Maximum Gain: The Ceiling on What an Assembly Pays

Maximum gain is the highest figure an assembly can reach, and the first job is to say what it is measuring. The ceiling on the payoff counts nothing that was paid to put the assembly on. The ceiling on the profit counts every premium, carried to the end date. The payoff ceiling and the profit ceiling are two different numbers on one assembly, and some assemblies have no ceiling in the arithmetic at all.

A picture that has nothing to do with contracts makes the point. A household water tank holds four hundred litres. How much rain falls this monsoon decides whether the tank fills, whether it half fills, or whether it stays empty. The rain never decides how big the tank is. Four hundred litres is a property of the tank, welded into it before a drop fell, and it can be read off the tank on a dry afternoon in April.

A ceiling on an assembly of option contracts works the same way. Once the legs are written down, the highest reading the assembly can produce is already fixed, and the price of the reference asset only decides whether that reading is reached. Which means the ceiling can be worked out before any price is chosen, and that is exactly what makes it worth working out. A ceiling is one of the very few figures in this subject area that needs no view about anything at all: no view on direction, no view on how far, no view on when.

A ceiling is also one of the most misquoted figures in this subject area, and for a reason worth stating up front. The phrase maximum gain is two words that between them do not say which of two quite different numbers is being reported, and do not say at which price the number is reached. A reader who writes the phrase down beside a figure has recorded roughly half of what they worked out. The rest of this guide is about the other half.

What exactly is the ceiling a ceiling on?

Maximum gainThe highest reading an assembly can produce, once it has been said whether that reading is a payoff or a profit. is a phrase, not a definition, and it points at two figures rather than one. The two figures are not close together, and the gap between them is not a rounding matter. Which of the two is meant has to be settled before anything is quoted.

The first is the payoff ceilingThe highest amount an assembly pays at the end date, before anything paid to put it on is counted.. The payoff ceiling is the highest amount the assembly pays at the end date, counting nothing that was handed over to put it on. The figure is pure arithmetic on the levels and the signs. No premium enters it, and a payoff ceiling can therefore be produced for assemblies whose cost nobody can work out.

The second is the profit ceilingThe payoff ceiling after every premium in the assembly is counted, each one carried forward to the end date at the financing cost for the period.. The profit ceiling is that same figure less every premium in the assembly, each one carried forward to the end date at the financing cost for the period. Premiums leave at the start and payoffs arrive at the end, so the two amounts are not comparable until they have been brought to the same date. Carrying the premium forward is not decoration and it is not conservatism. Without that step the subtraction means nothing.

Two numbers on one assembly. A reader who quotes one and means the other has overstated or understated by the whole of the financed cost. On the assembly worked all the way through this guide, that error is Rs 61.70/- on a figure of Rs 2,000.00/-, a little over three per cent. On an assembly whose premium is a larger share of what it can pay, the same error swallows the whole answer.

There is a third thing the phrase leaves out, and it does more damage than the first two put together. A ceiling is reached at a price, or across a stretch of prices, and the figure on its own says nothing about which. The tank holds four hundred litres whether the monsoon is a record or a failure, and quoting the capacity without saying what actually fell is how a household ends up planning around water it never had. The same sentence, written about an assembly, is how a reader ends up planning around a payoff attached to a price nobody mentioned.

How is the ceiling read off the legs rather than off a picture?

Most readers meet a ceiling as a flat bit at the top of a drawing. A drawing has an edge, and the edge is a decision somebody made about how wide to make the picture. Recognising a ceiling in a drawing is easy; working one out from a drawing is not. A line that runs off the right-hand side of a diagram looks exactly like a line that has stopped. So work the ceiling from the rows instead.

Every leg is written the same way. A signed primitiveOne leg written out as four fields: a plus or a minus, a call or a put, one level, and one end date. Primitive means it cannot be broken down any further. is a plus or a minus, a call or a put, one level, and one end date. Group the rows by level and by date, put the levels in order along the price line, and then the whole question reduces to what happens at the two ends of that line.

