Describe the short-term rate process under a model with time-dependent volatility. Calculate the short-term rate change and determine the behavior of the standard deviation of the rate change using a model with time-dependent volatility. Assess the efficacy of time-dependent volatility models. 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. Describe lognormal models with deterministic drift and mean reversion.
Questions:
23.5.1. The Cox-Ingersoll-Rand (CIR) model has the following dynamics:
Assume this CIR model is expressed by an interest rate tree where each step is one month, i.e., dt = 1/12. The initial rate is 7.00%, and the annual basis point volatility, σ = 130 basis points or 1.30%. The long-run rate, θ = 5.00%, and the mean-reversion parameter, k = 0.30.
Like the other models, dw captures the risk factor: dw is a normally distributed random variable with zero mean and sqrt(dt) standard deviation. If next month's move is a +2.0 standard deviation jump, where N^(-1)(97.725%) = +2.0, then which of the following is nearest to the change in the rate?
a. -.2.275%
b. +0.003%
c. +0.149%
d. +0.638%
23.5.2. For his term structure model, Peter considered the Vasicek model, but on reflection, he prefers the Cox-Ingersoll-Ross (CIR). In regard to the CIR, which of the following statements is TRUE?
a. A drawback of the CIR is that it does not mean revert to a long-term rate
b. In the CIR, the drift is time-dependent such that CIR is a no-arbitrage model
c. An advantage of the CIR is that the terminal distribution of the short rate is eventually (e.g., >10 years) normally distributed
d. Unlike the Vasicek model's constant volatility, the CIR's basis point volatility is not constant, and this ensures the short rate cannot become negative
23.5.3. Each of the following is true about the Gauss+ model EXCEPT, which is false?
a. In the cascade form of the Gauss+ model, the short-term rate mean reverts to the medium-term factor, which reverts to the long-term factor
b. Gauss+ has three state variables--that is, r(t), m(t), and l(t)--that denote the short-term interest rate, a medium-term factor, and a long-term factor
c. The long-term factor, l(t), is mean-reverting to a long-term constant level and aims to reflect long-term trends in production, demographics, technology, or other factor(s)
d. The plus ("+") in the Gauss+ model reflects the addition of a factor to ensure a downward-sloping term structure of volatility which matches empirically observed volatility term structures
Answers:
Questions:
23.5.1. The Cox-Ingersoll-Rand (CIR) model has the following dynamics:
Assume this CIR model is expressed by an interest rate tree where each step is one month, i.e., dt = 1/12. The initial rate is 7.00%, and the annual basis point volatility, σ = 130 basis points or 1.30%. The long-run rate, θ = 5.00%, and the mean-reversion parameter, k = 0.30.
Like the other models, dw captures the risk factor: dw is a normally distributed random variable with zero mean and sqrt(dt) standard deviation. If next month's move is a +2.0 standard deviation jump, where N^(-1)(97.725%) = +2.0, then which of the following is nearest to the change in the rate?
a. -.2.275%
b. +0.003%
c. +0.149%
d. +0.638%
23.5.2. For his term structure model, Peter considered the Vasicek model, but on reflection, he prefers the Cox-Ingersoll-Ross (CIR). In regard to the CIR, which of the following statements is TRUE?
a. A drawback of the CIR is that it does not mean revert to a long-term rate
b. In the CIR, the drift is time-dependent such that CIR is a no-arbitrage model
c. An advantage of the CIR is that the terminal distribution of the short rate is eventually (e.g., >10 years) normally distributed
d. Unlike the Vasicek model's constant volatility, the CIR's basis point volatility is not constant, and this ensures the short rate cannot become negative
23.5.3. Each of the following is true about the Gauss+ model EXCEPT, which is false?
a. In the cascade form of the Gauss+ model, the short-term rate mean reverts to the medium-term factor, which reverts to the long-term factor
b. Gauss+ has three state variables--that is, r(t), m(t), and l(t)--that denote the short-term interest rate, a medium-term factor, and a long-term factor
c. The long-term factor, l(t), is mean-reverting to a long-term constant level and aims to reflect long-term trends in production, demographics, technology, or other factor(s)
d. The plus ("+") in the Gauss+ model reflects the addition of a factor to ensure a downward-sloping term structure of volatility which matches empirically observed volatility term structures
Answers:
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