Hidden Size

Showing size costs money, so the people who have size do not show it, and the screen understates as routinely as it overstates. That makes hidden size a different object from lesson 26’s wall: when the detector fires it is an observation rather than a read, and no posterior is needed. The catch sits on the other side. In the hundred tests below it could not have fired at all on eighty of them, which is why a quiet book proves nothing — and every measurement of how much was hidden turns out to be a floor rather than a figure.

Prerequisites: Lesson 26, which priced a read of displayed size and said in its own bounds that the error runs the other way as well, and lesson 19, because almost everything below is a count and the counts are small.

A hundred tests of one level

One instrument, one price — a round number the market keeps coming back to — and a hundred occasions on which price traded there, logged over several months. The displayed quantity at that price was 200 contracts or near enough every time, which is what makes the hundred comparable. Each test is sorted by one question: was the display consumed, and did quantity come back?

What happened at the level	Tests	Could a reserve have shown itself?	Refill seen
Under half the displayed 200 traded	58	No	0
Over half traded, display not cleared	22	No	0
Display cleared, price moved on	6	Yes	0
Display cleared and refilled at the price	14	Yes	14

Read the third column before the fourth. Eighty of the hundred tests could not have produced a detection under any circumstances, because the displayed quantity was never fully taken. Whatever was or was not sitting behind those 200 contracts, the market did not ask, so the record does not know. That is not a weakness in the logging. It is what the detector is.

So the count is fourteen in a hundred, and fourteen per cent is not the number to write down. Among the twenty tests where a reserve could have shown itself, fourteen did, which is 70 per cent. Which of the two is the rate of hidden size at this level depends entirely on whether reserves are more or less likely to be sitting there on the occasions when a large order happens to arrive, and nothing in this record can settle that. What the record does settle is a bound. At least 14 of the hundred had hidden size behind the display, and at most 94 did, because all 80 of the blind tests could have had it and not one of them could have shown it. Fourteen per cent to ninety-four per cent — and 70 per cent is what you get by assuming the blind tests resemble the others.

The number a trader would actually write down after this exercise is one level in seven. That is not the middle of the range. It is the bottom of it.

How much was behind it

Now take the fourteen detections and ask the second question. Total traded at that price counts the display and every refill after it.

Refills seen	Total traded at the price	Episodes	Level broke afterwards
1	400	5	2
2	600	4	1
3	800	2	1
4	1,000	2	1
6	1,400	1	0

Nine thousand four hundred contracts traded at that single price across the fourteen episodes, against a book that never showed more than 200 of them at a time. The mean is 671 contracts, so the screen understated the quantity by a factor of about three and a third.

That mean should not be quoted, and the last column is why. In five of the fourteen the level eventually broke, which means the reserve was exhausted and the total is a total. In the other nine, price left while the display was still refreshing. Those nine figures are not measurements of anything. They are lower bounds. The largest of them — 1,400 contracts against a 200 display, seven times what was shown — is a floor with no ceiling anywhere in this data, because whatever remained behind it went home unfilled and unrecorded.

Read honestly, the second table says this. Five episodes finished, and those five averaged 640 contracts. Nine did not finish, and are known only to have exceeded 689 on average. Five is a sample from which lesson 19 permits approximately no conclusion, and the nine cannot be averaged in with the five without treating a floor as a figure. The factor of three and a third is the one number on this page that looks like a finding and is not one.

Why lesson 26’s table does not transfer

Lesson 26 wrote a read as a pair — 80/80 meaning eighty per cent right on each side — and showed that in odds terms a read is a single multiplier. That machinery needs both halves of the pair, and the reason it does not transfer is that here the two halves are nothing like each other.

The easy half first. When the detector fires there is no inference in it. You watched the quantity at a price get consumed and replaced, which is an observation of the kind lesson 28 made about a level holding, and it needs no posterior because there is nothing left to be uncertain about. Call that side 100.

The hard half. Of the seventy or so levels the estimate above puts hidden size behind, fourteen showed it. That is a sensitivity of about 20, and it inherits the assumption the seventy rests on — though the blind fraction pins it from the other direction too, since a detector that cannot fire on eighty tests in a hundred cannot be sensitive on more than twenty unless reserves prefer the occasions it can see. So the read is a 20/100, and there is no column for it in lesson 26’s table, because every read in that table is symmetric — which is precisely why one multiplier per column was enough there. For a symmetric read the two multipliers are reciprocals of each other. Here they are not, and the asymmetry is the finding.

A detection settles the question outright. A non-detection multiplies your odds of hidden size by about 0.8, against 0.43 for lesson 26’s 70/70 read and 0.25 for its flattering 80/80. Nothing in a quiet book entitles you to conclude the size is not there. You have not asked.

What a worked reserve leaves behind

One more thing, and it discharges a debt the last two lessons kept rolling forward. When a reserve is worked through, what does it leave on the pictures those lessons drew?

A tall bar at one price. Fourteen hundred contracts at a single price is a volume node, and a volume profile records it precisely. What the profile cannot record — because it is built by summing volume at a price and nothing else — is whether that quantity was ever displayed. A price that took 1,400 contracts in 200-lot slices out of a reserve and a price that took 1,400 contracts because a large buyer arrived and paid for them produce the same bar, at the same height, with the same claim on the point of control.

