# What is Arbitrage in FOREX?

Arbitrage, especially triangulation methods, is a perfect candidate for computer analysis and execution; it requires both deep and lightning-fast calculation. In general, arbitrage is the purchase or sale of any financial instrument and simultaneous taking of an equal and opposite position in a related market for the purpose of taking advantage of small price differentials between markets. Essentially, arbitrage opportunities arise when currency prices go out of sync with each other. There are numerous forms of arbitrage involving multiple markets, future deliveries, options, and other complex derivatives.

A less sophisticated example of a two-currency, two-location arbitrage transaction follows.

Bank ABC offers 170 Japanese Yen for one U.S. Dollar and Bank XYZ offers only 150 Yen for one Dollar. Go to Bank ABC and purchase 170 Yen. Next, go to Bank XYZ and sell the Yen for \$1.13. In a little more than the time it took to cross the street that separates the two banks, you earned a 13 percent return on your original investment. If the anomaly between the two banks' exchange rates persists, repeat the transactions. After exchanging currencies at both banks six times, you will have more than doubled your investment.

Within the FOREX market, triangular arbitrage is a specific trading strategy that involves three currencies, their correlation, and any discrepancy in their parity rates. Thus, there are no arbitrage opportunities when dealing with just two currencies in a single market. Their fluctuations are simply the trading range of their exchange rate.

In the subsequent examples, I refer to four Tables below of currency pairs consisting of the five most frequently traded pairs (USD, EUR, JPY, GBP, and CHF) with recent bid-ask rates.

[caption id="attachment_13350" align="aligncenter" width="515"] Combinations of the Five Most Frequently Traded Currencies[/caption]

 Formulas for Cross Currencies CHF/JPY = USD/JPY/USD/CHF 85.14 = 105.61/1.2402 85.1556 EUR/CHF = EUR/USD Ã USD/CHF 1.5676 = 1.2638 Ã 1.2402 1.567365 EUR/GBP = EUR/USD / GBP/USD 0.6915 = 1.2638/1.8275 0.691546 EUR/JPY = EUR/USD Ã USD/JPY 133.51 = 1.2638 Ã 105.61 133.4699 GBP/CHF = GBP/USD Ã USD/CHF 2.2666 = 1.8275 Ã 1.2402 2.266466 GBP/JPY = GBP/USD Ã USD/JPY 193.02 = 1.8275 Ã 105.61 193.002

[caption id="attachment_13351" align="aligncenter" width="515"] Calculations for Cross Currencies[/caption]
 Transaction Cost EUR/USD 2 USD/JPY +3 EUR/JPY +3

We omitted the other two majors, CAD and AUD, for the sake of simplicity and not because of lack of arbitrage opportunities in these two majors.

Example 1: Two USD pairs and one cross pair (multiply).

First, we must identify certain characteristics and distinguish the following categories:

USD is the base currency (leftmost currency in the pair):
 USD/CHF 1.2402/05 USD/JPY 105.61/64

USD is the quote currency (rightmost currency in the pair):
 EUR/USD 1.2638/40 GBP/USD 1.8275/78

Cross Rates (non-USD currency pairs):
 CHF/JPY 85.14/19 EUR/CHF 1.5676/78 EUR/GBP 0.6915/17 EUR/JPY 133.51/54 GBP/CHF 2.2666/74 GBP/JPY 193.02/10

The fact that the USD is the base currency in two of the pairs (USD/CHF and USD/JPY) and is the quote currency in two other pairs (EUR/USD and GBP/USD) plays an important role in the arithmetic of arbitrage. We begin our investigation with just the bid prices. (See Table above- Formulas for Cross Currencies.)

The criterion whether to multiply or divide the USD pairs in order to calculate the cross rate is simple:
• If the USD is the base currency in both pairs, then divide the USD pairs.
• If the USD is the quote currency in both pairs, then divide the USD pairs.
• Otherwise multiply the USD pairs.
To determine the deviation from parity for each cross pair, subtract the exchange rate from the calculated rate and convert the floating point decimals to pip values. (See Table- Transaction Cost.)

From Table above (Calculations for Cross Currencies), we can see that the EUR/JPY is out of parity by four pips. To determine if an arbitrage opportunity is profitable, we must first calculate the total transaction cost by adding the three bid-ask spreads of the corresponding pairs. (See Table above- Transaction Cost.)

