it a week later without changing the strike. Two hours, and you will have the first and third columns of the table above for an instrument you actually trade, which is more convincing than reading them here.
- Test the pin, on both sides. For ten expirations in one name, write down the strike with the largest open interest on the Thursday and the closing price on the Friday. Record the distance, signed. If price is drawn to the strike the distances will cluster near zero; if it is pushed away they will not; if the sign of the dealer’s gamma varies between expirations you will see both and the average will tell you nothing. An hour, and the third outcome is the most likely one, which is the finding.
- Price the amplification for your own instrument. Take the daily volume and the daily volatility you measured in lesson 59, and a rehedging figure of 100,000 shares per dollar as a stand-in for a busy strike. Put them through the square-root law. Ten minutes, and it tells you whether hedging flow in your name is a rounding error or a third of the move, which decides whether any of this is worth your attention at all.
Sources. Fischer Black and Myron Scholes, “The Pricing of Options and Corporate Liabilities” (Journal of Political Economy, 1973), for the replication argument that makes the hedge a function of the price and therefore makes its flow follow rather than lead. Nicolas Bollen and Robert Whaley, “Does Net Buying Pressure Affect the Shape of Implied Volatility Functions?” (The Journal of Finance, 2004), for evidence on which side of the option customers are actually on, which is the sign this page says you cannot read off open interest. Sophie Ni, Neil Pearson and Allen Poteshman, “Stock Price Clustering on Option Expiration Dates” (Journal of Financial Economics, 2005), for the measured pinning effect in single names and for how small it is. Guido Baltussen, Zhi Da, Sten Lammers and Martin Martens, “Hedging Demand and Market Intraday Momentum” (Journal of Financial Economics, 2021), for hedging flow measured against intraday returns rather than asserted.
A dealer’s hedge is a function of price, so its trading is a derivative and arrives after the move. On this course’s sixty closes, 10,000 contracts at 103 never need more than a million shares of hedge and take 3,079,024 shares of trading to hold, ending 555,304 from where they started. Gamma sets the rate and concentrates violently at the end: 57,861 shares a dollar at the strike with twenty bars left, 258,898 with one, and 91 six dollars away on that same last bar. Whether that flow amplifies the move or damps it depends on a sign that open interest does not report. And the dealer’s profit turns on where the movement landed rather than how much of it there was: sold at the series’ own 1.496 per cent a bar it made 556,654, because break-even was 1.378. Lesson 61 closes the module by asking what follows from all of it — how to think about a market in which every quantity you can see was put there by somebody who knew you would look.
What Hurry Costs
The square-root law this page uses to price what a rehedge does to the price.
Read Lesson →What the Spread Is Paying For
Quoting as inventory management, which is the same posture one derivative out.
Read Lesson →Volatility as a Quantity
The number the dealer is selling, measured rather than felt.
Read Lesson →Educational only. Trading involves substantial risk of loss. Not financial advice. Past performance does not guarantee future results.
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