$[8]$ Consider the $N-$period binomial lattice where, at time $t_n,$ the stock price $S_n^j$ can go up to $S_{n+1}^{j+1}=u{S}_n^j$ with probability $p^u$ and sown to $S_{n+1}^j=d{S}_n^j$ with probability $1-p^u.$ Denote by $r>0$ the constant risk free interest rate and denote by

$q^u=\frac{e^{rt}-d}{u-d}$

the risk neutral probability.

$($a$)[3]$ Provide an expression for all possible stock prices at $T=t*N.$

$($b$)[2]$ If $S_0$ is given, what is the risk neutral probability that, at time $T,$ the stock price has experienced exactly $k$ up moves?

$($c$)[3]$ Using risk neutral pricing, provide the expression, in terms of $T,q^u,K,r$ for the fair time $t=0$ value of a European straddle expiry $T$ and strike price $K.$ Recall from Question $3$ that such an option has payout max$\{K-S_N,0\}+$max$\{S_N-K,0\}.$ Justify your answer.