How scalping and triangular arbitrage keeps the currency market in arbitrage equilibrium
A stylized explanation of arbitrage in the foreign exchange market
There are many reasons why market actors need to convert their currency. Corporations may have workers, suppliers, customers, and investors all in different countries from and to which money needs to flow. Financial institutions may want to diversify their holdings by investing in international financial assets. Governments may want to amass foreign reserves to later balance payments or defend a currency peg. Tourists may want to travel to another country which uses a different currency. Migrant workers and immigrants may have family back home to which they send remittance.
When all these actors transact in the foreign exchange market, how can they be sure they are receiving the best deal? What if the quoted price significantly differs from an optimal price available? This is where arbitrageurs come in. By observing currency-pairs and currency-triangles, and quickly taking advantage of any discrepancies, they keep the market in arbitrage equilibrium. Meaning that, while turning a profit, arbitrageurs provide market actors with certain and optimal prices. They are typically employed by banks and foreign exchange trading firms.
Scalping between two currencies
Starting with the simplest case, suppose there are two currencies, alphas and betas, in the foreign exchange market. In addition, some prospective tourists in beta-land wish to exchange their betas for alphas at a rate, A, of alphas per beta. On the other side, in alpha-land, some tourists wish to exchange their alphas for betas at a rate, B, of alphas per beta. Or put more simply, in the beta market, buyers are willing to buy at a maximum price of B alphas (the bid price) and sellers are willing to sell at a minimum price of A alphas (the ask price). (On a currency exchange, this would be listed as β/α at B/A.)
For some reason, these tourists found themselves in a situation where the bid price (maximum buying price, B) is greater than the ask price (minimum selling price, A). Maybe they have not found each other to trade their currencies. So our observant arbitrageur buys some betas from the tourists in beta-land and sells them to the tourists in alpha-land, profiting the difference (B - A). Given enough of these transactions, the bid price will move down and the ask price will move up until they are approximately equal.
While our arbitrageur may have been motivated by profit, she helped our tourists by acting as an intermediary. In the end, everyone is happy. The tourists in beta-land can spend alphas when they travel, the tourists in alpha-land can spend betas, and future tourists are assured certain exchange rates — and the arbitrageur makes a nice profit.
Triangular arbitrage between three currencies
After hearing the news that such a profit can be made risk-free, arbitrageurs flock to the foreign exchange market and take advantage of such price differences. In a while, all arbitrage opportunities between currency-pairs have been exhausted to the point where they are in equilibrium. So our arbitrageur looks at the exchange rates between three currencies: alphas, betas, and gammas.
Since all currency-pairs are arbitrage-free, the rate for converting one currency to another is the same for converting it back to that currency. So tourists in alpha-land wishing to buy betas will buy at the same price, A, that tourists in beta-land wishing to go to alpha-land want to sell. Accordingly, the six possible transactions (buying and selling each currency-pair) are only at three rates. These exchanges are represented by the double-arrows in the diagram with three currencies.
Luckily, even though all the other arbitrageurs have been keeping currency-pair exchange rates arbitrage-free, our arbitrageur has spotted a discrepancy. She notices that converting an alpha to a gamma can take two paths: either by directly buying gammas with alphas (α → γ) or by first buying betas and then using those to buy gammas (α → β → γ). In the former, the price of a gamma would be C alphas, while in the latter it would be A × B alphas. If these two prices were different, our arbitrageur could buy at the lower price and sell at the higher price, pocketing the difference.
In this case, she spots that C is greater than A × B, so she buys betas with alphas, then gammas from those betas for a total cost of A × B per alpha, then finally sells those gammas for a total revenue of C per alpha and profits. Over time, given enough transactions, the laws of supply and demand kick in and A and B move upward and C moves downward until A × B = C.
An arbitrage opportunity between four currencies?
Once again, all the other arbitrageurs hear the news that risk-free profits can be made using triangular arbitrage. So they expand their monitoring from just currency-pairs to currency-triangles. Soon enough, all arbitrage opportunities between three currencies dry up as the exchange rates are kept in equilibrium.
Remembering how, even though arbitrage opportunities between two currencies disappeared, those between three currencies remained, our arbitrageur wonders if there is an opportunity between four currencies. Or five, or six. Unfortunately for her — though perhaps fortunately for us the public — there are not, once all pair and triangular exchanges are arbitrage-free, the whole foreign exchange market is.
To prove this, consider the price of a delta using alphas. This is depicted in the diagram with four currencies. A path utilizing all four currencies would be α → β → γ → δ with a price of A × B × C alphas. Directly exchanging (α → δ), a delta has a price of D alphas. The key insight is that, since the currency-triangle α-β-γ is kept in equilibrium, A × B = E. Similarly, for the currency-triangle α-γ-δ, E × C = D.
Applying these equations from the currency-triangles, our arbitrageur realizes that both the path using all four currencies and the direct-conversion path must have the same price. This is because the price of α → β → γ → δ is equal to the price of α → γ → δ (because of the α-β-γ triangle) which is equal to α → δ (because of the α-γ-δ triangle). With this knowledge, she learns that it is unnecessary to monitor for arbitrage opportunities between four currencies, because if one existed then one would also exist between three currencies as well.
The same logic could be applied to arbitrages between five, six, seven, and so on currency combinations. For example, consider the foreign exchange market between alphas, betas, gammas, deltas, and epsilons.
Using the same reasoning, our arbitrageur knows that the path α → β → γ → δ → ε is equivalent to α → δ → ε, since she just learned that an exchange between four currencies is in equilibrium (assuming that there are no scalping or triangular arbitrage opportunities). And since all currency-triangles are in equilibrium, α → δ → ε is equivalent to α → ε. So overall, α → β → γ → δ → ε is equivalent to α → ε in terms of price.
The messy real world
Of course, taking this simplistic theory into the real world raises many complications. For one, we assumed a fixed supply and demand for particular currency-pairs such that there is one optimal price for each currency-pair exchange rate. However, all sorts of events can affect the supply and demand and thus change the optimal price, introducing new opportunities for arbitrage. To take one example, the United States Federal Reserve may raise interest rates, causing more demand for the dollar to buy American bonds and thus raising the bid price. Arbitrageurs may then take advantage of this by buying and selling US dollars, profiting off the difference. Similar scenarios can emerge for currency-triangles.
Further complicating the matter, arbitrageurs use algorithmic electronic trading to almost instantaneously take advantage of slight discrepancies in currency-pairs and currency-triangles. These algorithms are so fast that a study found that 95% of arbitrage opportunities in the EUR-USD-CHF currency-triangle disappeared within five seconds. So arbitrageurs may only beat their competition when making advances in algorithmic trading that can navigate transaction fees and network and computation problems. Lastly, while currency arbitrage is ideally risk-free, there are slight risks as transactions cannot be guaranteed to be certain and instant.
When corporations, financial institutions, governments, tourists, immigrants, and migrant workers all participate in the foreign exchange market, they can rely on a certain and optimal quoted price. Lurking by, arbitrageurs watch like hawks for discrepancies. Closing off, similar logic can also be applied to all sorts of markets, such as the markets for stocks, bonds, and commodities. It can also be applied to differences in regulations: jurisdictional arbitrage! (Although jurisdictional arbitrage is a harmful form of law-skirting.)
