# Who is losing in an arbitrage?

1. When some entity takes advantage of an arbitrage opportunity, who is losing money? For example, when there are price differences across cryptocurrency exchanges and someone exploits an arbitrage opportunity, who is losing out?

2. A vaguer question: what are the microeconomic implications of arbitrage?

• The person who could've sold at the other exchange for more money? – user253751 Nov 27 '20 at 13:42
• In essence, arbitrageurs are paid to transmit information from one market to another. – Rodrigo de Azevedo Nov 27 '20 at 19:31
• Q2 is too broad for this site. – user18 Nov 28 '20 at 4:00
• I've read that crypto currencies suffer from a regulatory problem. Since regulatory bodies act on behalf of the people in a democracy, one might say that the people are losing out if crypto-currencies win out. In fact, Facebook was warned away from creating their own - libra - as I recall. – Mozibur Ullah Nov 28 '20 at 18:40

Nobody has to loose in an arbitrage. Economic relationships are not necessarily zero-sum (in fact often they will not be zero-sum). For example, if apples in city A are sold for $${\\\}5$$ and apples in city B can be sold for $${\\\}8$$, and we assume zero transaction cost there will be an arbitrage opportunity to earn $${\\\}3$$ riskless profit per apple by buying apples in city A and selling them in city B. But nobody looses in the transaction.

The apple producers in city A clearly value $${\\\}5$$ more (or at least indifferent) than an apple otherwise they would not trade it and just keep the apple. So giving them $${\\\}5$$ will make them better off (or at least not worse off). Then in city B people who buy apples must value apples more than $${\\\}8$$ they are paying for it otherwise they would just keep the money instead of buying them (or at least they must be again indifferent).

Consequently, when people take advantage of arbitrage opportunities generally speaking nobody looses anything. There must be some other issues present that would turn the problem into zero-sum game.

Regarding the microeconomic implications the main one is that people will take advantage of the arbitrage opportunities until the prices on the markets equalize - in the example above until the prices in both cities would become equal. The law of one price is based on the concept of arbitrage. There might be other implications depending on precise setting where arbitrage occurs, to explore all of them would be beyond the scope of SE answer.

• One could also argue that if the apple farmers sell an apple for \$5 in city A while they could have sold it for \$8 in city B (with 0 transaction costs), then they incur an opportunity cost of \$3 per apple sold. This makes the interaction zero-sum again. – VARulle Nov 27 '20 at 9:26 • @VARulle but in that case there would be no opportunity for arbitrage because then the farmers would not sell the apples for 5, if they are presented with choice to sell it in the other city directly. I think there has to be some extra problem there beyond arbitrage. – 1muflon1 Nov 27 '20 at 10:33 • @1muflon1 then why is arbitrage possible? Obviously the seller would sell the apples for$8 in city B so the situation is impossible....... right? – user253751 Nov 27 '20 at 13:42
• @user253751 you can assume that seller in city A cannot sell it in city B and then arbitrage is possible - but then there is no opportunity cost for seller A because opportunity cost can only exist if seller A could sell it at city B if they would want to – 1muflon1 Nov 27 '20 at 13:45
• @user253751 you have to impose that as an assumption if there is transaction cost there would not be any arbitrage at all if the transaction cost is equal to the difference in prices between A and B there is no possibility for risk-less profit or arbitrage. For example you can assume law prevents them from selling in city B – 1muflon1 Nov 27 '20 at 13:47

It's important to distinguish between the effects of arbitrage on: a) the direct parties to arbitrage transactions; b) other agents in the markets in which the arbitrage takes place.

Suppose arbitrageurs buy a good in market A in which its price is \$1 and sell in market B where its price is \$2. Assume further that in each market those prices have freely come about through the interaction of upward-sloping supply and downward-sloping demand curves. Sellers in market A will not lose by selling to an arbitrageur at \$1 or more, and buyers in market B will not lose by buying from an arbitrageur at \$2 or less. So there is scope for arbitrage to occur with no loss (and indeed some gain) to the direct parties to arbitrage transactions.

