LO 13.2: Calculate the short-term rate change and standard deviation o f the rate

LO 13.2: Calculate the short-term rate change and standard deviation o f the rate change using a model with normally distributed rates and no drift.
In Model 1, since the expected value of dw is zero [i.e., E(dw) = 0], the drift will be zero. Also, since the standard deviation of dw = V dt, the volatility of the rate change = a V dt. This expression is also referred to as the standard deviation of the rate.
In the preceding example, the standard deviation of the rate is calculated as:
1.2% x >/l /12 = 0.346% = 34.6 basis points
Returning to our previous discussion, we are now ready to construct an interest rate tree using Model 1. A generic interest rate tree over two periods is presented in Figure 1. Note that this tree is recombining and the ending rate at time 2 for the middle node is the same as the initial rate, rQ. Hence, the model has no drift.
Figure 1: Interest Rate Tree with No Drift
The interest rate tree using the previous numerical example is shown in Figure 2. One period from now, the observed interest rate will either increase with 30% probability to: 6% + 0.346% = 6.346% or decrease with 30% probability to: 6% – 0.346% = 5.654%. Extending to two periods completes the tree with upper node: 6% + 2(0.346%) = 6.692%, middle node: 6% (unchanged), and lower node: 6% – 2(0.346%) = 5.308%.
Figure 2: Numerical Example of Interest Rate Tree with No Drift
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Topic 13 Cross Reference to GARP Assigned Reading – Tuckman, Chapter 9