Introduction

If you read a scientific article on finance, you often find a phrase like "we assume an arbitrage-free market". On the other hand, you may have heard of people or trading firms that claim they are practicing arbitrage. Often, the people in firms that claim they do arbitrage happily read articles that assume there is no arbitrage. In this node, I will attempt to clarify this seeming contradiction.

What is an arbitrage-free market?

An arbitrage-free market is a market in which there is no arbitrage. Arbitrage is defined as the possibility to do a set of trades that leaves you with no position, but money in the pocket. I'll illustrate this with a simple example. Assume, for instance, that it is possible to buy yens at 100 per dollar. Assume furthermore that it is possible to buy euros for 150 yen a piece. If we can now buy more than 1.50 dollar per euro, say 1.51 dollar, we can make a profit. Say we start out with 1.50 dollar. We then buy 150 yen for this. This buys us precisely 1 euro. For this 1 euro, we buy 1.51 dollar, having gained 1 cent. This form of arbitrage is by the way known as triangle arbitrage. If we can do this as often as we want, we -or anyone!- could make any profit we want.

Mathematically, what we have just done is no different from stating that 1.50 = 1.51. This breaks math at a fundamental level; the "normal" definition of a number breaks down, because they essentially all have the same value. As such, we can't work with numbers anymore. So, one could argue that the main reason we need to assume an arbitrage-free market is that we otherwise couldn't do any math, or anything meaningful at all.

Arbitrage-free in practice

Of course, everyone wants to pick up this free penny. As such, a lot of people are searching the market for these opportunities. What they could do, for instance, is offering you to buy the dollars at for instance .67 eurocent. If you do, they - or rather, their computers - immediately buy 100 yen and convert these to 100 dollars, and they pocket something like .3 eurocent. In practice, the profit per trade can be even lower, as one incurs costs in trading; one has to pay the exchange, for instance. Furthermore, especially for a liquid set of currencies like this, the profit would be much smaller. In fact, one can count on it that as soon as the profit is higher than the costs, some computer will trade and the opportunity will be gone. On a whole, such razor-thin margin, high frequency arbitrage will make a handsome but not spectacular profit. It is noted that this introduces liquidity in the market.

In many cases, however, the so-called "arbitrage" that is practiced is not true arbitrage. One could, for instance, note that if for instance the Dow Jones Index goes up, European indices will usually close higher the next day. By consistently betting on this, money could be made. This is known as statistical arbitrage.

A very interesting form of arbitrage is arbitrage that isn't arbitrage after all. For instance, consider someone buying a structured product from a bank. As a simple example, this structured product pays 5% of interest every year the Eurostoxx 50 index is above 2000 (it's currently close to 3000). You borrow the cash for 4%, buy a 2000 put for less than that 1%, and presto, arbitrage!. Well, not really. See, the structured product claims the bank will pay you back your investment. If this bank fails, you will be left with a debt to the person you borrowed the cash from. You are left with credit risk. Of course, if you can convince your boss you made a profit, the only hedge you need is the O'Hare hedge.

Conclusion

On the one hand, academic literature often talks about arbitrage-free markets. On the other hand, trading firms claim they practice arbitrage. The reason for this discrepancy is on the one hand that the academic paper has to make this claim in order to be able to use numbers in the first place, so even if markets were not arbitrage-free, they'd still have to claim this. On the other hand, markets are mostly arbitrage-free: most so-called arbitrages are not total, true arbitrages, or are on such razor-thin margins that the profit per trade is almost negligible. As such, they do not meaningfully distort the market.

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