Learning Objectives: Describe the structure of the Gauss+ model and discuss the implications of this structure for the model’s ability to replicate empirically observed interest rate dynamics. Compare and contrast the dynamics, features, and applications of the Vasicek model and the Gauss+ model.
Questions:
25.7.1. The Gauss+ model describes the dynamics of the short-term rate (r_t), medium-term factor (m_t), and long-term factor (l_t) as follows:
Given the below parameters:
Which of the rates shows the adjustment factor with the least volatility?
a. Short Term Rate
b. Medium Term Rate
c. Long Term Rate
d. All adjustment factors = ρ
25.7.2. Which of the below charts most accurately conveys the volatility term structure shapes produced by the Vasicek and Gauss+ Models?
a. The Blue line describes the Vasicek Model; the Yellow line describes the Gauss+ model.
b. The Yellow line describes the Vasicek Model; the Blue line describes the Gauss+ model.
c. The Green line describes the Vasicek Model; the Yellow line describes the Gauss+ model.
d. The Blue line describes the Vasicek Model; the Green line describes the Gauss+ model.
25.7.3. In the Gauss+ model, the short-term rate rt is adjusted deterministically toward the medium-term factor mt according to the equation
Suppose that at time t = 0, the short-term rate is r_0 = 2%, and the medium-term factor is m = 3%. Also, assume that the mean reversion parameter a_r = 0.67. What is the expected short-term rate after 6 months?
a. 1.32%
b. 2.40%
c. 1.60%
d. 2.28%
Answers here:
Questions:
25.7.1. The Gauss+ model describes the dynamics of the short-term rate (r_t), medium-term factor (m_t), and long-term factor (l_t) as follows:
dr_t = -a_r (r_t - m_t) d_t
dm_t = -a_m (m_t - l_t)d + σ_m (ρ dW_t^1 + √(1 - ρ^2) dW_t^2)
dl_t = -a_l (l_t - μ) dt + σ_l dW_t^1
dm_t = -a_m (m_t - l_t)d + σ_m (ρ dW_t^1 + √(1 - ρ^2) dW_t^2)
dl_t = -a_l (l_t - μ) dt + σ_l dW_t^1
Given the below parameters:
- a_r = 0.5, a_m = 0.2, a_l = 0.05
- σ_m = 0.01, σ_l = 0.005
- ρ = 0.6
- μ = 0.04
Which of the rates shows the adjustment factor with the least volatility?
a. Short Term Rate
b. Medium Term Rate
c. Long Term Rate
d. All adjustment factors = ρ
25.7.2. Which of the below charts most accurately conveys the volatility term structure shapes produced by the Vasicek and Gauss+ Models?
a. The Blue line describes the Vasicek Model; the Yellow line describes the Gauss+ model.
b. The Yellow line describes the Vasicek Model; the Blue line describes the Gauss+ model.
c. The Green line describes the Vasicek Model; the Yellow line describes the Gauss+ model.
d. The Blue line describes the Vasicek Model; the Green line describes the Gauss+ model.
25.7.3. In the Gauss+ model, the short-term rate rt is adjusted deterministically toward the medium-term factor mt according to the equation
dr_t = - a_r (r_t - m_t) dt
Suppose that at time t = 0, the short-term rate is r_0 = 2%, and the medium-term factor is m = 3%. Also, assume that the mean reversion parameter a_r = 0.67. What is the expected short-term rate after 6 months?
a. 1.32%
b. 2.40%
c. 1.60%
d. 2.28%
Answers here:
