Arbitrage: What It Means Precisely, and Why It Is Rare

Arbitrage in its precise sense is the narrowest position a fund can take: buying and selling the same thing at two prices standing at the same moment, with both sides executed, so the difference is locked rather than expected. Such a position is rare because the difference has to be larger than the cost of taking it, and almost everything the word is used for has already dropped one of those conditions.

Arbitrage is a word before it is a position. Most approaches begin with a mechanism and work it. Arbitrage begins with a definition, and the definition is the mechanism. The same word is doing two completely different jobs in ordinary financial language. In one job it names a specific and very narrow thing. In the other it is a compliment paid to a position somebody likes the shape of. A reader who cannot tell which job the word is doing in a given sentence has no way of knowing what they have just been told.

A covered market makes the starting point concrete. Two vegetable sellers stand at opposite ends of the same covered market. Both have the same onions, off the same lorry, in the same condition, and both are open right now. One is asking Rs 30 a kilo and the other is asking Rs 32. The walk from one to the other takes ninety seconds carrying a sack. Two standing prices on one identical sack, and nothing else, is what the word arbitrageIn its precise sense, buying and selling the same thing at two prices at the same moment, both sides done. means when it is used precisely. Not that onions look cheap. Not that onions are likely to get dearer. The same sack, two prices, both standing at once, and a walk between them.

Everything else in this guide follows from taking that little scene seriously, including the two things people usually skip. The first is that the walk is not free: the ninety seconds, the sack, and the fact that both sellers can see the buyer coming are all real, and they come out of the Rs 2. The second is that if either seller has closed by the time the buyer arrives, what is held is not a locked difference at all. The buyer is holding onions.

What does the word mean when it is used precisely?

The word means a position in which the same thing is bought and sold at two prices that are standing at the same moment, with both sides actually executed. The sentence is a definition rather than a description, and definitions have conditions. Three of them, and every one has to hold at once.

Condition one is that it is the same thing. Not a similar thing, not a related thing, not a thing that has moved with it for as long as anybody has been watching. The identical claim. Whatever happens to one side happens in equal and opposite measure to the other. Condition two is that the two prices are standing at the same moment. Both quoted now, both available now, not one available now and the other expected later. Condition three is that both sides are actually done. Executed, filled, in the book. Not one side done and the second one intended, and not one side done and the second one waiting on somebody's decision.

Take any one of the three away and something remains, but it is not this. The remainder is a position whose outcome depends on something: on how two different things move relative to each other, on where the second price has got to by the time it is reached, or on whether the second side ever happens at all. Each of those dependencies has its own name, its own mechanism and its own way of losing money, and each is covered separately. None of them is what this word means when it is used precisely.

One thing is worth stating bluntly. A position that fails one of the three conditions is not worse, or careless, and nobody running one has made a mistake by running it. Such positions are different things, and the whole purpose of a definition is to keep different things apart. The three conditions are a test of category and never a test of quality. A position that fails all three can be perfectly well constructed and perfectly well understood. The position simply is not arbitrage in the precise sense, and calling it that hides the exact thing a reader most needs to see.

How much work is the phrase the same thing doing?

Almost all of it, and this is where the word is most often stretched. Back in the covered market: if one seller has onions and the other has onions, it is the same thing. If one has onions and the other has potatoes, nobody would call the Rs 2 difference a locked one. Onions and potatoes are simply two vegetables with two prices and no reason on earth to stay Rs 2 apart. Everybody sees that instantly with vegetables. The same sentence becomes much harder to see when both sides are financial claims that look alike on a screen.

The test that does the work is not whether the two things are similar. The test is whether they are the same claim on the same underlying obligation. Anything happening to one then happens in equal and opposite measure to the other, automatically, without anybody deciding anything. Two claims on the same thing, priced differently at one moment, is the precise sense. Two claims on two things that have moved together for years is not, however tight that relationship has looked. A relationship that has held is a description of the past and not a property of the position.

