In my role as advisor to Bancor one topic comes up from time to time – arbitrageurs. I want to take the opportunity to write down some key thoughts here as they relate to the role of arbitrageurs within and AMM eco system, and whether or not they are useful and should be attracted. We do not currently have a blog over at topaze.blue hence the slightly unusual location here at TheShortSTOry podcast.

Introduction and definitions

We assume that the reader has a good idea what an Automated Market Maker (“AMM”) is and how it operates. If not, there are a lot of resources on this on the Internet, including the economics paper we have written for the launch of the Bancor v2.1 protocol.

Very often in the crypto space it turns out that people have long winded and controversial discussions only to find out that they agree in substance, but that they use a different terminology. Therefore let us start with defining a few terms.

Arbitrage proper. Arbitrage proper (or simply arbitrage, if the context is clear) is defined here in the usual sense it is defined in finance. It means to execute a trading strategy - ie a series of transaction – that nets a risk free profit. Simplifying somewhat we can assume that all arbitrage transactions are cash to cash, ie they start and end with the numeraire NUM (which might be USD, or BNT, or any other numeraire asset). An arbitrage transaction is one that starts with $x$ units of the numeraire NUM and that ends with $y$ units of it, and we have $y\geq x$ in all possibly states of the world (and typically we also require that $x>y$ in at least on state of the world, otherwise just holding an asset would be an arbitrage; also we are ignoring discounting and interest payments here and simply assert that this can be dealt with if need be).

Statistical arbitrage. A statistical arbitrage is a trading strategy that (a) is expected to yield a profit (in the cash-on-cash sense defined above), and (b) is repeatable sufficiently often that the law of large numbers kicks in and that the expected profits are realised.

It is worth mentioning that in efficient markets there should be no arbitrages of either type – whenever you end up with an excess profit, so the theory goes, you have not accounted for a risk, typically a tail risk. Taleb referred to this as “picking up pennies in front of a steam roller” in one of his books: virtually always you make a small profit, but there are very rare but catastrophic states of the world where it hits so badly that overall the expected profits disappear.

Arbitrage proper

Coming back to the real world however markets are not efficient – as we can see by the fact that intermediaries exist (they would not be able to turn a profit in efficient markets). Let’s consider for example a dual listed stock, in London and New York. Fundamentally it should trade the same on both exchanges. However, short term prices are not driven by fundamentals but by supply and demand, and there is not reason to assume that London and New York supply and demand curves are the same, given that most players operate in one of the markets but not in both.

The way how in this case the London and New York markets equalise is by the presence of arbitrageurs which in this case are entities that operate on both markets. Whenever there is a price difference they buy in the cheaper market and sell in the more expensive one, making a riskfree profit in the process. They provide a useful service and hence they are able a earn a certain profit. This profit however will not be excessive as otherwise more players would set themselves up to operate in both markets (which typically comes at an additional cost).

This last point is crucial: in traditional markets there are often local pockets of liquidity in the same asset, and for most of the players operating in those pockets it is excessively costly to connect to anything but their local pocket (extra KYC, staff, regulation, capital requirements etc). The “wallet” the arbitrageurs can earn in in this case is limited by the savings the market participants by only connecting to their own, local pocket of liquidity.

Now let’s consider this in crypto space: we are thinking of arbitrageurs as people who arbitrage between different CEXes or DEXes, the same way they arbitraged London and New York in our examples. If we are talking about CEXes, the situation is indeed exactly the same: the local liquidity pockets at Binance, Kraken and Coinbase will be different, and for people who can trade at multiple of those it will be possible to make arbitrage profits, and the profits will be the bigger the more costly it is to establish a presence on multiple exchanges.

With DEXes however the story is fundamentally different: (Ethereum based) DEXes are a flat global eco system, and everyone (present on Ethereum) has full access to all DEXes, regardless where they are placed. This means that there should be no room for arbitrageurs to arbitrage between DEXes as end customers should route their trades along the optimal path.

This is not entirely true in the sense that even in the presence of DEXes there are still a number of situations where arbitrages arise, for example

  1. Some end customers may prefer to trade on CEXes because they do not like to custody their own keys, or because they want to trade against fiat

  2. Some end customers may trade in sizes so small that that gaz costs do not allow for them to trade on DEXes

  3. Some end customers may prefer to trade on DEXes with lower gas cost, but higher fee or slippage, again because of transaction size

All those situation may allow an arbitrageur to step in and serve as aggregator of those small trades and earn a modest profit out of this. We haven’t done a detailed quantitiative analysis of those opportunities, but our current hypothesis is that whilst they exist they are small.

Strategic implications

A priori there should be no need for an AMM to cater to arbitrageurs. Instead they should try to cater to end customers directly, except in circumstances like those identified above. Having said this – there are sometimes situations where end customers are willing to pay a premium, eg for a nice user experience, allowing for profitable market segmentation. In this case it may be more profitable to either work directly with those “arbitrageurs” (who in reality provide value added services to the end customers) or to replicate the valued feature, provided the price premium will not be lost in the process.

Statistic arbitrage and market makers

There are a second class of “arbitrageurs” in the market who run certain trading strategies that take on risk and who make money “on average”, ie they make statistic arbitrage. One such strategy would be that when big trades go into the AMMs (optimally routed, ie imbalancing all major AMM pools at the same time) they are rebalancing the AMM back towards the previous value, on the assumption that the AMM price move was bigger than warranted by the size of the trade.

It is useful to think of those operators less as arbitrageurs, and more as second level market makers, in the same way that reinsurers are second level insurance companies. The first level of AMMs only has a limited capacity – their programmatic nature means that the bigger they become the more Impermanent Loss starts to hurt, and also to poison the markets when big AMM pools create a correlation between their two constituent assets. This however means that slippage will always be a problem for bigger trades – and this is exactly where those second level market makers can come in. Contrary to AMMs they are not bound by a fixed asset allocation which means they do not run into Impermanent Loss and slippage problems the way AMMs do, and are in fact a highly usual addition.

Strategic implications

Contrary to arbitrageurs proper, statistic arbitrageurs – better named second level market makers – can be extremely useful in the AMM eco system as they will allow the system to return to “normal” more quickly after big market moves. This (a) allows for more confidence in executing bigger trades, and (b) allows market activity to resume more quickly after bigger trades happened, where otherwise participants may want to wait until prices return back into kilter.

Conclusion

We have locked at the role of arbitrageurs and statistical arbitrageurs in the AMM eco system. We have found that arbitrageurs are not particularly useful, except to aggregate volume in situations that we believe to be relatively minor in practical volume terms. Instead of catering for arbitragegeurs an AMM should try to understand where those arbitrageurs source their end customer flow, and should encourage those end customers to trade with the AMM directly. One exception maybe where the arbitrageurs is not really an arbitrageurs but offers value added services, eg a powerful UI. In this case it may make sense for the AMM to come to a profit sharing agreement with that arbitrageurs that caters to the end customers’ willingness to pay a premium.

Statistical arbitrageurs – whom we also called second line market makers – on the other hand have a very important role to play in the distribution of larger than usual trades. It is worth for AMMs to establish a synergistic relationship with them.