Problem $8(10pt)$

A trading desk holds the portfolio of over$-$the$-$counter options on the SP$500$ index shown in the table below.

$\backslash$table$[[$Type$,\backslash$table$[[$Number of$],[$options$]],\backslash$table$[[$Delta of$],[$Option$]],\backslash$table$[[$Gamma of$],[$Option$]],\backslash$table$[[$Vega of$],[$Option$]]],[$Call$,-500,0\mathrm{.}4,2\mathrm{.}1,1\mathrm{.}6],[$Call$,500,0\mathrm{.}6,0\mathrm{.}5,0\mathrm{.}1],[$Put$,1000,-0\mathrm{.}5,1,0\mathrm{.}5],[$Put$,-500,-0\mathrm{.}6,1\mathrm{.}8,1\mathrm{.}2]]$

Figure $1$: Option data for Problem $8.$

The desk would like to hedge the risks of this portfolio using an index option $O$ with delta $0\mathrm{.}5,$ gamma $3\mathrm{.}5$ and vega $0\mathrm{.}2.$

a$)$ What position in the traded option $O$ and in index futures would make the portfolio both gamma neutral and delta neutral?

b$)$ What position in the traded option $O$ and in index futures would make the portfolio both vega neutral and delta neutral?