Consider the risk$-$neutral Ho$-$Lee model for continuously compounded rates for $t=0,1,2$

$4$

and a step size of $=0\mathrm{.}25$ year $($today is denoted by time $0)$:

$r_0=0\mathrm{.}03$

$r_1=r_0+0\mathrm{.}01+0\mathrm{.}02{}_1^{**}$

$r_2=r_1+0\mathrm{.}015+0\mathrm{.}02{}_2^{**}$

where, under the risk$-$neutral measure, the shocks $)=(1,2,$ which are independent and identically distributed, are given by

${}_t^{**}=\{\begin{array}{ccc}+1& \text{w}\text{i}\text{t}\text{h}\text{p}\text{r}\text{o}\text{b}\text{a}\text{b}\text{i}\text{l}\text{i}\text{t}\text{y}& 0\mathrm{.}5\\ -1& \text{w}\text{i}\text{t}\text{h}\text{p}\text{r}\text{o}\text{b}\text{a}\text{b}\text{i}\text{l}\text{i}\text{t}\text{y}& 0\mathrm{.}5\end{array}$

$[5$ points$]$ What is the price of a $9-$month non$-$callable bond with principal $100$ and an annual coupon rate of $8\%?($To be precise, the coupons are paid at times $t=1,2,3,$ quarterly; while the principal repayment is at time $t=3).$

$[8$ points$]$ Now suppose that the above bond is callable. What is the value of the callable bond? What is the value of the implicit call option? In which nodes will the bond be called $($assume no call option initially$)?$

$[6$ points$]$ Compute the initial price of a $6-$month Asian interest rate call option with a strike of $r_K=0\mathrm{.}04$ and a notional amount of $N=100.$ That is$,$ the payoff of this option is given by $N$max$(\frac{r_0+r_1+r_2}{3}-r_K,0)$ and is paid at $t=2.$

$[2$ points$]$ What is the spot rate duration of the Asian option at $t=0?$

$[8$ points$]$ Consider a $6-$month Eurodollar futures. The underlying final cash payment is: $6-$month LIBOR $N.N=100$ is the contract size. The rates given in the HoLee interest rate tree, $\{r_{i,j}\},$ are continuously compounded; whereas the LIBOR rate with a given maturity $k$ in the Eurodollar futures is instead linearly compounded, i$.$e$.,$ LIBOR$(k,j)=\frac{1}{}(e^{r_{k,j}}-1).$ In the example, the $6-$month Eurodollar futures has $k=0\mathrm{.}5.$ Compute the futures price. $($Hint: you may want to use the convergence property and the law of iterated expectations of futures.$)$