where Cu, d are the values of the call in the up- and down-nodes of the tree one period into the future, and Tu, d are the corresponding Treasury prices. (Ken: Note that we will actually have Cu < Cd and Tu < Td, because the u and d refer to movements of the interest rate, rather than to movements of the Treasury price.) A similar formula holds for the put hedge ratio, put, with the corresponding up- and down-node put prices, Pu, d.

Putting in numbers for the two-year call on the 3-year Treasury, studied in the previous section, then we get the prices for the first time step, Cu - Cd = 0.74 – 3.15 = -2.41. And for the Treasury, Tu – Td = 91.33 – 98.79 = -7.46. Substituting these into the formula for the hedge ratio call, we get call = -2.41 / -7.46 = 0.32.

This number gives us the change in the call value per dollar change in the Treasury value, and thus lead to a strategy for hedging the Treasury with the call, and vice versa. (Ken: The authors don’t actually lay out that strategy here. My guess on what such a strategy might entail: Say you have a portfolio with some Treasuries and some calls. (Each quantity may be long or short, corresponding to positive/negative amounts of the securities.) Then you can keep the value of your portfolio constant by using the hedge ratio. Say the Treasuries go up in value, then you can keep the portfolio value constant by shorting some calls. Something like that.)

The authors observe that call > 0, because the call goes up in value when the Treasury goes up in value. Conversely, if you run the numbers, you will get put < 0, because the put will go down in value if the Treasuries rise.