The Triangular Arbitrage Opportunity

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 in 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 worth is undervalued with respect to the cross rate and sell X. Alternatively, if a dealer overvalues Z 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

Assume that 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 sides 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. However, 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. Further, note that 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 84.63/84.70. 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 84.63/84.70, 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 \(84.71-84.70=0.01\) 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

The correct answer is A.

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 8: Currency Exchange Rates: Understanding Equilibrium Value

LOS 8 (b) Identify a triangular arbitrage opportunity and calculate the profit, given the bid-offer quotations for three currencies.

Shop CFA® Exam Prep

Offered by AnalystPrep

Featured Shop FRM® Exam Prep Learn with Us

    Subscribe to our newsletter and keep up with the latest and greatest tips for success
    Shop Actuarial Exams Prep Shop Graduate Admission Exam Prep


    Daniel Glyn
    Daniel Glyn
    2021-03-24
    I have finished my FRM1 thanks to AnalystPrep. And now using AnalystPrep for my FRM2 preparation. Professor Forjan is brilliant. He gives such good explanations and analogies. And more than anything makes learning fun. A big thank you to Analystprep and Professor Forjan. 5 stars all the way!
    michael walshe
    michael walshe
    2021-03-18
    Professor James' videos are excellent for understanding the underlying theories behind financial engineering / financial analysis. The AnalystPrep videos were better than any of the others that I searched through on YouTube for providing a clear explanation of some concepts, such as Portfolio theory, CAPM, and Arbitrage Pricing theory. Watching these cleared up many of the unclarities I had in my head. Highly recommended.
    Nyka Smith
    Nyka Smith
    2021-02-18
    Every concept is very well explained by Nilay Arun. kudos to you man!
    Badr Moubile
    Badr Moubile
    2021-02-13
    Very helpfull!
    Agustin Olcese
    Agustin Olcese
    2021-01-27
    Excellent explantions, very clear!
    Jaak Jay
    Jaak Jay
    2021-01-14
    Awesome content, kudos to Prof.James Frojan
    sindhushree reddy
    sindhushree reddy
    2021-01-07
    Crisp and short ppt of Frm chapters and great explanation with examples.