Hickock Mining is evaluating when to open a gold mine. The mine has $60,000$ ounces of gold left that can be mined and mining operations will produce $7,500$ ounces per year. The required return on the gold mine is $12\%$ and it will cost $$14$ million to open the mine. When the mine is opened, the company will sign a contract that will guarantee the price of gold for the remaining life of the mine. If the mine is opened today, each ounce of gold will generate an after$-$tax cash flow of $$450$ per ounce. If the company waits one year, there is $60\%$ probability that the contract price will generate an after$-$tax cash flow of $$500$ per ounce and a $40\%$ probability that the after$-$tax cash flow will be $$410$ per ounce. What is the value of the option to wait based on the option valuation$/$decision tree method? Suppose the minimum required rate of return $($i$.$e$.$ the risk$-$free rate$)$ for Hickock Mining is $7\mathrm{.}5\%$ with a variance of $0\mathrm{.}1055.$ Verify the direction of your results if the Black$-$Scholes model is applied in the decision$-$making process.