Triangular Arbitrage & The Introduction of Liquidity to the Interbank Currency Markets: Evidence from GBP/MXN, JPY/MXN, and GBP/JPY Exchange Rates

Working through some archives, came across some old recordings that I never got a chance to sift through.  Below are the GBP/MXN, JPY/MXN, and GBP/JPY exchange rates recorded from the currency open on Sunday, October 2, 2011 through October 3, 2011.  Following the outright prices below are the spreads constructed from finding the synthetic exchange rates via triangular arbitrage and spreading them to their live equivalents.  

Because of the ease of this arbitrage and the automated nature of these trades, this relationship is a popular topic on this blog.  The market opening and subsequent introduction of liquidity into the interbank market has been particularly interesting to study. 

The Outrights.

  

 

 

These are the aforementioned spreads between the synthetic and live exchange rates.   Despite the volatility of the underlying rates, the spreads unsurprisingly remain remarkably consistent.  More detail on this type of trade can be found in the links below.

The Spreads.

  

 

Previous studies have touched on the following relationships:
CHF, JPY, and NZD
USD, SEK, and NOK
KRW, JPY, and HKD
USD, AUD, and HKD

Further Reading:
Heterogeneous Market-Making in Foreign Exchange Markets: Evidence from Individual Bank Responses to Central Bank Interventions (pdf)
Spread Components and Dealer Profits in the Interbank Foreign Exchange Market (pdf)

Triangular Arbitrage of CHF/JPY using NZD as a Medium

Full Trading Period

The CHF/NZD and JPY/NZD currencies were used to build a live synthetic CHF/JPY which was then spread to the live real CHF/JPY available on Idealpro (left axis).  The corresponding spread (right axis) is given by the orange/green combo hovering around parity (= 0). Prices were recorded Sunday, 2 October through Monday, 3 October, 2011.

The Opening.

Sunday Evening Opening: 4:06PM - Midnight CST

Sunday Evening Opening: 4:06PM - 7:00PM CST

A curious widening of the spread occurs right around 9:00am CST. Seconds after this loss of liquidity, a slight break occurs in the synthetic/real relationship (Figures 1 and 1.A), leading to the first visible arbitrage in the recorded series.  A 2nd break occurs just over an hour later (Figures 2 and 2.A). 

Liquidity Gap

Arbitrage 1.

In the first occurrence, the bid of the synthetic (Dark Blue) rises above the offered price available through the live CHF/JPY (Purple).  The synthetic could have been sold while the live CHF/JPY was purchased for an offsetting amount, resulting in a perceived profit subject to the minimum liquidity available across all the necessary legs of the trade.  Prices broke again in the opposite direction (Red below Green) a fraction of a second later.  After which, they quickly reverted to their "no-arbitrage" relationship. 

Figure 1

Figure 1.A

Arbitrage 2.

In the second (or third) finding, the opposite break in the relationship occurs.  The offered price of the synthetic (Red) falls below the bid price of the live CHF/JPY (Green) allowing the high-frequency trader to take advantage of the arbitrage, purchasing the synthetic and selling the live market.  Again, the availability of the trade is subject to the minimum quantity available on every leg of the trade. 

Figure 2

Figure 2.A

Triangular Arbitrage using high-frequency TAQ data on USD/HKD, AUD/USD, and AUD/HKD

Please note before reading: the watermarks (if you can call them that) on each picture in this post were from the original URL and authorship (a pseudonym) of this blog.  They have not been taken from another source and reposted - they are original works of this blog and this author.

Triangular arbitrage can be defined as the "price of a second currency expressed in terms of a third or an exchange rate calculated from two other rates." For instance, one can imply the synthetic AUD/HKD rate if one has the live USD/HKD and AUD/USD exchange rates.  The arbitrage relationship is inherent between the synthetic AUD/HKD traded against the live, executable AUD/HKD.

Fenn, Howison, McDonald, Williams, and Johnson (2009) note that "Triangular arbitrage represents one of the simplest arbitrage opportunities. However, there is, to our knowledge, no truly rigorous and robust study of triangular arbitrage in the finance literature.  We believe the main reason for this to be the lack of availability of datasets with prices which are sufficiently high-frequency and which are also executable."

Asynchronous data is a problem when trying to study any arbitrage relationship.  Be it stock to stock, stock to future, future to future, physical to cash, or speaking ubiquitously, any arbitrage relationship.  Studies using end-of-period (days, minutes, seconds) data recordings suffer from this issue whereby one variable may be said to have "this" price at the last known recorded interval, while the second variable similarly has "this" price at the last known recorded interval.  The problem arises because the two recorded prices, while they may be the last known price, cannot reliably be used due to their differences in time period.  This problem can only be eliminated using synchronized data in the most granular form available.

The Data.

The following data set was collected using a proprietary program developed by my IT team.  The software is connected to the interbank foreign exchange market via a direct access API.  In this case, the time and sales for the three pairs were recorded simultaneously and synchronously: any time a single variable changed, the entire data set was recorded as a whole.  The high frequency and synchronicity of the recording alleviates the problem that Fenn et al. (2009) discovered in their research of TriArb.

Below is the spread between the synthetic AUD/HKD and the live AUD/HKD.  As you can see, the arbitrage is remarkably consistent and very robust.  "Although some opportunities appear to exist for in excess of 100s," writes Fenn et al. (2009) in their study, "for both currency groups 95% last for 5 seconds of less and 60% for 1 second of less."

   (Click to enlarge)

There is a significant loss of liquidity beginning around 11:00 AM CST as evidenced by the increase in bid/ask spread of the three pairs, most notably the AUD/HKD.  The increase in bid/ask spread becomes evident in the construction of the synthetic/real arbitrage as referenced previously.

Bid/Ask Spreads
USD/HKD : Blue

AUD/USD : Red

AUD/HKD : Green

(Click to enlarge)

Trading the Opportunity.

At 1:33:55 AM on January 24, 2011, a small break occurs in the AUD/HKD synthetic and live relationship.  The break from parity is caused by an increase in the bid of the live AUD/USD from 0.98945 to 0.9897, an increase of .00052.  This bid-up caused an increase in the bid price of the synthetic AUD/HKD to 7.7154, while the live AUD/HKD rate was then offered at 7.7143, an arbitrage difference of .0011 AUD/HKD.

(Click to enlarge)

Notice: This break lasted only a single second.  While the opportunity undoubtedly lasted much less than a full second, the beta software version we were using to collect this data was not recording at the millisecond level, a problem that has since been eliminated.

The short duration of this break is an obvious barrier to entry.  Fenn et. al. (2009) cite "wider use of electronic trading platforms and trading algorithms" from 2003 to 2005 as the main cause of the opportunity's brevity.  They go on to note that, "From 2003-2005, the proportion of opportunities lasting less than 1 second increased from 40% to 62% for the JPY transactions and from 41% to 64% for the CHF transactions and the proportion of opportunities lasting in excess of 5 seconds halved for both sets of transactions."

Further Reading

Collective Dynamics of Small-World Networks
Hedging Currency Risk in International Investment and Trade
High Frequency Trading and the New-Market Makers