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

$2$ max$[(30)0]$ T $,$ S $,$ where T S 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$)$