Given the following three quotes: $$\frac{1\mathrm{.}1630}{?}$ SF; $$\frac{1\mathrm{.}0840}{}$;$\frac{1\mathrm{.}0960}{?}$ SF$.$ Show how an arbitrageur would exploit the triangular arbitrage opportunity implied in the three rates and how much profit they would make on a nominal sum of $$30$ million.

$($A$)$

Buy euros with dollars at $$\frac{1\mathrm{.}0840}{},$ buy SF with euros at $\frac{1\mathrm{.}0960}{?}$ SF$,$ buy back dollars with SF at $$1\mathrm{.}1630/$SF$,$ make $$646,534\mathrm{.}83$

Buy SF with dollars at $$\frac{1\mathrm{.}1630}{?}$ SF$,$ buy euros with SF at $\frac{1\mathrm{.}0960}{?}$ SF$,$ buy back dollars with euros at $$\frac{1\mathrm{.}0840}{},$ make $$646,534\mathrm{.}83$

Buy SF with dollars at $$\frac{1\mathrm{.}1630}{?}$ SF$,$ buy euros with SF at $\frac{1\mathrm{.}0729}{?}$ SF$,$ buy back dolRons with euros at $$\frac{1\mathrm{.}0840}{},$ make $$608\mathrm{.}77$

Buy euros with dollars at $$\frac{1\mathrm{.}0840}{},$ buy SF with euros at $\frac{1\mathrm{.}0960}{?}$ SF$,$ buy back dollars with SF at $$\frac{1\mathrm{.}1630}{S}F$ make $$459,876\mathrm{.}70$