Math Problem $($Show the detailed calculations$)$:

Suppose that you observe the following exchange rates:

$1\mathrm{.}15\frac{\text{\$}}{}$

$3\mathrm{.}15\frac{\text{\$}}{}$

$0\mathrm{.}45\frac{}{}$

Check if there is an arbitrage opportunity and calculate the profits if you start with $1,000,000.$ Assume that the total transactions costs are $2,000.$ Show all transactions and calculations involved in one round of transactions.