Multi-Day Structure

Every rule of the last eight lessons was built on a series in which consecutive bars always touch. That was not a market fact; it was how the series was constructed, and it means there is not one gap in it. Take the same sixty closes and remove the overlap, and nineteen of the thirty-nine transitions become gaps. The sample high and low do not move by a tenth. The efficiency ratio does not move at all. The swing count comes out at fifteen both times — and two of the fifteen points are at different bars. Meanwhile a stop resting inside one of those gaps is filled at a median of one and a half times the loss it promised, and at four and a half times in the worst case.

Prerequisites: Lesson 32, whose swing rule the discontinuity quietly rearranges, and lesson 21, which put the stop that the gap leaps over.

The same closes with the overlap taken out

Nineteen of the thirty-nine transitions between bars 21 and 60 become gaps. Six of them are down gaps and thirteen are up gaps, which is a property of a series that drifts upward and not a fact about markets. The largest is 0.8, at bar 43 and again at bar 53. The smallest is 0.1, at bar 35. Below, what that does to the quantities the last eight lessons taught.

Quantity	Bars that overlap	Same closes, no overlap
Gaps between bars 21 and 60	0 of 39	19 of 39
Sample high	107.4	107.4
Sample low	97.7	97.7
Efficiency ratio over the sample	0.082	0.082
Swing points, lesson 32’s rule	15	15
Summed bar range	81.0	53.1

The table sorts the module’s quantities into two piles by one criterion, and the criterion is whether the quantity looks at the highs and lows or only at the closes.

Everything computed from closes alone is exactly unchanged, and not approximately: the closes were not touched, so lesson 36’s ratio, lesson 37’s confirmation lengths and every reading in lesson 39’s agreement table come out identical to the last digit. That is a comforting result and it is also a warning. A measurement that cannot see a gap is a measurement that will report a smooth market through a discontinuity, because smoothness is all it was ever looking at.

The extreme survives too, and for a better reason. The sample high of 107.4 is a price at which trading actually happened, and removing the overlap did not un-happen it. Lesson 38 made the same point about regrouping; it holds here for the same reason and is the second time in three lessons that the levels have been the sturdy thing on the page.

Summed bar range collapses from 81.0 to 53.1. A third of what the module has been calling range was the part of each bar that re-covered ground the previous bar had already covered. That matters wherever you have sized something in bar ranges — a stop at one average range, a target at three — because the unit itself is a third smaller once the bars stop reaching backwards, and it changed without anything happening in the market.

Fifteen and fifteen is not the same fifteen

The swing count is the row that will fool you. Lesson 32’s rule finds fifteen swing points in both versions, and if the count were all you checked you would conclude the discontinuity had left the structure alone.

It has not. In the continuous series the swing highs sit at bars 6, 11, 21, 26, 30, 38, 53 and 56 and the swing lows at 8, 19, 24, 32, 40, 52 and 55. Remove the overlap and bar 21 is no longer a swing high and bar 32 is no longer a swing low, while bars 27 and 58 become swing lows that were not there before. Thirteen of the fifteen points are shared; two are gone and two are new, and the balance tips from eight highs and seven lows to seven and eight. The total matching is a coincidence of this series; the structure underneath it is not the same.

This is the general shape of what a discontinuity does to a rule that compares neighbours. It does not break the rule. The rule runs, returns an answer, and the answer is about a chart in which two of the bars being compared share no price at all. A structural point defined by bars that do not touch is a weaker thing than one defined by bars that do, and nothing in the output distinguishes them.

What a gap does to a stop

Lesson 21 put the stop below the structure. A stop is an instruction to trade at a price, and it is enforceable exactly as long as the market is willing to trade there. A gap is the market declining. The order is not cancelled; it becomes a market order at the other side of the hole.

That can be priced on this series. Take each of the nineteen gaps, suppose a position entered at the previous close with the stop a tenth beyond that bar’s extreme — below the low for a long, above the high for a short — and compare the loss the stop promised with the loss the best available fill on the gapping bar delivers.

Realised loss against the intended loss	Gap events, of 19	Share
under 1.5 times	8	42%
1.5 to under 2 times	3	16%
2 to under 3 times	5	26%
3 times or more	3	16%

The median event costs 1.5 times what the stop promised, the mean costs 2.0, and the worst two cost 4.5. One event in nineteen came in at exactly the intended loss. Read that last figure carefully: a stop that works as written is the exception among gap events, not the rule, and it is the exception by construction, because a gap event is defined as the case where the price was skipped.

