Consider the risk$-$neutral Ho$-$Lee model for continuously compounded rates for $t=0,1,2$ and a step size of $=1$ year:

$r_0=0\mathrm{.}02$

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

$r_2=r_1+0\mathrm{.}02+0\mathrm{.}01{}_2^{**}$

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 $3$ year non$-$callable bond with principal $100$ and an annual coupon rate of $4\%?($To be precise, the coupons are paid at times $t=1,2,3,$ while the principal repayment is at time $t=3).$

$5$

$[8$ 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$)?$

$[5$ points$]$ Compute the initial price of a three year Asian interest rate call option with a strike of $r_K=0\mathrm{.}03$ 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 $t=3.$

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

$[5$ points$]$ You would like to replicate the Asian option through a combination of $1$ year and $2$ year ZCBs $($both ZCBs have a face value of $100).$ Compute the number of units of $1$ year and $2$ year ZCBs in your replicating portfolio at $t=0.$

$[5$ points