Q$34)$ A financial institution has the following portfolio of options written on the same underlying, which currently sells for $$725$ per share and pays no dividends $($assume $1$ contract $=1$ share$).$

$(1)$ Option type: Call Position: long strike price: $780$ time:$0\mathrm{.}5$ years Number of contracts:$700$ Gamma:$1\mathrm{.}35$ Vega:$0\mathrm{.}25$

$(2)$Option type: put Position: long strike price:$750$ time:$0\mathrm{.}5$ years Number of contracts:$500$ Gamma:$1\mathrm{.}20$ Vega:$0\mathrm{.}5$

$(3)$Option type: Call Position: short strike price:$710$ time:$0\mathrm{.}25$ years Number of contracts:$250$ Gamma:$0\mathrm{.}95$ Vega:$0\mathrm{.}9$

$(4)$Option type: Call Position: long strike price:$700$ time:$0\mathrm{.}25$ years Number of contracts:$800$ Gamma:$1\mathrm{.}20$ Vega:$1\mathrm{.}3$ There are currently two traded options available in the market:

$$ Option A has a Delta of $0\mathrm{.}8,$ a Gamma of $1\mathrm{.}1,$ and a Vega of $0\mathrm{.}45.$

$$ Option B has a Delta of $1\mathrm{.}15,$ a Gamma of $0\mathrm{.}95,$ and a Vega of $0\mathrm{.}97.$ The risk$-$free rate is $10\%$ per annum and the volatility of the underlying asset is $40\%.$ Based on the information provided, answer the following questions $($show all the details of your calculations and present your results with four decimal places$)$:

a$)$ Calculate the Delta of each individual option and of the whole portfolio using the Black$-$Scholes formula.

b$)$ How can the financial institution make this portfolio delta$-$gamma neutral using traded option A only?

c$)$ How can the financial institution make this portfolio delta$-$gamma$-$vega neutral using traded options A and B$?$