2)

A second tree is given, showing prices for a 2-year zero-coupon Treasury. We know that at the last time on the tree, prices at all the nodes must be $100, since it is a riskless instrument.

3)

Now, we use this to get the price at the top node one period out. If we discount $100 by 11%, the rate on the top node one period, then we get $100 / (1 + 11%) = $90.09.

4)

Similarly, we get the price at the bottom node one period out, by discounting $100 by 9%, the rate at the bottom node one period out, to get $100 / (1 + 9%) = $91.74.

5)

So now put these two together: The price one year from now will be either $90.09 or $91.74, each with equal probability. So the expected price one year from now is the average of these two, or ½($90.09 + 91.74) = $90.92.

6)

We then find today’s price by taking this expected price, and discounting by today’s short rate, using the valuation formula from the last section, to get $90.92 / (1 + 10%) = $82.65.

This is a general procedure for finding the value of a zero of any maturity, provided the tree of short rates goes out far enough. By working each node as above, we can always make it through such a tree, back to the root, to find the value of such an instrument today.

Observations from Ken: What is actually varied over the nodes are the interest rates, then prices are obtained from the interest rate changes. Possibly an unimportant point, but the changes are thought of in terms of interest rates first, and only later are prices obtained from those varied interest rates.

Another observation: A crucial step in filling out the prices of the tree is the knowledge that at maturity of the bond, the price will be face value, for all nodes of the tree. This fact is used to get prices at all other nodes of the tree. Relating to earlier observation: say you were trying to calibrate to a tree of short rates for a corporation that was not risk-free. Perhaps then this step would be on more shaky ground – that is, putting in face value at all nodes at maturity.

Finding Short Rates from the Term Structure

This section starts an extended example, which shows how to build a binary tree. The short rates are not known beforehand, and the volatilities must be solved for.

Convention: The authors will use an annual yield in their discussions of interest rates. Specifically, we have that the annual yield y of the N-year zero, in terms of the price S, is given by:

S = 100 / [(1 + y)N]

And similarly, the yields yu and yd one year from now on the corresponding prices Su and Sd are given by

Su, d = 100 / [(1 + yu, d)N – 1]

Now, we want to find the short rates, such that the term structure of the model matches that of the market. The example term structure that will be used in the numerical example here is the following:

Maturity (years)Yield (%)Yield Volatility (%)

11020

21119

31218

412.517

51316