$($i$)$ Find the probability distribution function of $S_N.$ Please justify your answer. $[4]$

$($ii$)$ Find the expected values $E[S_N]$ and $E[ln(S_N)].$ If you use results on the expected

value of certain random variables, you need to justify them. Give your answers in

simplified and exact term in which the variable $N$ is allowed. $[$Hint: Use the

formulae $(1+x{)}^n={\displaystyle \underset{0=1}{\overset{n}{}}}([n],[i])x^i$ and $n(1+x{)}^{n-1}={\displaystyle \underset{i=1}{\overset{n}{}}}i([n],[i])x^{i-1}.]$

$($iii$)$ Suppose one period is $6$ months. The continuously$-$compounded risk$-$free rate is

$0\mathrm{.}12.$ A one$-$year European option has payoff function max$(\sqrt[\phantom{2}]{S_2},\frac{S_1}{10}),$ where $S_i$

is the stock price at the end of the $i$ th period, $i=1,2.$ Find the price of this

option. Give your answer to $3$ significant figures.