The payoff keeps rising as the price rises only where the bought calls outnumber the written calls, and it keeps rising as the price falls only where the bought puts outnumber the written puts. That is a counting rule. The counting rule needs no diagram, no software and no view. Counting the call rows by sign and the put rows by sign gives both ends.

Now the part almost everybody skips, and it is the part that decides whether a ceiling exists at all. The two ends of the price line are not the same shape as each other. A price can rise without any stopping point in the arithmetic. A price cannot fall below nil. So the two answers the counting rule gives are not symmetrical in what they imply.

Where the bought calls outnumber the written calls, the payoff keeps rising as the price rises and nothing anywhere stops it, so the arithmetic sets no ceiling. Where the bought puts outnumber the written puts, the payoff keeps rising as the price falls, but the fall runs out at a price of Rs 0.00/-, so there is a highest reading after all and it sits at that price. And where neither is true at an end, the payoff simply flattens there, and the flat reading joins the list of candidates.

A short procedure follows. The readings at every level are written down, the flat reading is added at each end that flattens, the reading at a price of Rs 0.00/- is added where the downward end keeps rising, and the largest of them is taken. Between two levels the payoff is a straight line, and a straight line reaches its highest point at one of its two ends, so nothing in the middle can beat every one of the readings collected. The largest of that short list is the payoff ceiling, exactly, with no picture drawn at any point.

Here is one everyday version of the counting rule, easier to trust once it has been felt. A rent agreement that caps the annual increase at a stated percentage has a ceiling written into the paper. A rent agreement with no such clause does not, and it does not acquire one because the landlord happened to ask for little last year. The cap is a feature of the document; the amount asked is a feature of the year. Reading the document settles which of the two situations a tenant is in, and reading last year's receipt does not.

Which assemblies have a ceiling, and which have none at all?

Run the counting rule across four assemblies and three different kinds of answer come back. The three answers differ in kind rather than in size, and sorting them by size is what hides the difference. Everything below is built on one invented reference asset priced at Rs 2,000.00/-, paying nothing at all while it is held, with financing at 6.50 per cent for the year and one year to the end date.

The first case: no ceiling in the arithmetic

Plus one call at Rs 2,000.00/- for one year, held on its own. Bought calls one, written calls nil, so the upward end keeps rising. As the price rises the payoff rises with it, one rupee for one rupee, and no term anywhere in the contract stops it. The formula has no upper stop at all, and a formula with no upper stop is not the same thing as a payoff waiting to be collected.

Careless words do real harm at this point, so the point is worth stating plainly and then leaving. The arithmetic sets no ceiling. A large payoff does not follow from that: it is not available, likely or worth chasing on the strength of an absent upper stop, and saying how likely one is would need a figure for how far the reference asset might move. Two sentences that sound alike say completely different things: the formula has no upper stop, and the position can pay a great deal. Only the first has been established.

The call has a premium of Rs 180.00/-, a given figure in this invented example rather than a market price. Carried at 6.50 per cent for the year it becomes Rs 191.70/-. Notice what that does to the profit ceiling: nothing. Subtracting a fixed amount from something with no upper stop leaves something with no upper stop. So this assembly has no payoff ceiling and no profit ceiling either, and the honest entry in both cells is that there is none.

The second case: an exact ceiling

Plus one put at Rs 2,000.00/- for one year, held on its own. Bought puts one, written puts nil, so the downward end keeps rising. But the fall stops. A price cannot go below Rs 0.00/-. At that price the put pays Rs 2,000.00/- less nil, or Rs 2,000.00/-. The payoff ceiling of the bought put is Rs 2,000.00/- exactly, and it comes from the level and the floor under prices rather than from anything anyone chose.

Upward, the count is nil bought calls against nil written calls, so that end flattens. A put pays nothing when the price finishes above its level, so the flat reading there is Rs 0.00/-. The list of candidates is therefore Rs 2,000.00/- at the downward end, Rs 0.00/- at the level itself, and Rs 0.00/- at the upward end. The largest is Rs 2,000.00/-. Every premium in this assembly is known, so a profit ceiling can be produced here too, and the next section works it.

