Take $N=3$ and let $S(0)=100,$ Up Move Rate, $U=35\%,$ down move rate, $D=-15\%,$ Discount rate, $R=5\%.$ Consider a European Call and American Call with exercise price $K=100$ at time $3.$ Determine the current value of the call price process $C(0).$

$(14$ Marks$)$

An analyst considers a $2$ period binomial lattice model. The first $3$ nodes of the European call option's nodes were as follows $25\mathrm{.}5,5\mathrm{.}3,28\mathrm{.}3$ Determine the options payoffs at nodes N$4,$ N$5,$ N$6$ given that for the call at node N$7$ is $54$ payoff.