$(20$ points$)$ A financial institution has the following portfolio of over$-$the$-$counter options

on pounds sterling:

A traded option is available with a delta of $0\mathrm{.}6,$ a gamma of $1\mathrm{.}5,$ and a vega of $0\mathrm{.}8.$

$($a$)$ What position in the traded option and in pounds sterling would make the port$-$

folio both gamma neutral and delta neutral?

$($b$)$ What position in the traded option and in pounds sterling would make the port$-$

folio both vega neutral and delta neutral?

$($c$)$ 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. How could the portfolio be made delta, gamma, and vega neutral?