$4$ A stock price is currently selling at $S_0=$$$55.$ Over each of the

next two $6-$month periods it is expected to go up by $1\%$ or up by

$3\%.$ The annual risk$-$free interest rate is $4\%.$ Let $d$ be a financial

derivative with payoff function $f(S_T,K)=$max$(S_T,K),$ with $S_T$

the final value of the stock and $K$ the strike price.

$($i$)$ Use a two$-$step tree to draw the payoffs of the financial derivative

$d$ for $K=$$$57$ between $T=0$ and $T=1$ year.

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

Elaborate your answer and draw the two$-$step tree of payoffs for

the American option.

$($iii$)$ Answer the following questions on Greek letters for the European

style option $d(S)$ in $($i$)$:

$($a$)$ Obtain the value of the Delta letter at $t=6$ months and

$t=1$ year. Interpret the numerical results.

$($b$)$ Obtain the value of the Gamma letter at $t=1$ year. Interpret

the numerical results.

$($c$)$ Obtain the value of the Rho letter at $t=0.$ Interpret the

numerical results.

$($N$.$B$.$ The risk$-$free interest rate is compounded every six months.$)$