Module 7 Quiz: The Other Side

This module turned round and looked at the people taking the other side of your fills, and found every one of them under an obligation rather than an opinion: a quoter who must cover adverse selection, a router steered by a published fee, a firm whose quote is an option it has written, a desk that owns a schedule rather than an order, and a dealer whose trading is the derivative of a hedge. Seven questions, all arithmetic. The last one takes the single 300-share round trip this module has been adding charges to since lesson 53 and finishes counting it.

Covers: Lessons 53 to 61, and the 300-share penny-wide round trip that lesson 53 priced at $3.00 and lesson 54 raised to $4.80.

Every question below hands you numbers and asks for a number back. Work all 7 with a calculator before you scroll to the answers; each answer shows the arithmetic, so a wrong result tells you which step to go back to rather than only that you were wrong.

The questions

1. What a spread would have to believe

Lesson 53’s zero-profit condition says a break-even spread is twice the share of arriving orders that are informed, multiplied by how far the price moves once what they know is public: s = 2aD. Turned round, a = s ÷ (2D).

Four spreads you can find on any screen: two cents, four cents, twenty cents and a dollar. Three sizes of informed edge: fifty cents, two dollars and eight dollars.

Ask. For each of the twelve pairs, what share of arriving orders would have to be informed for that spread to break even? Which cell refuses to answer, and what is it telling you?

2. The decision you never see and never pay for

A 500-share order. Taking liquidity is capped by rule at three tenths of a cent a share; assume the maker-taker venue sits at the cap and rebates 0.0022 to the resting side, and that an inverted venue pays 0.0020 to take and charges 0.0028 to rest. A wholesaler filling the order internally involves no exchange and no exchange fee.

Then the same 500 shares as an ordinary round trip: in and out with marketable orders, on an instrument quoted a penny wide, and again on one quoted two cents wide.

Ask. What does each of the five outcomes pay or cost on 500 shares, and how far apart are the best and the worst? What does the round trip cost once the access fee is added at each spread, and at what spread does the fee stop adding more than a fifth?

3. The delay at which speed starts to matter

A resting quote is an option written by whoever posted it, and its life is the time between the world changing and the cancellation landing. So the value of a delay is the value of an option with that life, which lesson 44 already taught you to compute: annual volatility divided by the square root of 252 gives a day, divided by the square root of 23,400 gives a second, and multiplied by the square root of the delay gives the rest.

Three instruments. One at $250 with 20 per cent annual volatility quoted a penny wide. The same instrument quoted five cents wide. And a small company at $12 with 55 per cent annual volatility quoted twenty-five cents wide.

Ask. What does each move in a second, and at what delay does a typical move reach the half-spread?

4. The clock, and where the two costs are equal

A desk has 400,000 shares to buy of a $60 stock that trades 2,000,000 shares a day at 35 per cent annual volatility and is quoted three cents wide. Lesson 57’s convention is to run at ten per cent of the day’s volume.

Two costs work against each other. The timing exposure is the daily standard deviation times the square root of the sessions used. The impact, by lesson 59’s square-root law, is that same daily standard deviation times the square root of the order’s days of volume divided by the sessions used. Add a willingness number for how much certain cost you will pay to shed a dollar of exposure, and the sum has one minimum.

Ask. How many sessions does the convention take, and what is the typical move over that span? Where is the minimum at a dollar of certain cost for a dollar of risk, what participation does it ask for, and what is true of the two terms there? And what willingness does the ten-per-cent convention imply?

5. A hedge is a position and its trading is the derivative

A dealer has sold options and holds shares against them. The hedge is a function of the price, so it cannot change until the price has. Here is the hedge, in shares, at ten successive rebalances: 440,000, 310,000, 520,000, 380,000, 660,000, 450,000, 720,000, 540,000, 830,000 and 1,000,000.

Then the rate at which it changes. On the same book, a one-dollar move obliges 46,000 shares of rehedging with twenty bars to expiry, 94,000 with five, and 212,000 with one. The underlying trades 8,000,000 shares a day at 1.5 per cent daily volatility and sits at 103.

Ask. How many shares did holding the hedge trade, how large did the position ever get, and how far did it end from where it started? And what impact does each of the three rehedges cause, as a share of the move that caused it?

6. The row every stop-hunting table leaves out

You are long at 240 with support at 239. The tight stop goes at 237, just under the level; the wide stop goes at 234, a volatility buffer below it. Three things can happen. The level holds and the price rallies to 246. Or the price runs to 236 and is reclaimed, then rallies to 246. Or the level genuinely breaks and the price goes to 232.

Then a count on your own instrument: 24 approaches, of which 18 held, 6 ran through the level, and 5 of those 6 were reclaimed.

Ask. Take both stops at 200 shares and use only the second and third states: at what run-and-reclaim rate are they equal? Now size them to the same risk and ask again. Then put the first state back and find the condition in full. And what does the count say?

7. One order, priced by every lesson in the module

The trade this module has been adding charges to. Three hundred shares of a $100 stock quoted a penny wide, in and out with marketable orders. The stock trades 2,000,000 shares a day and its daily standard deviation is 1.5 per cent. The access fee sits at the Rule 610 cap of three tenths of a cent.

Then the same trade at a hundred times the size: 30,000 shares, which is 1.5 per cent of the day’s volume.

Price the impact of the entry alone, because lesson 59 says part of the exit’s impact reverts and this page has no way to split it.

Ask. What does each trade cost in spread and access fee, what does the square-root law estimate for the impact, and at what order size do the two meet?

The answers

Each one is worked in full. Where a figure comes from a lesson rather than from this page, the lesson is named.

1. What a spread would have to believe

Quoted spread	Edge of 0.50	Edge of 2.00	Edge of 8.00
$0.02	1 in 50	1 in 200	1 in 800
$0.04	1 in 25	1 in 100	1 in 400
$0.20	1 in 5	1 in 20	1 in 80
$1.00	every order	1 in 4	1 in 16

Every cell is one division. Two cents against an eight-dollar edge is 0.02 ÷ 16 = 0.00125, which is one order in eight hundred. That is not a claim about how many clever people are in the market; it is what the quote implies if the quoter is breaking even, and it is why two cents is quotable against an earnings-sized move at all. The whole business is a very small edge earned very often against a very large loss taken very rarely.

Now the cell that refuses. A dollar spread against a fifty-cent edge gives a = 1.00 ÷ 1.00 = 1, meaning every single arriving order would have to be informed, which cannot be true of any market that trades at all. The model has been handed a spread it cannot explain, and the honest reading is not that the quoter is greedy. It is that adverse selection is not what most of that spread is paying for.

Which is the useful thing the formula does. It gives the most of a spread that information could possibly account for, and whatever is left over belongs to inventory risk and to fixed cost — the split lesson 5 said existed and could not measure. A wide spread in a thin name is mostly the price of being stuck with it.

One caution the table carries in its own headings. Every figure is an implication of an assumed edge, not an observation of one. Move along a row and the implied share of informed orders changes by a factor of sixteen without a single quote changing. The table asks what a spread would have to believe. It does not find out what it does believe.

Answer. From 1 in 800 at the top to 1 in 4 at the bottom, and the dollar spread against a fifty-cent edge returns every order, which is not a number about information at all.

2. The decision you never see and never pay for

Where the order ends up	Per share	On 500 shares
Maker-taker venue, order crosses	−0.0030	−$1.50
Maker-taker venue, order rests and is hit	+0.0022	+$1.10
Inverted venue, order crosses	+0.0020	+$1.00
Inverted venue, order rests and is hit	−0.0028	−$1.40
Internalised by a wholesaler	0.0000	$0.00

The top and the bottom of that column are 0.0052 apart, which is $2.60 on 500 shares, on one side of one trade. Notice which two rows are furthest apart. It is not a fast row against a slow row, because no such distinction appears anywhere in the table. It is the crossing row against the resting row, and then the venue type on top of that, and a fee schedule published months in advance settled all of it.

Then the round trip. A penny spread means half a cent a share crossed, which is $2.50 a side and $5.00 for the round trip; the access fee at the cap is $1.50 a side and $3.00 for the round trip, so the true bill is $8.00 and the fee has added 60 per cent. At two cents the spread cost doubles to $10.00 while the fee stays at $3.00, so the same fee adds 30 per cent.

That gives the crossover directly, because the fee is fixed per share and the spread is not. The fee adds a fifth when 0.0030 is a fifth of the half-spread, which puts the half-spread at 1.5 cents and the spread at three cents. At six cents it adds a tenth. So this is a penny-and-two-cent-instrument problem, and a reader who trades wider things has just found out that lesson 54 is not about them.

And the reason the schedule decides rather than you: on a commission-free account you neither pay the take fee nor receive the make rebate. The whole first table is somebody else’s profit and loss, which is exactly why it cannot move your behaviour and can move your broker’s completely.

Answer. A swing of $2.60 on one side of one trade; $8.00 and $13.00 for the round trips; and the fee adds a fifth at a three-cent spread.

3. The delay at which speed starts to matter

The first instrument moves 250 × 0.20 ÷ 15.87 = $3.1506 in a day, and $3.1506 ÷ the square root of 23,400 = $0.0206 in a second. Setting that equal to the half-cent half-spread and solving for the delay gives (0.005 ÷ 0.0206) squared = 0.0589 seconds, or 58.9 milliseconds. The third instrument moves 12 × 0.55 ÷ 15.87 = $0.4159 a day, which is 0.272 cents a second, and against a 12.5-cent half-spread that is 2,114 seconds.

Instrument	Move in one second	Delay at which it equals the half-spread
$250, 20 per cent, penny spread	2.060 cents	58.9 milliseconds
$250, 20 per cent, five cents wide	2.060 cents	1.47 seconds
$12, 55 per cent, twenty-five cents wide	0.272 cents	35.2 minutes

Read the first two rows against each other. Nothing about the instrument changed except the quoted spread, and the threshold moved by a factor of twenty-five, because the threshold goes as the square of the half-spread. Then read the third row: on a twelve-dollar company quoted a quarter wide, a typical move reaches the half-spread after thirty-five minutes. Nobody needs to be fast there and nobody is, because the entire quoting problem is inventory rather than latency.

Now place the participants on that scale. A retail order leaving a home connection takes somewhere between 50 and 200 milliseconds; a machine in the same building as the matching engine measures its round trip in tens of microseconds. On the first row the threshold sits between those two, which is the only row where the race is a race. On the third it sits eleven thousand times past the slowest participant, and the whole question stops being about speed.

What none of this says is that being faster wins more. The most that can be taken from a stale quote is the amount by which it is stale, and a quote standing for a few hundred microseconds is stale by hundredths of a cent. Speed buys a larger share of a fixed number of small prizes, not a larger prize.

Answer. 2.060 cents and 0.272 cents a second; and thresholds of 58.9 milliseconds, 1.47 seconds and 35.2 minutes.

4. The clock, and where the two costs are equal

The order is 400,000 ÷ 2,000,000 = 0.20 days of volume, and at ten per cent participation the sessions are the days of volume divided by the rate, which is 2.00. The daily standard deviation is 60 × 0.35 ÷ 15.87 = $1.3233, so the typical move over two sessions is $1.3233 times the square root of two, which is $1.8714, or 3.12 per cent of the price. Set that against the 1.5-cent half-spread and it is 125 times larger. At this size the spread is a rounding error and the clock is the bill.

Now the minimum. Differentiating the sum gives an optimum of the square root of the days of volume divided by the willingness, which at a dollar for a dollar is the square root of 0.20, or 0.4472 sessions. The participation that asks for is the days of volume divided by the sessions, which is 44.7 per cent — four and a half times the convention.

Schedule	Sessions	Impact a share	Exposure a share	Sum
Ten per cent participation	2.0000	$0.4184	$1.8714	$2.2898
The optimum at a dollar for a dollar	0.4472	$0.8849	$0.8849	$1.7698

Look at the two middle columns in the second row. The impact and the exposure are equal, at $0.8849 each, and that is not a property of these particular numbers. Whatever the volatility, whatever the volume, whatever the willingness, the schedule that minimises the sum is the one where the impact you pay equals the risk-weighted exposure you carry. You do not need either constant to use it: estimate both terms for the schedule you are running and the larger one says which way to move.

