Interest Rate Options

We can use an estimate of the volatility of an interest rate to create a set of possible rate paths for interest rates in the future called a binomial interest rate tree. The diagram in Figure 29.3 depicts a two period binomial interest rate tree.

Figure 29.3: Two Period Binomial Interest Rate Tree

The interest rate at each node is a one period forward rate. Beyond the root of the tree, there is more than one one period forward rate for each nodal period (i.e., at Year 1, we have two 1 year forward rates, i1,U and i1,D). The interest rates are selected so that the (risk neutral) probabilities of up and down moves are both equal to 0.5.

Given the two period interest rate tree shown next, what is the value of a two period European interest rate call option with an exercise rate of 5.50% and a notional principal of $1 million? (Assume that options cash settle at time T = 2.)

Solution

Given the exercise rate of 5.50%, the call option has a positive payoff for nodes $C^{++}$ and $C^{+-}$.

The value of the option at node $C^{++}$ can be calculated as:

Similarly, the value at node $C^{+-}$ can be calculated as:

Note that the discount rate is not constant.