Hold that next to lesson 29. That session’s point of control moved 9.5 ticks under a change of bin width because of one heavy price, 100.03, holding 1,080 contracts, and lesson 30’s value area was anchored down there for the same reason. Both lessons called it a print, which is what a profile lets you call it. Had it instead been a reserve worked out in slices, the profile would have looked identical and both lessons would have said exactly the same thing. That is not a defect in either — it is what a distribution of volume is — and it is why lesson 30 closed by saying the auction had one structural thing left to show you.

What this does not settle

That a refill proves a reserve. It proves size arrived at the price after the display cleared, which is compatible with one participant hiding and with two participants queueing. Separating them needs order-by-order data that most retail feeds do not carry, and the version you can establish — size kept arriving here — is the one the decision runs on anyway. Where this does matter is in the counting: two participants in succession will look like a bigger reserve than either of them placed.

That 70 per cent is your instrument’s number, or anyone’s. It is an estimate from a hundred tests at one level on one instrument, resting on an assumption the same data cannot check. The range those hundred tests actually permit runs from 14 per cent to 94 per cent, and the honest summary of the exercise is the range plus the reason it is so wide, not the point inside it.

That any of this is a reason to trade the level differently. It is not: nothing here establishes what price does after a reserve is detected, only that a reserve was there. That is the gap lessons 26, 28, 29 and 30 each ended on, every one of them having built something and then asked for a base rate nobody has produced. The third problem in lesson 30 is still the shape of the answer, and detecting hidden size does not shorten it.

That hiding is deception. It is not, and the distinction is the same one lessons 25 and 26 both insisted on. Everything above holds when every participant is acting in good faith and simply declining to write an option they are not paid for. Placing orders with no intention of filling them is a different act, it is an offence, and lesson 27 is where it lives.

That the blind fraction is a property of the detector. It is a property of the detector and the instrument together. Eighty in a hundred came out of how much volume arrives at that level relative to what sits displayed there, which is a fact about the market, and it will be a different number on a thinner instrument or at a busier hour. It is the first thing to measure and the last thing to assume.

The book overstates because a message costs nothing to send. It understates because meaning one costs something. Both are the same fact seen from two sides, and neither of them is a lie.

Problems

1. Build the record the arithmetic needs. Pick one level on one instrument — a round number, or the prior session’s extreme, the same kind of level every time — and log a hundred tests. For each one record three things: the quantity displayed when price arrived, the quantity that traded at that price, and whether the display was cleared and then refilled. Sort them into the four boxes of the first table. Log every test that met your definition rather than the ones that were interesting; that instruction came from lesson 26 and it is harder to obey here, because an interesting test and a detectable test are nearly the same event.
2. Measure your blind fraction before your hit rate. From that record, the number to read first is not fourteen in a hundred, it is eighty in a hundred. Work out what share of the tests cleared the display at all, because that share — not your attention, and not your platform — sets how often the question can even be put. If it comes out near a third, the detector is worth running where you trade. If it comes out at five in a hundred, you have learned that the book at your level is not something this method can measure, which is a real finding and costs a week.
3. Find out whether your feed can show you this at all. The detector needs depth at a price, updated in sequence, faster than the fills that consume it. Many retail feeds carry periodic snapshots instead, and a snapshot once a second cannot tell a display that was consumed and refilled from one that never moved. Open your provider’s documentation and establish which of three you are getting: order-by-order depth, aggregated depth by price with every update, or snapshots on a timer. Write down the answer. If it is the third, the first two problems are not available to you at any level of effort, and it is worth knowing that before a month of logging rather than after.

Where the problems above ask you to go and count something, How to Collect a Base Rate is the appendix that says how: define the observation, fix the criterion before you look, take consecutive cases rather than the memorable ones, and count into four cells.

Sources. Thomas E. Copeland and Dan Galai, “Information Effects on the Bid-Ask Spread” (Journal of Finance, 1983), for the result the whole lesson rests on: a resting quote is a free option written to whoever chooses to take it, so the quantity you display is the size of the option you have given away. Hendrik Bessembinder, Marios Panayides and Kumar Venkataraman, “Hidden Liquidity: An Analysis of Order Exposure Strategies in Electronic Stock Markets” (Journal of Financial Economics, 2009), for the trade-off measured on a market where traders choose exposure order by order — displaying gets the order done sooner and at a worse price, hiding costs less and may not get it done. Rudy De Winne and Catherine D’Hondt, “Hide-and-Seek in the Market: Placing and Detecting Hidden Orders” (Review of Finance, 2007), for the detector itself and for how much care its statement requires. Amber Anand and Daniel G. Weaver, “Can Order Exposure Be Mandated?” (Journal of Financial Markets, 2004), for the Toronto natural experiment: removing the ability to hide did not convert hidden depth into displayed depth.

This module has now spent seven lessons on what is at a price — what rests there unseen, what is displayed there, what trades there, and what was never shown. The next one changes the question. It stops asking what sits at one price and starts asking what the order of prices claims: a break of structure and a change of character are assertions about who is in control, and lesson 32 is about the fact that they are falsifiable, which is more than is usually said for them.

Educational only. Trading involves substantial risk of loss. Not financial advice. Past performance does not guarantee future results.