Given a current one-period spot rate of \(S_{0}=10 \%\), upward and downward parameters of \(u=1.1\) and \(d=0.9091\), and probability of the spot rate increasing in one period of \(q=0.5\) :

a. Generate a two-period binomial tree of spot rates.

b. Using the binomial interest rate tree from Question 5.a, determine the value of a two-period, option-free \(9 \%\) coupon bond with \(F=100\).

c. Using the binomial interest rate tree from Question 5.a, determine the value of the \(9 \%\) bond assuming it is callable at a call price of \(C P=99\). Use the minimum constraint approach.

d. Using the binomial interest rate tree, show at each node the call option values of the callable bond \((C P=99)\). Given your call option values, determine the values at each node of the callable bond as the difference between the option-free values found in Question 5.b and the call option values. Do your callable bond values match the ones you found in Question 5.c.

e. Comment on values of your call options being equal to the present value of the interest savings the issuer realizes from refunding the bond at lower rates.

f. Using the binomial interest rate tree, determine the value of the bond assuming it is putable in periods one and two at a put price of \(P P=99\). Use the maximum constraint approach.

g. Using the binomial interest rate tree, show at each node the put option values of the putable bond \((P P=99)\). Given your put option values, determine the values at each node of the putable bond as the sum of the option-free bond values found in Question 1.b and the put option values. Do your putable bond values match the ones you found in Question 5.f?