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# The Triangular Arbitrage Opportunity

Every bid-offer quote a dealer displays in the interbank FX market should possess the following properties to avoid the creation of arbitrage opportunity:

1. The bid should not be higher than the current interbank offer, and the offer should not be lower than the current interbank bid.

If this rule is broken, an arbitrage opportunity will arise, that is, a market participant will buy from a cheaper source and sell to a more expensive source. This will eventually bring the two prices back into line.

Example: Assume that the current spot USD/EUR is quoted as 1.3856/1.3858. Further, assume that a dealer’s quote is 1.3859/1.361. Market participants in the interbank market will pay for the offer by purchasing EUR at USD 1.3858 and subsequently hit the dealer’s bid by selling the EUR to them at 1.3859, making a riskless profit of 1 pip (1.3859 – 1.3858 = 0.0001).

2. The dealer’s cross-rate bids (offers) should be lower (higher) than the implied-rate offers (bids) available in the interbank market.

To illustrate this, consider the currency pairs X/Y and Z/Y. If we work out the cross-rate X/Z, it must be consistent with the X/Y and Z/Y rates. If this is not met, the arbitrageur will purchase currency Z from the dealer if its value is undervalued with respect to the cross rate and sell X. Alternatively, if Z is overvalued by a dealer with respect to the cross rate, then it will be sold, and consequently, X will be purchased. This is called triangular arbitrage.

To identify triangular arbitrage, learning how to calculate the market-implied bid and offer rates is of utmost importance. Consider the examples below.

#### Example: Calculating the Market-implied Offer Rate

Suppose the bid-offers in a certain interbank for USD/EUR are 1.3850/1.3851, and JPY/USD is 75.66/75.68. The market-implied bid-offer on the JPY/EUR cross rate is closest to:

#### Solution

The relationship between the quotes above is represented as:

$$\left(\frac{JPY}{EUR}\right)=\left(\frac{JPY}{USD}\right)\left(\frac{USD}{EUR}\right)$$

When calculating the offer rate, the numerators of each term (both left- and right-hand side of the above equation) are “sold” while the denominators are “bought.” For instance, the left-hand side implies “sell JPY, buy EUR.” That is, to get the implied cross-rate, we multiply the bid-rates of the involved currencies (left-hand side of the above equation)

So,

\begin{align*} \left(\frac{JPY}{EUR}\right)_{\text{offer rate}} & =\left(\frac{JPY}{USD}\right)_{\text{offer rate}}\left(\frac{USD}{EUR}\right)_{\text{offer Rate}} \\ & =75.68\times1.3851 \\ & =104.824 \end{align*}

#### Example: Calculating the Market-implied Bid Rate

To compute the market-implied bid rate, we adopt a similar approach, but the numerators of each term (both left-hand and right-hand sides of the cross-rate equation) are “bought” while the denominators are “sold.” For instance, the left-hand side implies “Buy JPY, Sell EUR.”

Therefore,

\begin{align*} \left(\frac{JPY}{EUR}\right)_{\text{bid rate}} & =\left(\frac{JPY}{USD}\right)_{\text{bid rate}}\left(\frac{USD}{EUR}\right)_{\text{bid rate}} \\ & =75.66\times1.3850 \\ & =104.789 \end{align*}

As expected, the implied cross-rate bid should be less than the offer rate.

#### Example: Calculating the Market-implied Bid and Offer Rates Via Inversion

Assume that the USD/GBP is 1.5846/1.5848, and the USD/EUR is 1.3850/1.3851. Calculate the implied GBP/EUR cross rate.

#### Solution

It is easy to see that:

$$\left(\frac{GBP}{EUR}\right)\neq\left(\frac{USD}{GBP}\right)\left(\frac{USD}{EUR}\right)$$

We need to invert the first term on the right-hand side so that:

$$\left(\frac{GBP}{EUR}\right)=\left(\frac{1}{\frac{USD}{GBP}}\right)\left(\frac{USD}{EUR}\right)=\left(\frac{GBP}{USD}\right)\left(\frac{USD}{EUR}\right)$$

So,

\begin{align*} \left(\frac{GBP}{EUR}\right)_{\text{bid rate}} & =\left(\frac{GBP}{USD}\right)_{\text{bid rate}}\left(\frac{USD}{EUR}\right)_{\text{bid rate}}\\ & =\left(\frac{1}{1.5848}\right)\left(1.3850\right) \\ & =0.8739 \end{align*}

And

\begin{align*} \left(\frac{GBP}{EUR}\right)_{\text{offer rate}} & =\left(\frac{GBP}{USD}\right)_{\text{offer rate}}\left(\frac{USD}{EUR}\right)_{\text{offer rate}} \\ & =\left(\frac{1}{1.5846}\right)\left(1.3851\right) \\ & =0.8741 \end{align*}

Note that arbitrage constraints on the implied cross-rates also apply to the spot rates, and any violations of these constraints will cause arbitrage opportunities, which will naturally disappear in a short time.