The third case: a ceiling that cannot be turned into a profit

Plus one call at Rs 2,000.00/- with minus one call at Rs 2,200.00/-, both for one year. Bought calls one, written calls one, so the upward end flattens, and the flat reading is the distance between the two levels, Rs 200.00/-. Puts nil against nil, so the downward end flattens at Rs 0.00/-. Readings at the levels are Rs 0.00/- at Rs 2,000.00/- and Rs 200.00/- at Rs 2,200.00/-. The payoff ceiling is Rs 200.00/-.

Then the profit ceiling stops dead. Rs 2,200.00/- is a declared levelA level chosen here so that a shape can be drawn. No premium attaches to it, because working one out would need a figure for how far the reference asset might move., chosen so that a shape could be drawn, and it carries no premium. The one level in this worked example that carries premiums is Rs 2,000.00/-. So this assembly has an exact payoff ceiling of Rs 200.00/- and no profit ceiling at all, and the second half of that sentence is said out loud rather than quietly left off. Why the premium cannot simply be supplied is worked in full further down.

The fourth case: a ceiling of nil

Four legs, all at declared levels: plus one put at Rs 1,600.00/-, minus one put at Rs 1,800.00/-, minus one call at Rs 2,200.00/- and plus one call at Rs 2,400.00/-. Calls one bought against one written, so the upward end flattens. Puts one bought against one written, so the downward end flattens. Both flat readings work out to minus Rs 200.00/-. The readings at the four levels are minus Rs 200.00/- at Rs 1,600.00/-, Rs 0.00/- at Rs 1,800.00/-, Rs 0.00/- at Rs 2,200.00/- and minus Rs 200.00/- at Rs 2,400.00/-.

The largest reading on that list is Rs 0.00/-, so this assembly's payoff ceiling is Rs 0.00/-, and the arithmetic says its payoff is never above nothing at any price. The payoff is nil across the whole stretch from Rs 1,800.00/- to Rs 2,200.00/- and below nil everywhere outside that stretch. A reader who has only ever seen the phrase maximum gain attached to a cheerful number should stop here for a moment. The arithmetic is perfectly capable of returning nil, and returning it is not a fault in the working.

What do the four ceilings look like worked end to end?

Here is the whole of it in one place, with every figure labelled as a payoff or a profit. Read the last two columns as a pair, never one without the other. The reference asset here is a teaching construction, the two premiums are given figures rather than market prices, and the levels other than Rs 2,000.00/- are declared so that shapes can be drawn.

One comparison in that table is worth making carefully. The careless version of it is wrong. Compare the two-level assembly with the bought call alone. Below Rs 2,000.00/- both of them pay nil, so at those prices they are equal, not different. Between Rs 2,000.00/- and Rs 2,200.00/- the written leg has not started working, so the two are still equal. Only above Rs 2,200.00/- do they part company. So the correct statement is that the two-level assembly never pays more than the bought call at any price, and pays less at every price above Rs 2,200.00/-. Saying it pays less everywhere is a different claim, and a wrong one. A comparison that says strictly less at every price is wrong wherever both legs pay nothing, and that is a large part of the price line.

Three numbers in this guide are all Rs 2,000.00/-, and they agree for three different reasons rather than because one was copied into another. The price of the reference asset is Rs 2,000.00/-. The pair carrying the two premiums is struck at the moneyWhere the level written into a contract and the current price of the reference asset are the same number., and at the money means exactly that the level and the price are one number, so that level is Rs 2,000.00/- too. And one unit of a reference asset priced at Rs 2,000.00/- refers to that much value, so the exposureThe value of the reference asset a contract is written on. Nobody has paid it and nobody holds it as cash. Exposure is simply the quantity the contract refers to. on one unit is Rs 2,000.00/- as well. None of the three is the same quantity as either of the others: one is a price, one is a level written into a contract, and one is a value nobody has paid. The bought put's payoff ceiling comes out at Rs 2,000.00/- for a fourth reason again. A price cannot fall below nil, so the most a put at that level can pay is the level itself.

