Consider the risk$-$neutral Ho$-$Lee model for continuously compounded rates with a step size of $=0\mathrm{.}5$ year:

$r_{t+0\mathrm{.}5}=r_t+{}_t^{**}+\sqrt[\phantom{2}]{}{}_{t+0\mathrm{.}5}^{**},$

$5$

where, under the risk$-$neutral measure, the shock ${}_{t+0\mathrm{.}5}^{**},$ which is realized at time $t+0\mathrm{.}5,$ is given by

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

You observe the following information at time $t=0$ :

Volatility is $=0\mathrm{.}02.$

The price of a half year ZCB with a face value of $100$ is $98\mathrm{.}04.$

The price of a one year ZCB with a face value of $100$ is $94\mathrm{.}26.$

Using the above information, calibrate your interest rate tree. What is $r_0,{}_0^{**}$ and $r_{0\mathrm{.}5,+-}=r_0+{}_0^{**}+-?$