Suppose a put option with $1-$year maturity on a $$1,000$ par, $1\mathrm{.}5-$year zero coupon bond is to be valued using the bond$-$pricing tree below. This put option has a strike price of $$978.$

A$.$ What are the payoffs of the put option at nodes D$,$ E$,$ and F$?$

B$.$ What are the risk$-$neutral probabilities for the up and the down states implied by the prices of the zero$-$coupon bonds between time $0$ and $0\mathrm{.}5?$

C$.$ What are the risk$-$neutral probabilities for the up and the down states implied by the prices of the zero$-$coupon bonds between time $0\mathrm{.}5$ and $1?$

D$.$ Using the risk$-$neutral probabilities, what are the values of the put option at nodes B and C$?$

E$.$ Using the risk$-$neutral probabilities, what is the value of the put option at node A$?$ Time $0$

Time $0\mathrm{.}5$

Time $1$