Notice also what the optimum does not contain. The volatility cancels and so does the price. The order enters only through its ratio to the day’s volume, which is the same quantity the schedule reduced to in the first place.

Then run it backwards. Two sessions is the optimum when the square root of 0.20 divided by the willingness equals 2.00, which puts the willingness at 0.2236. The convention is optimal for a desk that values a dollar of timing risk at twenty-two cents, and weighting the raw $1.8714 exposure by that figure gives $0.4184, which is exactly the impact in the same row. The convention is not the model’s answer. It is a choice, and anybody presenting ten per cent as the output of an optimisation is presenting a preference as a calculation.

Answer. Two sessions and $1.8714, against an optimum at 0.4472 sessions and 44.7 per cent participation where both terms are $0.8849; and the convention implies a willingness of 0.2236.

5. A hedge is a position and its trading is the derivative

The nine differences are 130,000, 210,000, 140,000, 280,000, 210,000, 270,000, 180,000, 290,000 and 170,000, which total 1,880,000 shares. The largest position the hedge ever holds is 1,000,000, and the net change from first reading to last is 560,000. So the trading was 1.88 times the largest position and 3.36 times the net change.

That ratio answers every headline of the form: dealers have eleven billion dollars of hedges to unwind. Eleven billion is a position. What reaches the market is its derivative, and a hedge that does not move trades nothing at all.

Bars to expiry	Shares rehedged by a $1 move	Impact	Share of the move
20	46,000	0.1137%	11.7%
5	94,000	0.1626%	16.7%
1	212,000	0.2442%	25.2%

Now the second half. A dollar on a 103 instrument is a 0.97 per cent move. Lesson 59’s law puts the impact of rehedging at the daily volatility times the square root of the shares over the day’s volume, so 212,000 shares against 8,000,000 gives 0.015 times the square root of 0.0265, which is 0.2442 per cent — a quarter of the move that caused it.

Read the column downwards and the honest statement is a range with an input in it: hedging flow adds something between a ninth and a quarter of a move here, and the figure moves with the volume, the distance to expiry and the distance from the strike. It does not create the move. It cannot, because it arrives afterwards.

And one thing the arithmetic cannot supply. Reverse the sign — a dealer long the options rather than short — and the same 212,000 shares damp the move instead of amplifying it, because rehedging a long position sells into strength. Open interest counts contracts, not sides, so a strike showing a large number tells you the last hour will be busy near it and tells you nothing whatever about the direction.

Answer. 1,880,000 traded against a largest position of 1,000,000 and a net change of 560,000; and impacts of 11.7, 16.7 and 25.2 per cent of the move.

6. The row every stop-hunting table leaves out

Start where the literature starts. At 200 shares each, the tight stop loses $600 in both of the two states drawn, because 236 is below 237 either way. The wide stop makes 200 × 6 = $1,200 if the run is reclaimed and loses 200 × 6 = $1,200 if it breaks. Setting −600 equal to 2,400p − 1,200 gives p = 0.25, and that is the familiar answer.

It is also an artefact of the sizing. The tight stop risks $600 and the wide one risks $1,200, so they are not the same bet. Lesson 9 says to compare them at the same risk, which puts the wide stop at 100 shares. Its payoffs halve to +$600 and −$600, and it now beats the tight stop at any run-and-reclaim rate above zero at all. The tight stop’s expectation contains no p, because it loses $600 in both columns, which makes it a refusal to bet rather than a cheaper bet.

Which proves too much, and that is the tell. A rule that is right whatever the world does has usually been asked the wrong question, and the wrong question is that the table has two columns.

Outcome	Tight stop, 200 shares	Wide stop, 100 shares
Level holds	+$1,200	+$600
Run, then reclaimed	−$600	+$600
Real breakdown	−$600	−$600

The missing state is the one where nothing happens: the price approaches, the level holds, no run occurs, and the rally arrives with neither stop touched. There the tight stop is long 200 shares into a six-point rally and makes $1,200 while the wide stop is long 100 and makes $600. It is the state in which being tight pays, and it is left out of the literature because the literature is about being hunted.