Here is the household version of that distinction. A person buys the same brand and weight of cooking oil in two shops on the same street for Rs 180 and Rs 174, and carries a tin from one to the other. Two shops, one identical tin, two prices standing at once. A person buys cooking oil and sells groundnut oil, on the reasoning that the two have kept about Rs 20 apart for as long as they have been shopping, is doing something else entirely. The reasoning might be sensible. The reasoning might be well judged. The position is still not a locked difference, and the Rs 20 is a habit of two prices rather than a property of one thing. When the habit breaks, and habits do break, there is nothing structural to pull it back.

What changes when the two prices are not standing at the same moment?

Everything, and it changes quietly. Quiet change is what makes this condition the easiest one to lose without noticing. The two prices have to be available at the same moment because that simultaneity is what turns a difference into a locked one. A purchase now and a sale in an hour is not a difference at all. The two legs are a purchase, and then, separately, a sale at whatever price exists in an hour.

Consider a wedding caterer who agrees a price for 400 kilos of rice with a buyer on Monday and plans to buy the rice on Thursday. The caterer has not locked anything. The caterer has a fixed obligation on one side and an open exposure on the other, and three days in which the rice price can do whatever it does. If rice falls, the caterer keeps more than expected. If it rises, the caterer can lose more than the whole margin on the booking. The caterer might well take that on with their eyes open, and caterers do it every season. The booking is not the same shape as walking a sack across a market where both sellers are open right now.

The moment the two legs are separated in time, the second one becomes a forecast, and a forecast is exactly the thing the precise sense was defined to exclude. This is also why the precise sense is short-lived wherever it does exist. A difference that only holds while both prices are standing is a difference that stops existing the instant one of them moves, and prices move.

What is the difference between both sides done and one side intended?

The difference is a whole risk, and it is the condition people forget because it sounds like paperwork. Condition three says both sides are actually executed. Not agreed in principle, not queued, not conditional on somebody else. Done.

Why this matters is easiest to see half done. The sack of onions is bought at Rs 30 and carried across the market, and the second seller has shut for lunch. The buyer now holds onions at Rs 30. The holding is not a difference of Rs 2 between two prices. The outcome from here is the onion price, in full, however far it goes. The position intended and the position actually held are not related to each other, and the change happened at the moment the second side did not fill.

Where the side left uncovered is a sale of something the seller does not hold, the loss on it is not bounded by anything. There is no ceiling on what the thing can cost to buy back, and how a sale of something not held is arranged is covered separately. A position with one side done and one side outstanding is a single position on a price, whatever the trade ticket was supposed to be.

One more thing sits inside condition three, and a reader is least likely to have thought about it: settlementThe completion of a trade, when the instrument and the cash actually change hands.. Doing both sides and having both sides complete are two different events separated by days, and the gap between them is not free. The settlement gap is where the second half of the arithmetic comes from, and it is the reason a difference that looks locked on a screen is smaller than it looks.

What does taking a gap actually cost?

Now the arithmetic, and it is the half of the subject that gets skipped. The definition is interesting and the subtraction is not. The subtraction is the whole reason the precise sense is rare, so it is worth doing slowly and in full.

Everything below belongs to Nilgiri Absolute Return Fund, invented, an open-ended vehicle managed by Nilgiri Alternatives Advisors Private Limited, whose operational counterparty is Marudhar Securities Private Limited. The two rates used are that fund's own contracted terms with that counterparty and nothing else. Those two rates are contracted terms between one fund and one counterparty, and a fund on different terms would face a different floor entirely.

The set-up. The same instrument is available at Rs 100.00 in one place and at Rs 100.40 in another, at the same moment. The gapThe difference between the two prices, which is what the position starts with before costs. is Rs 0.40 a unit. On a price of Rs 100.00 that is 0.40 per cent. Buy at Rs 100.00, sell at Rs 100.40, both sides done. All three conditions hold. The position is the precise sense, and every other position named below has dropped one of the three conditions.

