LO 14.2: Calculate the short-term rate change and determine the behavior o f the

LO 14.2: Calculate the short-term rate change and determine the behavior o f the standard deviation o f the rate change using a model with tim e dependent volatility.
The relationships between volatility in each period could take on an almost limitless number of combinations. For example, the volatility of the short-term rate in one year, cr(l), could be 220 basis points and the volatility of the short-term rate in two years, cr(2), could be 260 basis points. It is also entirely possible that cr(l) could be 220 basis points and cr(2) could be 160 basis points. To make the analysis more tractable, it is useful to assign a
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Topic 14 Cross Reference to GARP Assigned Reading – Tuckman, Chapter 10
specific parameterization of time-dependent volatility. Consider the following model, which is known as Model 3:
dr = \(t)dt + ere atdw
where: a = volatility at t = 0, which decreases exponentially to 0 for a > 0
To illustrate the rate change using Model 3, assume a current short-term rate, rQ, of 3%, a drift, \ , of 0.24%, and, instead of constant volatility, include time-dependent volatility of ae_0-3t (with initial a = 1.30%). If we also assume the dw realization drawn from a normal distribution is 0.2 (with mean = 0 and standard deviation = Vl /12 = 0.2887), the change in the short-term rate after one month is calculated as:
dr = 0.24% x (1/12) + 1.5% x e-0-3(1/12) x 0.2
dr = 0.02% + 0.29% = 0.31%
Therefore, the expected short-term rate of 5% plus the rate change (0.31%) equals 5.31%. Note that this value would be slightly less than the value assuming constant volatility (5.32%). This difference is expected given the exponential decay in the volatility.
M odel 3 Effectiveness

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