The euro and U$.$S$.$ dollar, S$($USD$/$EUR$),$ spot exchange rate is $1\mathrm{.}0703.$ The Japanese yen and U$.$S$.$ dollar S$($JPY$/$USD$)$ spot exchange rate is $147.$ If the market is quoting the Japanese yen and euro, S$($JPY$/$EUR$)$ spot exchange rate at $160,$ determine if triangular arbitrage exists. If it does, how much can you profit $($in US dollar$)$ if you can borrow up to $$1,000,000$ without any transaction cost?

$($EUR: euro; JPY: Japan yen; USD: U$.$S$.$ dollar$)$

Be sure to answer the following questions:

$($a$)$ What is the no$-$arbitrage exchange rate between the Japanese yen and euro, S$($JPY$/$EUR$)?$

$($b$)$ Is there an arbitrage opportunity? Why or why not?

$($c$)$ If there is an arbitrage opportunity, state clearly each step necessary to capture the profit. In each step, state which currency you would buy and sell and include the amount of each currency with its currency symbol.

$($d$)$ How much is the arbitrage profit in US dollar?