Practical Limits of Multi-Leg FX Arbitrage Chains

# Usefulness of simultaneously buying triangular and multiple arbitrages on the Forex


# Usefulness of simultaneously buying triangular and multiple arbitrages on the Forex












I have a limited financial background but I'm trying to figure out the usefulness of buying size-n arbitrages (n > 3), and I wonder the kind of risks - if any - associated with such a strategy.

Say that I simultaneously detect :

- a triangular arbitrage with currencies A, B and C (one intermediate currency) ;

- a quadrangular arbitrage with currencies A, B, C and D (two intermediate currencies) ;

- a quintangular arbitrage with currencies A, B, C, D and E (three intermediate currencies) ;

Now suppose that I decide to buy all these arbitrage opportunities at the same time and buy, for each of them, the same amount of the first currency. If I get I right, I'll actually be buying 3 times the first size-3 arbitrage, 2 times the second one and 1 time the last one. Despite triangular arbitrage being theoretically risk-free, we know that we may encounter potential issues here when facing real market conditions, ie. latency, not to mentions transaction costs and finite (limited) margin.

Plus, buying the quadrangular arbitrage would also mean that we could possibly be buying up to 4 different triangular arbitrages opportunities instead (i.e. one, some or all of the following permutations : A, B, C / B, C, D / A, C, D / A, B, D).

The reasoning goes on for all size n arbitrages opportunity where I could potentially be simultaneously buying at least n - 2 arbitrage opportunities.

Both theoretically and practically speaking, would it be wise to filter out some (or all) of these size > 3 arbitrage opportunities, and if so, why ? Or, in the contrary, would a strategy actually benefit from simultaneously buying all these arbitrage opportunities ? The maths behind this puzzles me !

## Answer by Matt Wolf (score 7, accepted)

https://quant.stackexchange.com/a/3677

I am not sure why your question had so many upvotes because in currency markets anything else but triangular arbitrage does not exist. What is a quadrangular arb, I have never heard of it despite having traded fx among other asset classes for over ten years now.

Think about it: Lets say you observe the price of EUR/USD. You can build triangular arbs by trading in 2 other pairs that flatten your exposure in EUR and USD in total, such as through EUR/CHF, and USDCHF. If trading in the other two pairs yields a synthetic price of EUR/USD where the bid is higher than the offer in outright EUR/USD and those prices are tradable and you can get filled without slippage (all grand assumptions) then selling the synthetic and buying the outright would yield you an instantaneous arbitrage profit. Likewise if the synthetic offer was lower than the outright bid then buying the synthetic and selling the outright would yield an arb profit. Its a very simple concept, whether you spot such opportunities often is an entirely different question. (I can tell you with confidence that it rarely ever happens, at least in a way that you not only spot the opportunity but also be able to successfully execute it)

I would say, judging from your question, that you think of this concept as way too complicated. Given you are trading arbs you should engage in as many as possible (given you can execute it with an edge) because by definition you will always be hedged. Do not make it more complicated than it is, I do not care whether some "coaches" or "books" tell you that quintangular arb exists or not, it does not, but it sounds fancy. Fact remains especially in quant modeling, pricing, and quant trading: Always break concepts down to the simplest elements and level then it all makes sense. Of course it can happen that you spot two arbs at the same time and that this would entail to buy 2x EUR/USD or for that matter a multiple of any currency pair if this is what the sum of your arb trades dictates.

Not sure what puzzles you, because I think it shames the Mathematicians to call this "math". So to answer: YES, engage in as many arb trades as you possibly can, if you spot more than 1-2 a day which you can actually execute in size then you are my hero and I will worship you.

Hope this makes sense.

## Answer by Ryogi (score 6)

https://quant.stackexchange.com/a/3692

While triangular arbitrages exists, they are a rare, short lived, and shallow. In several academic datasets they are very rarely seen, mainly for two reasons, market efficiency aside: (1) the time resolution of the data is not tick by tick but aggregated at some level (for example at 1 second intervals), (2) the dataset doesn't include all available quotes and transactions, but only those at a particular venue.

A overview and some analysis of the problem is done by

D. Fenn, S. Howison, M. Mcdonald, S. Williams, and N. Johnson. The mirage of triangular arbitrage in the spot foreign exchange market. International Journal of Theoretical and Applied Finance, 12(08):1105– 1123, 2009.

They document that some triangular arbitrage opportunities do exist, but that have short durations (they disappear quickly, either because they are traded or because quotes are adjusted) and small magnitudes (the volume at the arbitragable quotes is thin). Fleeting duration is a vague term that depends on the time period considered. If in the late 90s this opportunities lasted 10s on average, as reported by

Y. Aiba, N. Hatano, H. Takayasu, K. Marumo and T. Shimizu, Triangular arbitrage as an interaction among foreign exchange rates, Physica A 310(3–4) (2002) 467–479;

more recently these numbers are significantly lower (maybe 0.2 secs in 2010). As to the depth, very rarely one can trade more than one or two lots (ballpark \$1M) on platforms such as EBS. Assuming no inventory exposure, an arbitrage of 1 pip (net of transaction costs and bid-ask spreads and latency risk) will pocket you $100.

Bottom line, if its there its elusive.

On the topic of arbitrages involving more than 3 currency pairs, the first and foremost worry are trading costs, in particular the number of times that as a taker of liquidity you have to pay the spread. The direct consequence of having to pay higher effective trading costs is that the number of tradable arbitrage opportunities is much lower.

That being said, there are some theories (most notably the work of Rod Ross et al) that suggest that arbitrage chains in currency markets involving 4 or more pairs might display a periodicity pattern. These claims are, to the best of my knowledge, completely academic and with no empirical support.

In practice, the situation you describe is not realistic. Even in case of contemporaneous arbitrage opportunity, the amount of liquidity available is fairly limited and constrains your actions to trying to complete one arbitrage trade sequence as quickly as possible.

## Answer by Jordan (score 0)

https://quant.stackexchange.com/a/68907

I know the question is old but I'm not sure why the response had so many upvotes because quadrangular arbitrage exist.

https://www.researchgate.net/publication/228228520_A_Matrix_Approach_to_Asset_Pricing_in_Foreign_Exchange_Market

"the arbitrage pattern must involve all four currencies and four locations, which means fully quadrangular arbitrage. An arbitrageur borrow currency jmin in location jmin to buy currency imin, and sell it in location imax for currency imin,then sell it in location jmax for currency imax, then buy currency jmin in location imax..."

https://blog.ylxdzsw.com/2017-10-coinflow-proposal.pdf "a “quadrangle arbitrage” with all theorems the same as triangular arbitrage except for there being 4 nodes."

example : buy BTC with EUR, sell BTC for ETH, sell ETH for USD, sell USD for EUR <<- 4 nodes ->> eur to btc, btc to eth, eth to usd, usd to eur. triangular arb would be : usd to lira, lira to btc, btc to usd.