Question

Homework 1 on Triangular Arbitrage with Bid-Ask Spread

S (€ / £) = S ($ / £) / S ( $ / €)

Suppose

Citibank’s quote: $1.5445 – 1.5460 / €

Barclay’s quote: $1.9443 – 1.9453 / £

Dresdner’s quote: €1.2789 – 1.2799 / £

Cross-rate between Citibank and Barclay should be € 1.2589 / ₤, compared to the actual Dresdner quote of €1.2789 / ₤.

Is triangular arbitrage possible if an investor starts with € 1 million and follows the following two strategies independently?

(a) Euros to Dollars to Pounds to Euros.

(b) Euros to Pounds to Dollars to Euros.

If there are arbitrage gains or losses in either case, how much?

Answer 1

Option 1:

1M € to $ = 1M€ X 1.5445 $ = 15,44,500 (Rate is 1€ = $ 1.5445 i.e. Euros to be sold to bank hence bank will charge Bid rate).

Step 2 $ 15,44,500 to be sold to Get Pound i.e $ 15,44,500/1.9453 $ = Pound 7,93,965 (Rate is 1 Pound = 1.9453 because client has pound hence it is to be sold to bank to get US $ but the quote is available for Pound i.e. bank will exchange at Ask Rate)

Step 3 Pound 7,93,965 to be sold to get € i.e. Pound 7,93,965 x 1.2789 € = 10,15,402 Euros

Hence the gain is € 15,400 which is indicating the possible arbitrage opportunity is there.

Option 2

Applying the same logic above;

Step 1 € to Pound € 10,00,000/1.2799 Pound = 7,81,311 Pounds

Step 2 Pound to $ Pound 7,81,311 x 1.9433 =15,19,103$

Step 3 $ to € 15,19,103/1.5460 € = 9,82,602 Euros

Hence there is no possible arbitrage opportunit in this option.

Note: Logic for selecting the exchange rate is when one currency needs to exchange we have to go either to buy or sell that currency to bank. If we have to buy that then bank will charge higher rate i.e. Ask Rate but if we have to sell currency then bank will give us lesser rate i.e. Bid rate. Accordingly logic is apply to corresponding available currency for exchanged.