triangular arbitrage w/ spreads

can someone walk me thru this question, i understand cross rates and triangle arb, but i have no idea how to even approach triangle arb w/ spreads. --------------------------- Spot Rates Bid Price:Ask Price Euro/US$ €1.0000 : €1.0015 British Pound/US$ ₤2.0000 : ₤2.0100 Euro/British Pound €0.3985 : €0.4000 Using the appropriate bid or ask rates for the Euro/US$ and the British pound/US$, what will be the profits from triangular arbitrage, starting with $1,000? A) $248.46. B) $245.65. C) $243.78.

2 routes possible Approach 1: 1. Sell USD buy Euro 2. Sell Euro buy BP 3. Sell BP buy back USD This route -> 1000/1.0015 * .4 * 2 --> leaves you with 798.8$ a loss. Approach 2: 1. Sell USD Buy BP 2. Sell BP buy Euro 3. Sell Euro buy USD This route --> 1000/2.01 --> /0.4 --> / 1.0 --> gives you 1243.78 a Profit. Hence route 2 is the way to go --> leaving you with C as the answer

^ agree w/ CPK- profit of 1243.781095 - 1k original = 243.781095, answer C you just go around the triangle and see if you make or lose $$. with spreads, you always make yourself worse off- so you multiply by the lower # or divide by the bigger # always to get you around the triangle. that’s the only other little thing to remember when spreads are involved.

* edit though- CPK i don’t think you did your loss right (you multiplied by .4 not .3985) 1000 USD x 2 = 2k british x .3985 x 1 = a 797 loss small, but we don’t want to confuse anything- agree on 797 loss side? i’d get the gain as the 1k x 1E = 1k euros, divide by .4 = 2500 british pounds, divide by 2.01 to get back to USD.

my calculation: 1000*2*0.3985*(1/1.0015)=$795.81. A loss of $204.19

My question is: how do you know when to multiply or divide by the bid/ask given a spread?

bid is when someone buys, ask is when someone sells… so use that to decide number to use. and whether to divide or multiply is dependent upon the source currency and the target currency…

if you simply remember “up the bid, down the ask” and construct the currency arb triangle the solution will fall into place. When you “up the bid” you multiply by the bid price and when you “down the ask” you divide by the ask price. let see if i can do this here. each corner of the triangle represents the currency and each leg (side) shows the appropriate exchange rate moving from one currency to the next (bid/ask) …USD …/…\ …/…\ 1.000-1.0015 EUR/USD…/…2.000-2.01 GBP/USD …/…\ …/…\ …EUR---------GBP …0.3985-0.400 EUR/GBP for me this helps keep things straight and helps minimize foolish errors. so moving counter clockwise with $1000 initial investment $1000*1.000 (moving from USD to EUR so “up the bid” therefore multiply bid price) then $1000/0.400 (moving from EUR to GBP so “down the ask” therefore divide by ask price) then $2500/2.01 (moving from GBP to USD again “down the ask” therefore divide by ask) =1243.7811 less the 1000 initial yields 243.7811

best way to keep it straight joy is to write the currency out and you go around. so if you have $1000 USD, you have to go around back to your original currency one way or the other. what are your options here to start? Either you could dive in multiplying by the Euro/USD to get euros or you could start multiplying by british pounds/USD and go that way. just keep your numerators and demonimator currencies straight and you’ll be fine. with spreads involved, always multiply by the lower # or divide by the bigger one. so $1000 USD x 1E/USD = 1,000 Euros. (see the USD’s cxl out) 1000 euros / .4 euro/GBP = 2,500 GBP 2,500 GBP / 2.01 GBP/USD = $1,243.781 USD. you’re back to USD. subtract the original 1k USD and you’re there. i finished residual income today, started into alts. i see the light at the end of the equity tunnel! hope to finish most of alts tomorrow before superbowl. we’ll see if that happens.

char – a question here – is your UP and DOWN purely based on the direction? What helps you decide UP or Down? What would you do if you went clockwise in the above scenario?

cpk up/ down is strictly based on direction you are going. i.e given bid ask for USD/EUR going from EUR to USD is “up the bid” and USD to EUR is “down the ask”… obviously things change is you are given EUR/USD (the numbers change but my “rules” will get you to the same place) if you go clockwise you see that there was a loss…gives the same number as you pointed out above.

Thanks for the thorough walkthrough and that amazing looking triangle Char. So with your method, going clockwise would mean $1000 * 2 * 0.3985 / 1.0015 = $795.81, a loss of $204.19. Is this correct?

Joy of CFA~ Wrote: ------------------------------------------------------- > Thanks for the thorough walkthrough and that > amazing looking triangle Char. > > So with your method, going clockwise would mean > $1000 * 2 * 0.3985 / 1.0015 = $795.81, a loss of > $204.19. Is this correct? yes

Use the midpoint when dealing with spreads in a triangular arbitrage and then go around the triangle as described above. You will avoid a lot of headaches that way.

clama midpoint you would end up getting an answer of 250$ – which would be closer to ans a and would then be wrong.

I just did this and you’re right … taking the midpoint gives you the wrong answer. I swear I saw that somewhere in the book … or am I confusing myself w/ another concept. Will have to take another look. Thanks for your clarification.

Probably you got confused with triangular arbitrage and taking midpoints of bid-ask spreads while calculating forwards premiums/ forwards discounts?

Good recap guys. Great post.

if is the base currency in one exchange rate, and is counter currency in the other exchange rate, would we approach this problem differently? would we approach from the base currency quote?

Do you have a sample problem to put up here and we can walk you through of how it’s done?