Why is the ceiling on the payoff a different number from the ceiling on the profit?

Work it on the one assembly in this example that can be costed from end to end: plus one put at Rs 2,000.00/- for one year, premium Rs 57.93/-, a given figure here rather than a market price. Everything below is arithmetic on that single row.

The payoff ceiling is Rs 2,000.00/-, established above. The premium left the buyer's hands at the start of the year. The payoff, if it arrives at all, arrives at the end of the year. The premium and the payoff sit twelve months apart, and subtracting one from the other while they sit apart is the same mistake as adding this year's rent to last year's rent and calling the total a monthly figure. So the premium is carried forward at 6.50 per cent for the year first, and only then subtracted.

Carried forward, the premium becomes Rs 61.70/-. The financed premiumA premium carried forward from the day it was paid to the end date at the financing cost, so that it can be compared with a payoff arriving on that date. is what actually stands against the payoff. The profit ceiling is therefore Rs 2,000.00/- less Rs 61.70/-, or Rs 1,938.30/-. The two ceilings on this one assembly differ by Rs 61.70/-.

The gap looks small and matters a great deal. Rs 61.70/- against Rs 2,000.00/- is 3.0850 per cent, so the two figures round to the same thousand and read almost identically at a glance. The near-identity is exactly why the mistake survives: a reader who quotes the wrong one of the two is not obviously wrong on the face of it, and nobody checking the note can see the error in the number itself.

Why these figures hold to the paisa and never exactly

The put premium carried in this worked example is Rs 57.93/-, and it is a rounded figure. The parity relationship that ties it to the call premium of Rs 180.00/- produces Rs 57.9342723/- and the digits do not stop there. The rounding travels into everything derived from it, so the wording throughout is to the paisa rather than exactly, and the difference is printed rather than hidden.

Every one of those differences is under half a paisa, and not one of them is nil. The second fact is the whole reason for printing them. The profit ceiling of Rs 1,938.30/- is right to the paisa; worked from the rounded premium instead it would read Rs 1,938.30455/-. Writing exactly where to the paisa is meant would teach a reader to stop checking, and a reader who stops checking on a difference of half a paisa will not start again on a difference of six hundred rupees.

At which price does a ceiling actually bind?

A ceiling bindsThe price, or the stretch of prices, at which a ceiling is actually reached. A ceiling that binds at one price only describes a single point of the price line. somewhere, and where it binds is half the fact. Quote the figure without it and the reader supplies their own answer. The supplied answer is almost always wider than the arithmetic supports. So the habit is fixed and short: the figure, the label, the price, every time.

The bought put's payoff ceiling of Rs 2,000.00/- binds at one price and one price only, a price of Rs 0.00/- for the reference asset. Not near nil. Not below some level. At nil, and nowhere else. At Rs 1.00/- the put pays Rs 1,999.00/-, a rupee short of the ceiling, and at every price above that it falls away rupee for rupee. So the ceiling describes a single point on a line that runs from nil upwards without limit.

The two-level assembly is a different animal. Its payoff ceiling of Rs 200.00/- binds at every price at or above Rs 2,200.00/-, a stretch of prices rather than a point. Above that level the bought leg gains a rupee for every rupee the price rises and the written leg owes a rupee for the same rise, so the pair cannot move, and it sits at Rs 200.00/- forever. The four-leg assembly's ceiling of Rs 0.00/- binds across the stretch from Rs 1,800.00/- to Rs 2,200.00/-, a stretch with two ends rather than one open end.

Two ceilings of the same size can therefore describe completely different situations, and two ceilings of different sizes cannot be compared without knowing where each one binds. A ceiling that binds at one point and a ceiling that binds across a wide stretch are not the same kind of statement, and the figure on its own does not say which of the two is in hand.