Now take the costs off, every one of them named. Transaction costWhat each execution costs, charged on each side separately. at this fund's contracted 0.05 per cent of value on each execution is Rs 0.05 a unit, and there are two executions rather than one. A position with two sides is charged on both. The two together are Rs 0.10. Then financingInterest on the cash used between doing the trade and settling it.: the cash is out for the two days between doing the trade and settling it, and at this fund's contracted 8.50 per cent a year on Rs 100.00 those two days come to Rs 0.0466, written as Rs 0.05 throughout. Total cost Rs 0.15 a unit, and every figure below rests on that Rs 0.15.

Rs 0.40 less Rs 0.15 is Rs 0.25 a unit. The position keeps Rs 0.25 a unit. Against a difference of Rs 0.40, the kept amount is 62.5 per cent of what the two prices were showing. Put another way, before anything at all has gone wrong, and with all three conditions holding perfectly, more than a third of the visible difference has already been spent on taking it.

How big is that profit against the position that produced it?

Rs 0.25 a unit is a number without a size until it sits next to what had to be committed to get it. On 25,00,000 units a side, taken at Rs 100.00 a unit, the buy is Rs 25,00,00,000 and the sale is another Rs 25,00,00,000. The gross positionBoth sides added together, which is what the profit is measured against. is Rs 50,00,00,000. Both sides are real, and both had to be financed, margined and settled.

The profit is Rs 6,25,000. Against a gross position of Rs 50,00,00,000 that is 0.125 per cent. The whole result of a position that met all three conditions perfectly, on a difference of 0.40 per cent between two prices, is an eighth of one per cent of the money involved. That is not a complaint and it is not a criticism of the arithmetic. The shape of the position is simply that, and it is the reason positions of this kind are described by size and repetition rather than by how much any one of them made.

Drawing it to scale is the only way to feel it. On a bar the width of the screen, the profit is about the width of a pencil line. The figure below draws the real scale first and then magnifies the relevant sliver two hundred times, rather than quietly redrawing the bar to make the profit visible.

What gap has to be cleared before the position makes anything?

Rs 0.15, the total cost of putting the position on. Rs 0.15 is the single most useful number in the whole subtraction, and it deserves its own name: the break-even gapThe gap at which the position makes exactly nothing, being equal to the total cost.. Below it, the two prices genuinely differ and the position still loses money.

The claim is not intuitive, and it is the centre of the whole subject. At a gap of Rs 0.12, the same instrument really is available at two different prices at the same moment, both sides really can be done, all three conditions really do hold, and taking it loses Rs 0.03 a unit. Nothing has gone wrong. Nobody has been unlucky. The difference was simply smaller than the cost of collecting it. At a gap of Rs 0.15 exactly, the position makes nothing at all, having done everything correctly.

Plotted against the gap, the profit is a straight line. Both the gap and the cost are per unit, and the cost does not move. The line starts below zero at minus Rs 0.15 when the gap is nothing at all. Doing this trade for no gap whatsoever costs Rs 0.15 a unit, being Rs 3,75,000 on 25,00,000 units. The line crosses zero at Rs 0.15 and rises at one rupee of profit for every rupee of gap after that.

Why is the precise sense rare?

Because of what taking a difference does to it, and because of what is left over when the ones worth taking have been taken. Those two facts are the whole argument, and both are arithmetic rather than a position about how markets behave.

The covered market shows it again. Two sellers, same onions, Rs 30 and Rs 32. When a sack is bought at Rs 30 to sell at Rs 32, two things happen to the two prices at once. Stock comes off the cheap seller, who now has less to sell and no reason to keep asking Rs 30, and stock is added at the dear end, where a buyer at Rs 32 now has a choice. The act of taking a difference is the act of closing it, and the second person to notice the same difference finds it smaller than the first person did. Do it at scale and the difference is gone before the tenth sack.