However, if the volume of arbitrage is sufficient to move the prices in the two markets, so that the price in market A becomes more than \$1, and that in market B becomes less than \$2, then there are also consequences for agents who are not parties to the arbitrage transactions, namely, buyers in market A and sellers in market B. The higher price in market A reduces the consumer surplus to its buyers, while the lower price in market B reduces the producer surplus to its sellers. Thus there is a welfare loss to both these groups.

This is not to make a case that arbitrage is undesirable or a zero-sum game: its overall effect may well be to raise welfare. It is only to assert that, even so, it may lower welfare for some groups.

The main point (already made by 1muflon1) is that no one needs to lose. (The presumption that someone must lose in any transaction or exchange is an example of the zero-sum fallacy. This is a common mistake by non-economists.)

The comments to 1muflon1's answer seem to contain some objections/confusion. To clear these up, here's an example where everyone wins and no one loses:

Example. Each nail clipper usually trades for \$1 in city A and \$1.05 in city B (10000 km away). On a particular day, Bob happens to be driving in his car from A to B to visit a dying relative. His car can load up to 10000 nail clippers. So, his plan is to buy 10000 nail clippers in A and sell them when he arrives at B.

It would however take him some time to buy and sell the 10000 nail clippers if he simply offers the usual prices. So, he decides to offer to buy the nail clippers for \$1.01 each in A and sell them for \$1.04 each in B. By doing so, he is able to quickly buy and sell the 10000 nail clippers in A and B.

Altogether, everyone wins and no one loses:

• Bob makes \$300, which exceeds his costs (time spent buying and selling, loading/unloading his car, additional gasoline). • Sellers in city A make an additional \$100 (compared to what they'd usually have made).

• Buyers in city B save \$100 (compared to what they'd usually have paid). Now, if there are many individuals who regularly drive from A to B, then we'd expect the \$0.05 price difference to be arbitraged away. But if Bob's drive from A to B is a rare occurrence, then this price difference can persist over time, because driving from A to B is costly.

I want to also emphasize a second point:

The costs and benefits of arbitrage may differ across individuals, so that only particular individuals may find it worthwhile to execute the arbitrage.

In the above Example, for most individuals, the costs (time and money spent driving 10000 km, time taken to buy and sell 10000 nail clippers) outweigh the benefits (\\$300).

But for Bob though, he has the additional benefit of visiting his dying relative, so that benefits happen to outweigh costs. It may be that this is true only of Bob on this particular day and that this is the only occasion where the nail-clipper arbitrage is ever executed.

Other points:

• Information -- it may be that Bob is the only one aware of the price discrepancy, so that even though many others regularly drive from A to B, only Bob knows to take advantage of it.
• Alertness -- it may be that many are aware of the price discrepancy and regularly drive from A to B, but only Bob is "alert" and proactive enough to take action. (This point about "entrepreneurial alertness" was emphasized by the Austrian-school economist Israel Kirzner. I think it's an important point that's been neglected as it's not easily formalized, modeled, quantified, tested, and so not very useful for pumping out papers.)

I would note that there is a dispute about the approaches to answer this question. I will give an answer based on finance theory, which may or may not be better placed to be on the quantitative finance board. That said, “finance” is a tag here, so the answer appears to appropriate. Note that I am using an idealised definition of profit that is used in finance and accounting, which might not align with that of economics.

I am using the formal definition of arbitrage - an opportunity to simultaneously enter trades that generate risk-free (supernormal profits). Somewhat equivalently, the ability to enter into a position with zero initial cost, and in all (future) states of the world, the future value of the portfolio is greater than or equal to zero - with strictly positive profits in some states. (My reference is page 80 of Rebonato’s “Interest-Rate Option Models,” but should be in any text on mathematical finance.) Many people use a less formal version of “arbitrage,” but I am referring to the formal definition.

