you are making a market in nat gas options. on $27$ march $2024$ you sold put options on $100$ july, $2024$ natural gas futures contracts. the options are struck at $$2\mathrm{.}10$ MMBTU. The relevant volatility is $43\mathrm{.}52$ percent, and the relevant interest rate is $5$ percent. the options expire on $25$ june $2024.$ the current july $2024$ NG futures price is $$2\mathrm{.}285/$MMBTU$.$ each contract is for $10000$ MMBTU.

A$)$ what is the price of the options sold? B$)$ what are the gamma, delta and vega of the entire position in the options you sold?

C$)$ what are the risks associated with this transaction?

D$)$ Devise a delta hedge for this transaction. How many futures contracts should you trade? forr what delivery month? should you buy or sell? E$)$ what risks do you face when you merely delta hedge? how can you mitigate these risks?

F$)$ you have an opportunity to purchase calls on july $24$ NG struck at $$2\mathrm{.}5/$MMBTU$.$ devise a transaction in the $$2\mathrm{.}5$ stike july $24$ calls and july $24$ futures that offsets the delta and gamma risks of your original sale of puts. How many $2\mathrm{.}5$ calls should you trade? should you buy or sell? how many june $24$ futures should you trade? should you buy or sell? what is the vega of your gamma and delta hedged position?