So consider what can still be standing by the time anybody else looks. Only three kinds of difference survive being noticed. There are the ones too small to be worth the walk, which on this fund's contracted rates means anything under Rs 0.15 a unit, and there are a great many of those. The cost is a floor that does not move. There are the ones that existed for so short a time that nobody reached both sides while both were standing, and a difference nobody could reach is not one anybody collected. And there are the ones that are not actually locked at all, where one of the three conditions has quietly failed and what looked like a difference between two prices for one thing is really a relationship between two prices for two things.

Those three kinds are the entire explanation. Small, brief, or not really locked. Whatever survives being taken is, by definition, whatever was not worth taking, and the cost is what decides which is which. None of this establishes that prices are correct, that differences do not exist, that markets process information well, or that anybody who looks for such positions is wasting their time. Each of those is a contested claim about how markets work, and one subtraction on one fund's contracted rates is no basis for any of them.

What is the word used for the rest of the time?

Three things, mostly, and each of them is a real approach with a real mechanism, covered separately and in full. The useful question is not what each one is. The useful question is which condition each one has dropped, and the answer is the shortest route to what the position actually depends on.

The first is a book with equal long and short sides, built so that the market direction cancels out. The two sides are not the same thing. The two sides are two different things chosen so that a common influence on both is removed. Removing a common influence is a real and useful construction, and it is not arbitrage in the precise sense. Condition one has gone, and what is left is a position on how the two sides differ from each other.

The second is the nearest neighbour to the precise sense and the one most often called by this word, so the boundary deserves stating exactly. A trader takes a position in the difference between two related prices and holds it because the difference is expected to close. The line between that and the precise sense is the line between a difference that can be locked and a difference that is expected to narrow. A locked difference is collected the moment both sides are done, at prices that were both standing. An expected one is collected later, at whatever the second price has become, if the position is still open when it happens. The difference can widen first, and widening costs money for as long as the position is carried, and a position can be closed for reasons that have nothing to do with whether the expectation was right. The mechanism is real and has its own arithmetic, and it is set out where that approach is covered.

The third is a position taken around an announced corporate event, held for the difference between the price now and the price the announcement implies. Both sides are rarely both done. The second one completes only if the event completes, and whether it completes is somebody else's decision. Condition three has gone. The remaining position depends on a decision that can be refused, delayed or replaced, and that approach too is covered separately.

Why does the distinction matter to somebody reading about a fund?

Because the word carries the precise sense with it wherever it goes, including into the places where it does not apply, and a reader who accepts the word stops asking the only question that would have told them anything.

What does somebody reading a factsheet actually do with this?

Most people who meet this word never place a trade in their lives. Readers meet it in a paragraph describing what a fund does, in a note an adviser has forwarded, in a clause of an offering document, or in a sentence somebody said confidently in a meeting. The practical value of the three conditions is that they turn a word into three questions, and the three questions can be asked by somebody who has no access to anything and no intention of buying anything.

An analyst reading a description of an approach uses them in that order. A person sitting on a committee that has to understand a proposal uses the same three. And the household version is the same shape at a smaller scale: the person deciding whether the tin of oil on the next street is really Rs 6 cheaper, once the walk and the time are counted, is doing the entire subtraction worked here, in their head, correctly.

The last row is the one most worth keeping, and it survives long after the definition has faded. Whatever the position turns out to be, the difference it starts with is not the difference it keeps, and the distance between those two is the cost of taking it. On this invented fund's own contracted rates that distance was Rs 0.15 a unit against a gap of Rs 0.40, which is more than a third of everything the two prices were showing, and none of it had anything to do with being right.

One more use, and it is the least glamorous. The three conditions are a vocabulary check. When a description of an approach uses this word and then, two sentences later, explains that the position is closed when the gap narrows, those two sentences contradict each other, and noticing that is worth more than any amount of arithmetic. A difference collected when both sides are done and a difference collected when something narrows are not the same object. The word does not say which one is in front of the reader. The three questions do.