Question

A corporate treasury working out of Austria with operations in New York corresponds with two banks simultaneously Well Fargo (NYSE:WFC) in New York City and Deutsche Bank (NYSE:DB) in Frankfurt. The banks give the following bid and ask quotes for the euro. Well Fargo NYC quotes for euro Deutsche Bank Frankfurt quotes for euro $1.1872/€ (bid) $1.1884/€ (ask) $1.1843/€ (bid) $1.1861/€ (ask)

(a) How many dollars is Deutsche Bank offering in exchange for 3,000 euros?

(b) How many euros is Wells Fargo NYC offering in exchange for 7,000 dollars?

(c) Using $1 million (or its euro equivalent), describe what actions could the corporate treasury take in order to make an arbitrage profit facing these quotes.

(d) Suppose the banks update their quotes to the following: Well Fargo NYC quotes for euro Deutsche Bank Frankfurt quotes for euro $1.1862/€ (bid) $1.1881/€ (ask) $1.1846/€ (bid) $1.1866/€ (ask) Is there still an arbitrage opportunity? Why or why not?

Answer 1

EURUSD rate offered by:

1) Wells Fargo -> 1.1872 (Bid) & 1.1884 (Ask) [Given as USD per EUR]

2) Deutsche Bank -> 1.1843 (Bid) & 1.1861 (Ask) [Given as USD per EUR]

Let,

Wells Fargo EURUSD Bid Rate = EURUSD_WF(B) = 1.1872

Wells Fargo EURUSD Ask Rate = EURUSD_{WF(A)^{​​​​​​​}} = 1.1884

Deutche Bank EURUSD Bid Rate = EURUSD_{DB(B)^{​​​​​​​}} = 1.1843

Deutsche Bank EURUSD Ask Rate = EURUSD_{DB(A)^{​​​​​​​}} = 1.1861

Part A

Dollars offered by Deutsche Bank for 3,000 Euros

= 3000 x EURUSD_DB(B)

= 3000 x 1.1843

= 3,552.9 USD

Part B

Euro offered by Wells Fargo for 7,000 Dollars

= 7000 x USDEUR Bid Rate of Wells Fargo

= 7000 x 0.8415

= 5,890.5 USD

Note: USDEUR Bid Rate for Wells Fargo = 1 / EURUSD_WF(A) = 1 / 1.1884 = 0.84146752

Part C

Arbitrage Opportunity exixts whenever one can Buy at a Lower rate and Sell at a Higher rate. As a market taker, the corporate treasury can Buy Euro at Ask Rate & Sell Euro ay Bid Rate only. Thus, Arbitrage Opportunity exists if any available Ask Rate is lower than any available Bid Rate.

We see, EURUSD_{DB(B)^{​​​​​​​}} = 1.1843 < EURUSD_{DB(A)^{​​​​​​​}} = 1.1861 < EURUSD_{WF(B)^{​​​​​​​}} = 1.1872 < EURUSD_{WF(A)^{​​​​​​​}} = 1.1884

Therefore, as an Ask rate is lower than a Bid rate, an Arbitrade Opportunity exists and corporate treasury can buy from Deutsche Bank at $1.1861/Euro and sell it to Wells Fargo at $1.1872/Euro.

If initial capital is $1,000,000 -

1) Buy Euro worth $1 Mn from Deutsche Bank at $1.1861/Euro

2) You will get (1000000/1.1861) Euro = 843099.2328 Euro

3) Sell back the Euro to Wells Fargo at $1.1872/Euro

4) You will get (843099.2328 x 1.1872) Euro = 1,000,927.4092 Dollars

5) Arbitrage Profit of $927.4092 has been made

Part D

New Rates,

Wells Fargo EURUSD Bid Rate = EURUSD_{WF(B)^{​​​​​​​}} = 1.1862

Wells Fargo EURUSD Ask Rate = EURUSD_{WF(A)^{​​​​​​​}} = 1.1881

Deutche Bank EURUSD Bid Rate = EURUSD_{DB(B)^{​​​​​​​}} = 1.1846

Deutsche Bank EURUSD Ask Rate = EURUSD_{DB(A)^{​​​​​​​}} = 1.1866

Arbitrage Opportunity exixts whenever one can Buy at a Lower rate and Sell at a Higher rate. As a market taker, the corporate treasury can Buy Euro at Ask Rate & Sell Euro ay Bid Rate only. Thus, Arbitrage Opportunity exists if any available Ask Rate is lower than any available Bid Rate.

We see, EURUSD_{DB(B)^{​​​​​​​}} = 1.1846 < EURUSD_{WF(B)^{​​​​​​​}} = 1.1862 < EURUSD_{DB(A)^{​​​​​​​}} = 1.1866 < EURUSD_{WF(A)^{​​​​​​​}} = 1.1881

We see that both Bid rates are lower than both Ask rates, thus the corporate treasury cannot buy at a lower rate and sell at a higher rate. Thus, no Arbitrage Opportunity exists.