Triangular arbitrage

Three currencies give three exchange rates — but only two of them are independent. The third should equal the cross rate implied by the other two. When it does not, there is a free lunch, at least briefly.

The workbook's definition: triangular arbitrage involves identifying and exploiting the arbitrage opportunity resulting from price differences among three different currencies in the forex market. It takes three trades:

1. Exchange the first currency for a second.
2. Exchange the second for a third.
3. Exchange the third back into the first.

If you end with more of the first currency than you began with, you have captured the misalignment.

The workbook is candid that this is very rarely profitable in practice. As soon as such an opportunity appears, traders exploit it and prices adjust until it disappears. Even when it appears, the disparity may be around 1 basis point, too small after transaction costs and taxes.

The test. Compute the implied cross rate from two quotes and compare it with the third quoted rate.

Implied EURINR = EURUSD × USDINR

If the quoted EURINR is below the implied rate, EUR is cheap in rupees: sell EUR for USD, USD for INR, then buy EUR back cheaply with INR. If the quoted EURINR is above the implied rate, go round the triangle the other way.

Risks (Chapter 5, section 5.5):

• Adverse price movement while the position is still being set up.
• Time gap between legs even on electronic systems.
• An illiquid leg may not execute, leaving a naked position instead of a hedged one.
• Transaction costs — brokerage, stamp duty, margin cost and impact cost — must be counted before entering.

Triangular arbitrage is one reason the workbook says that at any moment the exchange rates of major currencies are virtually identical across active financial centres, leaving very little scope for arbitrage.

Implied cross rate (EURINR) = EURUSD × USDINR
Opportunity exists if quoted EURINR ≠ implied EURINR (after costs)

The workbook's rates: EURUSD 1.09, USDINR 82.89, EURINR 90.11. Start with EUR 1,000.

Implied EURINR = 1.09 × 82.89 = 90.3501
Quoted EURINR  = 90.11 → EUR is cheap in rupees

The three legs:

1. Sell EUR 1,000 for USD at 1.09      → USD 1,090
2. Sell USD 1,090 for INR at 82.89     → Rs 90,350.10
3. Buy EUR with Rs 90,350.10 at 90.11  → EUR 1,002.67

Arbitrage profit = 1,002.67 − 1,000 = EUR 2.67

In rupees at the quoted 90.11, that is about Rs 240 on a EUR 1,000 round trip.

The workbook adds: had EURINR been trading at 90.35, there would be no arbitrage; with these EURUSD and USDINR rates, any EURINR below 90.35 offers one.

Scale and costs. Illustrative. The profit is 0.267% of the starting amount. On EUR 10 lakh the same misalignment gives EUR 2,670 before costs — but real misalignments, the workbook notes, are often nearer 1 basis point, i.e. EUR 100 on EUR 10 lakh, easily consumed by charges.

Chapter 5, section 5.5 (Arbitrage), works the EUR-USD-INR triangle above and lists the risks carried by arbitrageurs. Chapter 1, section 1.4, supplies the cross-rate arithmetic the test depends on. Expect a three-leg calculation or a question on why such opportunities are rare.

• Three currencies, three trades, ending in the currency you started with.
• Compare the quoted rate with the implied cross rate, not with yesterday's rate.
• Direction matters. Quoted below implied → the target currency is cheap, buy it last; quoted above → go round the other way.
• Rarely profitable — gaps are tiny (around 1 basis point) and close fast.
• An unexecuted leg leaves a naked position; arbitrage is only riskless if all legs fill.
• Transaction costs and impact cost are ignored in the textbook example and cannot be ignored in practice.