Please sove the following questions. A financial institution plans to offer a security that pays off a dollar amount equal to $S_T^7$ at time $T,$ where $S_t$ is the price at time $t$ of a stock that pays no dividends,

the current stock price is $S_0=0\mathrm{.}6,$ the risk$-$free interest rate is $5\%$ per annum and the time to maturity is $3$ months. The stock price follows a geometric Brownian

motion with an expected return $9\%$ per annum and a volatility $22\%$ per annum.

$(1)$ The price of the security is

$(2)$ The delta of the security is

$(3)$ The gamma of the security is

$(4)$ The vega of the security is