Maverick Manufacturing PLC must purchase gold in three months for use in its operations. Maverick$$s management has estimated that if the price of gold were to rise above $$875$ per ounce, the firm would go bankrupt. The current price of gold is $$850$ per ounce. The firm's chief financial officer believes that the price of gold will either rise to $$900$ per ounce or fall to $$825$ per ounce over the next three months. Management wishes to eliminate any risk of the firm going bankrupt. Maverick can borrow and lend at the risk free APR of $16\mathrm{.}99\%.$

a$)$ Should the company buy a call or a put option on gold? To avoid bankruptcy, what time to expiration would the company like this option to have?

b$)$ Using risk neutral probabilities method, how much should such an option sell for in the open market? Show how Probability$($rise$)=0\mathrm{.}7868$