Push the control all the way down to Rs 1,400.00/- and read the diagram again. The payoff climbs to Rs 600.00/- and the profit to Rs 538.30/-. Neither reading gets larger anywhere the control can go. The two dashed markers have not shifted by a pixel. The whole distance still separating the reading from the ceiling is the lime stretch of line running away to the left, and at any price at or below Rs 2,000.00/- that distance is exactly the price itself. At Rs 1,400.00/- the payoff is Rs 1,400.00/- short of its ceiling. At Rs 400.00/- it would be Rs 400.00/- short. Only at nil is it not short at all, and nil is off the end of the control on purpose.

Pushed the other way, past Rs 2,000.00/-, the put pays nothing at all. So the payoff goes to nil and stays there, and the profit sits at minus Rs 61.70/- and stays there, the financed premium being all that is left. The ceilings still have not moved. One sentence is worth carrying away from the whole treatment: the price decides whether a ceiling is reached and never where it sits.

What can be said about the ceiling of an assembly whose cost is unknown?

Most of the assemblies anybody meets sit in this position, so it is worth being precise about what survives and what does not. The two halves of the answer rest on completely different footings, and treating them as one is where invention starts.

A payoff needs no premium at all, so the payoff ceiling is exact. The payoff ceiling comes from the levels, the signs and the floor under prices, all of which are written into the contracts. For the two-level assembly the payoff ceiling is Rs 200.00/-, simply the distance between Rs 2,000.00/- and Rs 2,200.00/-. Nobody needs to know what either leg cost to produce that figure, and nobody needs a view about anything.

The profit ceiling cannot be produced at all. Producing it needs the premium at Rs 2,200.00/-, and no premium at that level was ever set down. Nor can one simply be worked out. Producing a premium at a level away from Rs 2,000.00/- would need a figure for how far the reference asset might move, and that figure belongs to a different subject. So the honest entry is a blank with a reason beside it, not an estimate.

The temptation at this point is enormous and worth naming. One cell in the table is empty, the shape of the answer is obvious, and a plausible number would make the table look finished. A plausible number would also make every figure derived from it an estimate wearing the clothes of arithmetic, and a reader would have no way to tell which of the two columns they were reading. So the table carries the payoff ceiling, carries no profit ceiling, and says which is which in the same sentence rather than letting the exact figure quietly stand in for the missing one.

The household version of the same discipline runs like this. A hall booked for a wedding holds four hundred people, and the capacity is written on the booking form. The caterer's charge per head is not known, and the quote has not come back yet. The capacity can be stated exactly and the bill cannot be stated at all, and writing down a guessed bill next to an exact capacity produces a document where nobody can see which number is which. The fix is not to guess better. The fix is to leave the cell empty and say why.

Does a ceiling say anything about how often it is reached?

No, and this is where the word gain does its damage. Gain sounds like something that happens. A ceiling is something that is written down. How often a ceiling is reached is a question about which prices actually arrive, and no count of levels and signs answers it.

The missing ingredient is worth stating exactly rather than vaguely. Working out how often a price arrives would need a record of how the reference asset has behaved, or a statement of how far it might move and how likely each move is. A historical series, a distribution, a statement of probability: each of those is an input of a different kind from a level and a sign, and no amount of arithmetic on levels and signs produces one. The answer to how often belongs to the study of how prices move, a separate subject.

The bus at the end of the street has fifty-two seats. Fifty-two is a fact about the bus, printed on the registration papers, true at midnight when the bus is empty and true at nine in the morning when it is full. Counting the seats says nothing whatever about how many people ride it. Anyone who quoted fifty-two as the ridership would be quoting a real number that answers a different question, and that is exactly what quoting a ceiling as an expected outcome does.

The second half of this matters more, and it is where readers who have understood everything so far still go wrong. A ceiling has two properties that are independent of each other: its size, and the width of the stretch of prices over which it binds. The bought put's ceiling is Rs 2,000.00/- and binds at one price only. The two-level assembly's ceiling is Rs 200.00/- and binds across every price at or above Rs 2,200.00/-. The four-leg assembly's ceiling is Rs 0.00/- and binds across the stretch from Rs 1,800.00/- to Rs 2,200.00/-.

