$($Ch $12)(10$ points$)$ A stock $($paying no dividends$)$ price is currently $$30.$ During each twomonth period for the next four months, it is expected to increase by $8\%$ or reduce by $10\%.$ The risk$-$free interest rate is $5\%$; this rate is annualized and to be continuously compounded.

$($a$)$ Use a two$-$step tree to calculate the value of a derivative, called "power option," that pays off max$[(30-S_T),0{]}^2,$ where $S_T$ is the stock price in four months? $[$Hint: Use a binomial tree and follow risk$-$neutral valuation by computing $u,d,$ and $p$ based on tradeable assets.$](6$ points$)$

$($b$)$ If the derivative is American$-$style, should it be exercised early? $(4$ points$)$