#### Example: Identifying a Triangular Arbitrage Opportunity and Calculating the Profit

Consider the following spot rates in an interbank market.

$$\begin{array}{c|c} \textbf{Currency} & \textbf{Quotation} \\ \hline \text{SEK/USD} & 6.7738/6.7740 \\ \hline \text{JPY/USD} & 80.86/80.88 \\ \hline \text{CAD/USD} & 0.9543/0.9545 \\ \hline \text{USD/EUR} & 1.35458/1.3560 \end{array}$$

Assume that an inexperienced dealer quotes a bid-offer rate of JPY/CAD as 84.63/84.74. To identify the triangular arbitrage, we need to calculate JPY/CAD:

$$\frac{JPY}{CAD}=\frac{JPY}{USD}\times\frac{USD}{CAD}=\frac{JPY}{USD}\times\left(\frac{CAD}{USD}\right)^{-1}$$

But,

$$\frac{USD}{CAD}{=\left(\frac{CAD}{USD}\right)}^{-1}=\frac{\left(\frac{1}{0.9545}\right)}{\left(\frac{1}{0.9543}\right)}=\frac{1.04767}{1.04789}$$

So that, USD/CAD is quoted as 1.04767/1.04789 and:

\begin{align*} \left(\frac{JPY}{CAD}\right)_{\text{Bid}} & =80.86\times1.04767=84.71 \\ \left(\frac{JPY}{CAD}\right)_{\text{Offer}} &=80.88\times1.04789=84.75 \end{align*}

The implied interbank cross-rate for JPY/ CAD is now 84.71/84.75. Going back to the dealer’s quote of 84.63/84.74, the dealer is offering to sell CAD at a lower price (below the interbank quoted rate, 84.71). A prudent market participant would utilize this triangular arbitrage by purchasing CAD from the dealer and selling it in the interbank market, making a profit of $$84.71-84.63=0.08$$ per CAD involved.

## Question

Consider the following spot rates in an interbank market.

$$\begin{array}{c|c} \textbf{Currency} & \textbf{Quotation} \\ \hline \text{SEK/USD} & 6.7738/6.7740 \\ \hline \text{JPY/USD} & 80.86/80.88 \\ \hline \text{CAD/USD} & 0.9543/0.9545 \\ \hline \text{USD/EUR} & 1.3558/1.3560 \end{array}$$

Using the table above, the SEK/EUR implied cross-bid (offer) rate is closest to:

1. 9.1839/9.1855.
2. 9.1825/9.1839.
3. 9.1849/9.1865.

#### Solution

We can write the equation for the SEK/EUR spot rate as:

$$\frac{SEK}{EUR}=\frac{SEK}{USD}\times\frac{USD}{EUR}$$

So that:

\begin{align*} \left(\frac{SEK}{EUR}\right)_{\text{bid rate} } & =\left(\frac{SEK}{USD}\right)_{\text{bid rate}}\times\left(\frac{USD}{EUR}\right)_{\text{bid rate}} \\ & =6.7738\times1.3558 \\ & =9.1839 \end{align*}

And

\begin{align*} \left(\frac{SEK}{EUR}\right)_{\text{offer rate}}& =\left(\frac{SEK}{USD}\right)_{\text{offer rate}}\times\left(\frac{USD}{EUR}\right)_{\text{offer rate}} \\ & =6.7740\times1.3560 \\ & =9.1855 \end{align*}

Therefore SEK/EUR implied cross-bid (offer) rate is 9.1839/9.1855.

Reading 6: Currency Exchange Rates: Understanding Equilibrium Value

LOS 6 (b) Identify a triangular arbitrage opportunity and calculate the profit, given the bid-offer quotations for three currencies.

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