i$=0$ : r$0=0\mathrm{.}04$

i$=1$ : r$1,$u $=0\mathrm{.}0868$ and r$1,$d$=0\mathrm{.}0268$

i$=2$ : r$2,$uu$=0\mathrm{.}1299$ and r$2,$ud$=0\mathrm{.}0723$ and r$2,$dd$=0\mathrm{.}0147.$ You have estimated the risk neutral interest rate tree for the continuously compounded interest rate as in the table above. There is equal risk neutral probability to move up or down the tree and each interval time represents $6$ months, that is$,=0\mathrm{.}5.$

Compute the discount factors for all maturities.

Compute the price of a security that pays $$100$ at time $i=2$ if the continuously compounded interest rate at that time is less than $5\mathrm{.}00\%$ and zero otherwise.

Compute the price of a $1\mathrm{.}5-$year floor, paying semi$-$annually, with strike rate $r_K=4\mathrm{.}50\%,$ and notional $N=300.$