An eight-pip transaction cost to earn a four-pip profit is counterproductive (it amounts to a four-pip loss). If the parity deviation (the number of pips by which the three currency pairs are out of alignment) were greater, say 30 pips, then a definite arbitrage opportunity exists.

The trading mechanism to take advantage of this anomaly requires some consideration. First, determine what market actions are necessary to correct this anomaly. Assume that the EUR/JPY rate is currently trading at 133.51 and the calculated rate using the current EUR/USD and USD/JPY pairs is 133.81 (a 30-pip deviation). Parity between the three currencies will be restored if the following price action occurs:
• The EUR/JPY pair rises to 133.81
• The product of the EUR/USD and USD/JPY pairs drops to 133.51
Therefore, the following trades are required to lock in the 30-pip profit:
• Buy one lot of the EUR/JPY pair.
• Sell one lot of the EUR/USD pair.
• Sell one lot of the USD/JPY pair.
• Liquidate all three trades simultaneously when parity is reestablished.
Warning: Executing only one, or even two, legs of the three trades required in an arbitrage package does not guarantee a profit and may be quite dangerous. All three trades must be executed simultaneously before the locked-in profit can be realized.

Example 2: Two USD pairs and one cross pair (divide)

The previous example uses the product of the two USD currencies to calculate the cross rate. An example of the ratio of the two USD currencies follows. Assume the EUR/GBP cross pair is currently trading at 0.6992 and that the ratio between the EUR/USD and GBP/USD pairs is calculated as 0.6952, a 40-pip deviation. Parity will be restored when the following price actions occur:
• The EUR/GBP pair drops to 0.6952
• The ratio of the EUR/USD and GBP/USD pairs rises to 0.6992
For the second action to rise, either the EUR/USD pair must also rise or the GBP/USD pair must decline (this differs from the previous example). Therefore, the following trades are required to realize a 40-pip profit:
• Sell one lot of the EUR/GBP pair.
• Buy one lot of the EUR/USD pair.
• Sell one lot of the GBP/USD pair.
• Liquidate all three trades the moment parity is reestablished.
Example 3: Three non-USD cross pairs

Technically, the arbitrage strategy can also be performed on three non-USD currency pairs. In this example, we examine a straddle between the three European majors (EUR, GBP, CHF), where we focus on the EUR/CHF pair in respect to the two GBP currency pairs (GBP/CHF and EUR/GBP).

Assume the current rates of exchange are:
• EUR/CHF = 1.5676/78
• EUR/GBP = 0.6915/17
• GBP/CHF = 2.2604/12
and their relationship is:

Thus, the calculated value for the EUR/CHF rate is 0.6915 Ã 2.2604, or 1.5631. The deviation from parity is â.0045 (1.5631 â 1.5676), or 45 CHF pips, since CHF is the pip currency in the EUR/CHF pair. The trading strategy is:
• Sell one lot of EUR/CHF.
• Buy one lot of EUR/GBP.
• Buy one lot of GBP/CHF.
• Liquidate all three when parity is reestablished.
If all three trades are executed successfully, a profit of 45 CHF pips is realized. Subtract the three bid-ask spreads for the transaction costs (2 + 2 + 8 = 12) to see a net profit of 33 CHF pips. Now convert CHF pips to dollars (33 divided by USD/CHF rate 1.2402) to obtain 27 USD pips.

It should be noted in all the examples presented here that only three currencies are analyzed simultaneously. It is possible to add a fourth, or even a fifth, currency to the mix, although this is normally left to the very serious arbitrage strategists.

The methodology for examining four (or even five or six) currencies at one time is to calculate every possible three-currency combination among the currencies selected. Rearrange them in magnitude of deviation from parity. Examine the deviations closely to see if there is a single anomaly or possibly even a double anomaly among the four currencies. This type of scrutiny will then determine if a four-currency arbitrage opportunity exists.

Specialized software is definitely required when dealing with four or more currencies in a single arbitrage package. The approach here requires instantaneous calculation of arbitrage values across multiple pairs using transitivity algorithms. Extreme low latency is required for it to work.
By Michael Duane Archer
Michael Duane Archer has been an active futures and FOREX trader for more than 35 years. He has worked in various advisory capacities, notably as a commodity trading advisor, registered SEC investment advisor, and branch manager for Heinold of Hawaii. He currently trades FOREX and futures and is involved in several technical analysis research projects.

Copyrighted 2020. Content published with author's permission.

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