Now the sizing consequence, which is the whole practical content of this lesson. If your risk budget is two per cent of the account and you want that to be what a bad morning actually costs rather than what you intended it to cost, you cannot size the position so that the stop distance is two per cent. On this series you would size it at two divided by four and a half, which is 0.44 per cent, if you wanted the worst observed event to land on your budget; or at two divided by two, which is one per cent, if you were content for the average event to land there. Somewhere between a fifth and a half of the size the stop distance suggests.

The familiar advice to use half a per cent overnight where you use two intraday is that arithmetic, arrived at from the other end. It is not a rule of thumb about caution. It is the number you get when you divide your intended risk by how far past the stop the market actually goes, and the reason it deserves a lesson is that almost nobody has measured the divisor on their own instrument.

Which of yesterday’s levels are still worth marking

Sort them by whether a price traded there. The high and the low of the previous session are prices at which business was done, and lesson 25 already said what sits at them; they survive the gap in the same way the sample extreme survived the regrouping. The close is a traded price and survives on the same grounds. So do the levels lesson 30 built from volume, because volume is a count of trades and a gap adds none.

What does not survive the same way is anything defined by adjacency. A swing point that needed two lower highs either side is now resting on bars that may never have overlapped. A sweep that needed price to go beyond a level and come back needs a bar to have reached across it, and no bar reaches across a gapped stretch: a level sitting inside one was never wicked at all, because price was below it and then above it with no bar spanning the two. Lesson 35’s careful distinction between a wick beyond and a close beyond has no purchase on a gap at all, and a sweep counter that does not test for the discontinuity will count one anyway.

And the gap creates a level of its own that has the opposite property. The empty stretch is a range at which no business was done in either direction, so nobody is positioned inside it, nobody is trapped there, and none of the arguments lesson 25 made about what sits at a level apply. Whatever a gap fill is, it is not a return to somewhere participants have unfinished business, because there is no business there to finish. It may be a real tendency and this lesson has not measured it; it is not the same kind of object as a level with trades behind it, and the two get drawn on charts with the same line.

Problems

1. Count the holes in your own data. Take two hundred consecutive bars of the instrument and interval you actually trade and count the transitions where the next bar’s low is above the previous bar’s high, or its high below the previous low. That count divided by 199 is how often your tape is discontinuous. If you trade a daily chart the answer will be large and if you trade five minutes inside a session it may be zero, and the two cases call for different sizing on the same account.
2. Measure your own overshoot. For each of those gaps, work out what a stop placed a tick beyond the previous bar’s extreme would have promised and what the best fill on the gapping bar would have delivered, and divide the second by the first. Write down the median and the worst. Those two numbers are the divisor in the sizing arithmetic above, and until you have them your risk-per-trade figure is a statement about your intentions rather than about your account.
3. Sort your own rules into the two piles. Go through the rules you actually use and mark each one according to whether it reads only closes or reads highs and lows. The first pile is blind to discontinuities and will keep reporting through them; the second pile changes meaning at every one. Neither is a reason to abandon a rule, and both are a reason to know which pile a given number came from before you act on it. How to Collect a Base Rate is how the count is kept honest.

Sources. Robert C. Merton, “Option Pricing When Underlying Stock Returns Are Discontinuous” (Journal of Financial Economics, 1976), for the founding treatment of exactly this problem — the whole paper exists because a discontinuity cannot be traded through, so no amount of adjusting the position while the market is open protects you from a price that never printed. Ole E. Barndorff-Nielsen and Neil Shephard, “Econometrics of Testing for Jumps in Financial Economics Using Bipower Variation” (Journal of Financial Econometrics, 2006), for the formal version of this lesson’s two piles: a method that splits observed variation into the part a continuous path could have produced and the part only a jump could. Yacine Aït-Sahalia and Jean Jacod, “Testing for Jumps in a Discretely Observed Process” (The Annals of Statistics, 2009), for how hard the last bound above really is — telling a jump from a fast move in sampled data is a testing problem, and the answer depends on how finely you sampled.

The discontinuity arrives at a boundary, and the boundary has a name. A session opens, runs and closes; the gap sits between the close of one and the open of the next, and the sessions themselves are not uniform inside — the first minutes and the last minutes behave differently from the middle, and they do so for reasons that are structural rather than psychological. Lesson 41 is about the shape of the day: what a session actually is, why its edges concentrate activity, and which of the quantities in this module have to be recomputed separately for the open, the middle and the close.

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