In the three$-$period model of Figure $1,$ let the interest rate be $r=\frac{1}{4},$ and $u=2,$ and $d=\frac{1}{3}.$

fi$)$ Find the stock prices at different states $,$

$\{$:$S_3(HHH),S_3(THH),S_3(HTH),S_3(HHT),S_3(TTH),S_3(HTT),S_3(THT),S_3(TTT)).$

ii$)$ Find the risk$-$neutral probability $p.$

iii$)$ Determine the price at time zero of the American put that expires at time three and has intrinsic

value $g_P(s)=(5-s{)}^{+}.$

iv$)$ Determine the price at time zero of the American call that expires at time three and has intrinsic

value $g_C(s)=(s-5{)}^{+}.$