2013 FRM Calendar

# P2.T5.307. Cox-Ingersoll-Ross (CIR) interest rate process

14 Feb 2013   by Suzanne Evans

Exam Relevance: Optional

AIMs: Describe the short-term rate process under the Cox-Ingersoll-Ross (CIR) and Lognormal models. Calculate the short-term rate change and describe the basis point volatility using the CIR and Lognormal models. Summarize the application of a lognormal model with deterministic drift and a lognormal model with mean reversion

### Questions:

307.1. Donald the analyst is employing the Cox-Ingersoll-Ross (CIR) model for the short-term rate process:

His assumptions include (see above):

- The time-step is monthly, dt = 1/12
- Today's initial rate, r(0) = 1.00%
- The annual basis point volatility, sigma = 2.50%
- The long-run rate, theta = 8.00%
- The strength of reversion, k = 0.60

For the first month, the random uniform value is 0.71 such that the random standard normal is 0.5534 and dw = 0.5534*SQRT(1/12) = 0.160. What is the short-rate in the first month under this CIR process, r(1/12)?

1. -0.250%
2. 0.444%
3. 1.390%
4. 2.172%

307.2. Peter the analyst is constructing a binomial tree according to Tuckman's lognormal model (without mean reversion). Here are his assumptions and partial tree:

His assumptions include:

-The time-step is monthly, dt = 1/12
-Today's initial rate, r(0) = 2.00%
-Annual drift is constant at 50 basis points
-The annual basis point volatility, sigma = 9.00%

What is the rate at node [1,1]?

1. 1.98665%
2. 2.05350%
3. 2.38794%
4. 3.12550%

307.3. Each of the following is true about lognormal models of the short-term interest rate process EXCEPT for:

1. In the lognormal models, the the natural logarithm of the short rate is normally distributed
2. Like the Cox-Ingersoll-Ross (CIR), the lognormal models offer the advantage of not allowing negative rates
3. The lognormal model without mean reversion is similar to the Ho-Lee model but based on the natural logarithm of the short rate instead of on the short rate itself
4. The Black-Karasinski Model is similar to the Ho-Lee Model but instead is an equilibrium model with constant drift