10% in the tree, so potential buyers will offer less to get a 10% instrument.)

The options to be valued are a European call and a European put, each with maturity two years, and strike price $95. In order to value these options, we must consider the possible bond values at that time.

To get the possible bond prices, do the following:

1)

From the trees worked out previously, the short rate in two years time will be one of 19.42%, 13.77%, or 9.76.

2)

With these short rates, then the known 3-year payoff of $110 will be worth one of $92.11, $96.69, or $100.22, respectively.

Continuing on, get the option values:

3)

With the given bond prices, then the call will be worth $0, $1.69, or $5.22.

4)

Carry out the averaging/discounting procedure: Specifically:

(a)

Average $0 and $1.69 and to get $0.845, and then discount by the “up” short rate in one year’s time, 14.32%, to get

(1/2)($0 + $1.69) / (1 + 14.32%) = $0.74

(b)

Average $1.69 and $5.22 to get $3.455, and then discount by the “down” short rate in one year’s time, 9.79%, to get

(1/2)($1.69 + $5.22) / (1 + 9.79%) = $3.15

(c)

Given steps (a) and (b), then we have all the possible values for the option in one year’s time. So, if we average and discount one more time, then we’ll get the present value for the option. The spot short rate is 10%, so this calculation is given as

(1/2)($0.74 + $3.15) / (1 + 10%) = $1.77

which is the final answer: $1.77 is the price of the European call.

Similar calculations give a value of $0.57 for the put.

The procedure given above is the general strategy for valuing European options. Find the value of the option at any node as the discounted expected value from one step in the future. American options require just a little more effort. At each node, you must compare the value obtained if held – that is, the discounted expected value as worked out above – with the value if exercised early. (For the early exercise value in the case of the call, take the bond’s value at that node from the lattice, and subtract off the strike price.) The value at the node will be given by the greater of the two quantities.

Option Hedge Ratios

In this section, the authors define the hedge ratio , and discuss how to obtain it from the trees of the previous sections. The hedge ratio gives the relation of how much option prices change in response to changes in the price of the underlying bond.

The formula for the hedge ratio is

call = (Cu – Cd) / (Tu – Td)