Write q for a clean hold and p for a run and reclaim. The tight stop is worth 1,800q − 600 and the wide one 1,200q + 1,200p − 600. Subtract, and the difference is 1,200p − 600q, so the wide stop wins when p is greater than q over 2. Not a quarter, not zero, but half the probability that the level simply holds — and both terms are countable on your own instrument.

So count them. Eighteen holds in 24 approaches is q = 0.750, and five reclaimed runs is p = 0.208. The threshold is 0.375 and p does not reach it, so on these counts the tight stop is the better of the two. Notice which term decided it. Not the run-and-reclaim rate the whole argument fixates on, but the hold rate, which the argument never mentions.

Answer. 0.25 unsized on two states, zero when they are sized to the same risk, and p greater than q over 2 once the third state is there — which the count fails, 0.208 against 0.375.

7. One order, priced by every lesson in the module

The two charges with published numbers are the ones lessons 53 and 54 counted. Half a cent a share crossed, twice, is a cent a share, or $3.00 on 300; the access fee at the cap, twice, is 0.006 a share, or $1.80. Both scale exactly with size, so at 30,000 shares they are $300.00 and $180.00.

The impact does not scale that way, and that is the whole of the result. The law puts the fractional move at the daily volatility times the square root of the order over the day’s volume, so 300 shares against 2,000,000 gives 0.015 times the square root of 0.00015, which is 0.0184 per cent. On a $100 stock that is 1.84 cents a share, or $5.51. At 30,000 shares the fraction is ten times larger and the shares are a hundred times more numerous, so the dollar cost is a thousand times larger: $5,511.

Charge	300 shares	30,000 shares
Spread, both crossings	$3.00	$300.00
Access fee at the cap, both crossings	$1.80	$180.00
Subtotal, the two charges with published numbers	$4.80	$480.00
Estimated impact of the entry alone	$5.51	$5,511.35
Impact as a multiple of the subtotal	1.15	11.48

Set the two per-share figures equal and the crossover falls out. The spread and the fee come to 1.6 cents a share for the round trip; the impact is 100 × 0.015 times the square root of the order over 2,000,000. They meet at 228 shares, which is below the trade in the left-hand column. On the numbers as written, this order is already past the point where the law says impact is the larger charge.

Which is exactly where lesson 59’s concession has to be taken seriously rather than skipped. The law carries a coefficient that published calibrations put anywhere between a half and one and a half, and the crossover moves as the inverse square of it: at a coefficient of a half the two charges meet at 910 shares instead of 228. So the answer to whether impact is your largest charge depends on a constant nobody publishes, and a reader who takes the 228 as a fact has taken the wrong thing from the arithmetic. What is not in doubt is the shape: the first two charges are linear in size and the third goes as size to the power of one and a half, so there is a crossover, and above it the two charges this module could price stop being the bill.

And when the fill prints, lesson 56 has the last word on what anyone else can read from it. The tape carries the price, the size and whether it happened away from an exchange. The one field it never carries is the side. Every charge on this page was paid by somebody whose direction the print does not record.

Answer. $4.80 against $5.51 at 300 shares and $480 against $5,511 at 30,000; the crossover is 228 shares at a coefficient of one and 910 at a half.

What this quiz was testing

Whether you can read the other side as a set of obligations with numbers attached. Handed a spread, you invert it and find what it would have to believe, and notice when it refuses to answer; handed a fee schedule, you price a decision that is made about you and paid for by somebody else; handed a price and a volatility, you find the delay at which speed is a factor and the instruments where it is not; handed an order too large for a day, you balance the clock against the impact and find the two equal at the bottom; handed a hedge, you take its derivative; and handed a payoff table, you check whether a state is missing from it before you solve it.

Module 8 turns all of it inward. Lesson 62 takes one ordinary observation and judges it twice: six wins in nine trades beats a coin and gets adopted, and the same six in nine against a base rate of 54.24 per cent settles nothing whatever. The difference is not the data. It is what the observation was compared against, and naming that in advance is what turns a noticing into a hypothesis.

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