LO 14.3: Calculate the short-term rate change and describe the basis point

LO 14.3: Calculate the short-term rate change and describe the basis point volatility using the C IR and lognormal models.
.Another issue with the aforementioned models is that the basis-point volatility of the short- term rate is determined independently of the level of the short-term rate. This is questionable on two fronts. First, imagine a period of extremely high inflation (or even hyperinflation). The associated change in rates over the next period is likely to be larger than when rates are closer to their normal level. Second, if the economy is operating in an extremely low interest rate environment, then it seems natural that the volatility of rates will become smaller, as rates should be bounded below by zero or should be at most small, negative rates. In effect, interest rates of zero provide a downside barrier which dampens volatility.
A common solution to this problem is to apply a model where the basis-point volatility increases with the short-term rate. Whether the basis-point volatility will increase linearly or non-linearly is based on the particular functional form chosen. A popular model where the basis-point volatility (i.e., annualized volatility of dr) increases proportional to the square root of the rate (i.e., (Wr) is the Cox-Ingersoll-Ross (CIR) model where dr increases at a decreasing rate and a is constant. The CIR model is shown as follows:
dr = k(0 r)dt + a Vr dw
As an illustration, lets continue with the example from LO 14.2, given the application of the CIR model. Assume a current short-term rate of 3%, a long-run value of the short-term rate, 6, of 24%, speed of the mean revision adjustment, k, of 0.04, and a volatility, a, of 1.30%. As before, also assume the dw realization drawn from a normal distribution is 0.2. Using the CIR model, the change in the short-term rate after one month is calculated as:
dr = 0.04(24% – 5%)(1/12) + 1.5%
S %x 0.2
dr = 0.063% + 0.067% = 0.13%
Therefore, the expected short-term rate of 5% plus the rate change (0.13%) equals 5.13%.
2018 Kaplan, Inc.
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Topic 14 Cross Reference to GARP Assigned Reading – Tuckman, Chapter 10
A second common specification of a model where basis-point volatility increases with the short-term rate is the lognormal model (Model 4). An important property of the lognormal model is that the yield volatility, O’, is constant, but basis-point volatility increases with the level of the short-term rate. Specifically, basis-point volatility is equal to or and the functional form of the model, where a is constant and dr increases at err, is:
dr = ardt + crrdw
For both the CIR and the lognormal models, as long as the short-term rate is not negative then a positive drift implies that the short-term rate cannot become negative. As discussed previously, this is certainly a positive feature of the models, but it actually may not be that important. For example, if a market maker feels that interest rates will be fairly flat and the possibility of negative rates would have only a marginal impact on the price, then the market maker may opt for the simpler constant volatility model rather than the more complex CIR.
The differences between the distributions of the short-term rate for the normal, CIR, and lognormal models are also important to analyze. Figure 1 compares the distributions after ten years, assuming equal means and standard deviations for all three models. As mentioned in Topic 13, the normal distribution will always imply a positive probability of negative interest rates. In addition, the longer the forecast horizon, the greater the likelihood of negative rates occurring. This can be seen directly by the left tail lying above the x-axis for rates below 0%. This is clearly a drawback to assuming a normal distribution.
Figure 1: Terminal Distributions
5% Rate
——–CIR ——– N o r m a l——-Lognormal
In contrast to the normal distribution, the lognormal and CIR terminal distributions are always non-negative and skewed right. This has important pricing implications particularly for out-of-the money options where the mass of the distributions can differ dramatically.
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2018 Kaplan, Inc.
Topic 14 Cross Reference to GARP Assigned Reading – Tuckman, Chapter 10