jacksparrow

This is my first post on this forum. So far this forum was extremely helpful.
I hope someone finds my first attempt helpful as well.
This approach should be useful to find the profit on a triangular arbitrage opportunity starting with X amount of currency Y.

Let's assume problem states that you start with GBP.

We start with 3 quotes:
GBP:CHF 2.20279 (GBP 1.00 = CHF 2.20279)
USD:CHF 1.2755
GBP:USD 1.62699

First, you have to line up exchange rates in such a way that you start with GBP and finish with GBP:

GBP:CHF -> USD:CHF -> GBP:USD (GBP to CHF to USD to GBP)

Ending currency and beginning currency for adjacent exchange rates (linked by arrow ->) in the chain should match in order to cancel each other. To accomplish that you need to flip exchange rates (take the reciprocal of the original exchange rate).
In our example we need to flip last two exchange rates to get a chain like this:

GBP:CHF -> CHF:USD -> USD:GBP

2.20279 (1/1.2755) (1/1.62699)

Now take the product of these rates to find how much in GBP we will get starting with GBP 1 :

GBP 1 -> 2.20279*(1/1.2755)*(1/1.62699) = GBP 1.06147

We made GBP 0.06147 risk free starting with GBP 1.
If question asked how much you could make starting with let's say GBP 1MM,
then multiply 1,000,000 by 0.06147 to get GBP 61,470 profit.

If resulting product of exchange rates is less than 1 you need to reverse the chain and start with GBP:USD exchange rate instead.
If product equals to 1 then there is no arbitrage.

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For problems with spreads you need to remember the rule "up the bid and down the ask":

Let's modify our example to include spreads:

GBP:CHF 2.20274 - 2.20284
USD:CHF 1.2750 - 1.2760
GBP:USD 1.62694 - 1.62704

We start with chain of exchange rates as before:

GBP:CHF -> USD:CHF -> GBP:USD

We need to flip two last rates to cancel the intermediate currencies (the adjacent currencies in chain have to match).

The resulting product

(GBP:CHF) * 1/(USD:CHF) * 1/(GBP:USD)

gives you how much in GBP we will get staring with GBP 1 .

Think about this product as a fraction with GBP:CHF exchange rate in the numerator (up)
and USD:CHF and GBP:USD rates in the denominator (down).

The question is should we use bids or asks to calculate the product?

Now the rule "Up the Bid and Down the Ask" comes in handy.

If rate is in the numerator of the fraction you have to use bid price of the quote (up the bid) and if rate is in the denominator you have to use the ask (down the ask).

This rule makes a lot of sense, since every time you convert currency you get the worst price from FX dealer. If rate goes to numerator you should use smaller number, bid of the spread. If rate goes to denominator you should use larger number, ask, to get lower reciprocal.

Finally, calculate the product of exchange rates applying the rule to pick bid or ask:

2.20274(up the bid) * 1/1.2760(down the ask) * 1/1.62704(down the ask) = 1.060997

We made extra GBP 0.060997 starting with GBP 1 .

BTW, notice that in example with spreads we made slightly less than in
first example where we used midpoint quotes (FX dealers pocket the difference by making spread on exchange trades).