A U$.$S$.-$based bank$$s position in options on the EURUSD exchange rate has a delta of $30,000$ and a gamma of $80000,.$ The EURUSD $($dollars per euro$)$ exchange rate is now $1\mathrm{.}07.$

$($a$)$ Explain how these numbers can be interpreted. $(6$ points$)$

$($b$)$ What position would you take to make the position delta$-$neutral? $(3$ points$)$

$($c$)$ After a short period of time, the EURUSD exchange rate wildly moves to $1\mathrm{.}11.$ Estimate the new delta. $(6$ points$)$

$($d$)$ Continuing from part $($c$),$ what additional trade is necessary to keep the position delta neutral? $(3$ points$)$

$($e$)$ Assuming the bank did set up a delta$-$neutral position originally, has it gained or lost money from the exchange$-$rate movement? $(3$ points$)$