The authors start by getting the price of a one-period zero today by taking its expected value one period in the future, and then discounting by today’s (known) spot rate, 10%. Now, the price of this zero one period in the future is 100 in all cases, because one period in the future represents maturity for this bond. So the authors work out
90.91 = [ (1/2)100 + (1/2)100 ] / (1 + 0.10)
so 90.91 is today’s price for a one-period zero.
Next the goal is to find the short rates one year from now, using the yield and vol for a 2-year zero from the Table above.
Specifically, we now have a 2-step binomial tree. We wish to find the short rates ru and rd to put into the nodes at the beginning of the 2nd step of the tree. And we must meet the constraint of matching the 2-year zero price and vol, as listed in the Table above.
So, we have the following situation. We have a 2-step tree, and we know the price at all 3 of the final nodes. (There are 3 final nodes, because it is a recombinant tree.) The price at all final nodes is face value, $100. We need to find the prices at the nodes of the intermediate step, and at the initial node. Follow the procedure:
We know that today’s short rate is 10%.
Guess that the short rate in the up- and down-states of the intermediate step are ru = 14.32 and rd = 9.79. Again, don’t worry about how they got these numbers too much, they’re just trying to show the tree can be made to match the market prices and vols. In reality, you would have to do some kind of numerical solver to get these values for ru and rd.)
Use the valuation formula – the formula given before that said S = [(1/2)(Su + Sd)] / (1 + r), to get the prices of the 2-year zero at each of the intermediate nodes. Use the fact that the zero is equal to $100 at all final nodes, and along with the rates ru, rd assumed above, we get
$87.47 = (1/2)($100 + $100) / (1 + 14.32%)
$91.08 = (1/2)($100 + $100) / (1 + 9.79%)
as the prices of the 2-year zero at the up and down intermediate time nodes.
Use the prices obtained in the last step to get the spot price of the 2-year zero:
$81.16 = [(1/2)$87.47 + (1/2)$91.08] / (1 + 10%)
where we averaging the two possible 1-step prices, $87.47 and $91.08, and discounting at today's known short rate of 10%, in order to get $81.16, the spot price of the 2-year zero.
Finally, we check that the annualized yield on the two-year zero, obtained by solving with the price given above and the yield formula stated previously, agrees with the yield quoted in the market. So, we have
$81.16 = $100 / [(1 + 11%)2]
which comes out correctly. Here we are applying the previously stated formula of S = 100 / [(1 + y)N], with S = $81.16, the 2-year zero price obtained from the 2-step tree, y = 11%, the two-year annualized yield taken from the market, and N = 2, the maturity of the transaction.