A large ceiling that binds at one extreme price and a small ceiling that binds across a wide stretch cannot be ranked against each other at all. Ranking them would need exactly what is missing: a statement about which prices arrive and how often. A single figure invites a reader to read only the size, and reading only the size makes a ceiling misleading rather than merely incomplete.

How does somebody reading a position statement actually use a ceiling?

Four habits, and every one of them is a direct consequence of something above rather than general good practice. Somebody reviewing a set of positions, a lender looking at what a borrower has written against a holding, an analyst reconciling a note against the contracts behind it and a household reading its own record all do the same four things, and the failure being guarded against is the same one every time.

A figure on its own invites the reader to supply the two missing parts from imagination. So the most useful thing anyone does with a ceiling is refuse to write it down until all three of its parts are on the line.

1. Label the figure before recording itPayoff or profit, in the same cell as the number. On the bought put at Rs 2,000.00/- those are Rs 2,000.00/- and Rs 1,938.30/-, and a statement that shows one of them with no label has produced a number nobody can use. The two are not close enough to treat as one and not far apart enough to tell apart by eye.
2. Write the price beside it, and say whether it is a point or a stretchRs 2,000.00/- at a price of Rs 0.00/- and nowhere else is a completely different statement from Rs 200.00/- at every price at or above Rs 2,200.00/-. The first describes one point; the second describes an open stretch. A note that records only the figure has thrown away the part that decides how it should be read.
3. Count the legs by type and sign before trusting any flat bit in a pictureBought calls against written calls settles the upward end and bought puts against written puts settles the downward end. A picture whose line runs off the right edge looks identical to a picture whose line has stopped, and the counting rule tells the two apart in seconds without opening anything.
4. Leave the profit ceiling blank where a premium is missing, and write the reason in the cellNot an estimate, not a dash, and not the payoff figure moved across. The two-level assembly here has an exact payoff ceiling of Rs 200.00/- and a blank profit ceiling, and the blank carries more information than a plausible number would. The blank tells the next reader precisely which figure to go and find.

Notice what none of those four is. Not one of them is a view about whether the reference asset will rise or fall, whether the assembly is worth placing, or whether one shape is preferable to another. All four are acts of reading, and every one of them can be done on a Tuesday afternoon with nothing but the contracts and a calculator.

Should a reader choose an assembly by its ceiling?

No, and the reason is substantive rather than a matter of caution. The reason sits in the block above: a ceiling is a fact about a formula, and choosing between two formulas needs facts about the world that no formula contains.

An open question without a list is just a shrug, so the three things that would have to be known first are worth naming. Not one of the three appears above. First, a view on how far the reference asset might move and how likely each move is: only such a view lets anyone weigh a large ceiling reached at one price against a small one reached across a wide stretch. Second, the circumstances of whoever holds the assembly. The same obligation means different things to different holders. Third, what each assembly costs to place, to hold and to unwind, a cost that for most of the assemblies above cannot be produced at all, as the missing premium showed at length.

The pull in the other direction is worth being blunt about. A ceiling drawn clearly has already made the thing feel available, and the word gain does the rest. A drawn ceiling is not evidence that it is likely to be reached, that a capped shape is preferable to an uncapped one, or that any of the four assemblies is worth holding. The order of the treatment follows from that: the legs first, then what they oblige, then what they pay, then what they cost, and last what cannot be reached at all.

A ceiling is a fact about a formula, and a fact about a formula is not a reason to take on the obligation underneath it. That sentence is the whole of the answer, and the four assemblies above are there so that the sentence has something concrete standing behind it.

The reference asset, its price of Rs 2,000.00/-, the financing cost of 6.50 per cent for the year, the two premiums and every level used above are invented.
Educational material. Not advice on any investment, tax, budget or market position.