Update: Varian uses the definition I use (or the methamtical equivalent): link to Varian article “The Arbitrage Principle in Financial Economics”.

(The zero entry cost definition of arbitrage eliminates the issue of defining “excess risk free profits”: one can buy a default-risk free money market instrument and get a “risk-free return”. An arbitrage has to generate a higher return. The zero cost definition eliminates the issue; you sell short the money market instrument to get zero initial investment.)

We then assume that we have observed market quotes at all time, and that we can find a set of theoretical prices from an arbitrage-free model that best fits those prices. For simplicity, I will assume that these are mid-prices. Note that observed prices might have arbitrage opportunities, and there are presumably many fittings. All that matters is that we can find one fitting.

We can then do a mark-to-market of all instruments versus that theoretical fitting at all times. Any time an entity trades away from that mid price, they get a corresponding mark-to-market gain/loss, which is mirrored by their counterparty.

Note that this is an idealised version of how financial accounting treats mark-to-market of securities. In the real world, everything is marked to market (bid, mid, depending on convention) at the close. In this example, the pricing is continuous - to match the simultaneous transactions. It’s also pricing versus an arbitrage-free model that is calibrated/fit against benchmarks, which is also standard practice for the derivatives that represent the bulk of the arbitrage trading. Finally, all entities use the same prices for this hypothetical mark-to-market, which is needed for trading activity to be zero sum.

When an entity enters into an arbitrage trade, it is clear that the gains/losses of all trades have to add up to a profit. The counter parties will in aggregate generate a loss, but the exact distribution is obviously unknown (unless a single counterparty was willing to arbitrage itself).

This framework also explains how market-makers earn a profit: they hope to continuously be trading at prices on the “correct” side of the mid price, and thus generate continuous trading profits.

In the real world, mark-to-market is not done instantaneously, it is done at the end of the day. Furthermore, the pricing used for the mark-to-market procedure used by different entities need not be the same. (In fact, some entities will not do a daily mark-to-market, such as in held-to-maturity accounting.) This implies that actual trading profit and loss will no longer be exactly zero sum.

• This answer does not use the word arbitrage in the meaning it has in economics. Following Mankiw Principles of Economics, by definition arbitrage is: " arbitrage, the process of buying a good in one market at a low price and selling it in another market at a higher price to profit from the price difference." In general I dont know of any definition of arbitrage within economics (finance of course can have different definition) that would count in profits/losses generated by accounting not trades themselves. – 1muflon1 Nov 27 '20 at 18:33
• As noted, I am using the formal definition from academic finance. Since there is a “Finance” tag, it seems unusual to argue that it is inadmissable. – Brian Romanchuk Nov 27 '20 at 18:38
• but MTM profit/losses exist because of the fair value accounting under GAAP they are not caused by arbitrage... they are not real profits and they are not caused by arbitrage they exist in any trades especially high frequency trades even if there is no arbitrage – csilvia Nov 27 '20 at 19:02
• @BrianRomanchuk we also have tag for agriculture. As far as I understand tags are often descriptive and I am sure you are well aware that there exists two strands of literature - one I would call finance proper and finance or financial economics. Originally it was all finance but I am sure you are aware they diverged in last 80 years or so. Also, I dont have problem with you giving answer with finance proper perspective but I also think you are not making clear distinctions here that this is not an economics answer to the issue as OP might think finance is just subset of economics. – 1muflon1 Nov 27 '20 at 19:16
• For example, I dont know if the other comment here is correct or not because my understanding of accounting is based on 1 mandatory undergraduate class, but it even seems that here you are not even using profit in the same meaning as economists do. In economics accounting profit and economic profit are different. Yet you are not warning users about that and since this is economics site I think by default people will assume you are using economics definition of profit. I think if you want to take the finance route here that might be good to explain differences in terminology in general – 1muflon1 Nov 27 '20 at 19:17