A stock price is currently $$30.$ During each $2-$month period for the next

$4$ months it will increase by $8\%$ or reduce by $10\%.$ The risk$-$free interest

rate is $5\%.$ Use a two$-$step tree to calculate the value of a derivative that

pays off $[$max $(30$ ST $,0)]2$

$,$ where ST is the stock price in $4$ months. If

the derivative is American$-$style, should it be exercised early?