$($Ch $17)(31$ points$)$ Assume that all implied volatilities change by the same amount so that vegas can be aggregated. IBC, a U$.$S$.-$based financial institution, has the following portfolio of over$-$the$-$counter options on the Great British pound $($GBP$)$:

$\backslash$table$[[$Type$,$Position,Delta of Option,Gamma of Option,Vega of Option$],[$Call$,-1,000,0\mathrm{.}50,2\mathrm{.}2,1\mathrm{.}8],[$Call$,-500,0\mathrm{.}80,0\mathrm{.}6,0\mathrm{.}2],[$Put$,-2,000,-0\mathrm{.}40,1\mathrm{.}3,0\mathrm{.}7],[$Call$,-500,0\mathrm{.}70,1\mathrm{.}8,1\mathrm{.}4]]$

Another available option in the market to be traded has a delta of $0\mathrm{.}6,$ a gamma of $1\mathrm{.}5,$ and a vega of $0\mathrm{.}8.$

$($a$)$ Based on the information in the table, what are the delta, gamma, and vega of the IBC's existing options portfolio? $(9$ points$)$

$($b$)$ What position in the traded option and in GBP amounts would make the portfolio both gammaneutral and delta$-$neutral? $(6$ points$)$

$($c$)$ What position in the traded option and in GBP amounts would make the portfolio both veganeutral and delta$-$neutral? $(6$ points$)$

Suppose that a second traded option with a delta of $0\mathrm{.}1,$ a gamma of $0\mathrm{.}5,$ and a vega of $0\mathrm{.}6$ is available.

$($d$)$ How could the portfolio be made delta, gamma, and vega neutral? $(10$ points$)$