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Answer: Under the Gauss+ model, the short-term rate mean reverts to a medium-term factor, the medium-term factor mean reverts to a long-term factor, and the long-term factor is modelled with time-dependent drift and time-dependent volatility.
## Explanation The correct answer is **A** because it accurately describes the cascade structure of the Gauss+ model. ### Key Features of the Gauss+ Model: 1. **Cascade Structure**: - Short-term rate mean reverts to a medium-term factor - Medium-term factor mean reverts to a long-term factor - Long-term factor has time-dependent drift and volatility 2. **Why Other Options are Incorrect**: - **Option B**: Incorrectly attributes the model's hump-shaped volatility to Fed policy pegging, when it actually arises from the cascade structure itself - **Option C**: Incorrect - in the Gauss+ model, factors are typically assumed to be independent, not correlated - **Option D**: Incorrect - the model allows for different mean reversion speeds among factors, which is actually supported by empirical evidence ### Mathematical Structure: The Gauss+ model typically has the form: - Short rate: dr = κ₁(θ - r)dt + σ₁dW₁ - Medium factor: dθ = κ₂(φ - θ)dt + σ₂dW₂ - Long factor: dφ = μ(t)dt + σ₃(t)dW₃ This cascade approach helps capture the hump-shaped term structure of volatility observed in real markets, where medium-term rates exhibit higher volatility than both short-term and long-term rates.
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Which of the following statements correctly describes the dynamics and implications of the cascade form of the Gauss+ model?
A
Under the Gauss+ model, the short-term rate mean reverts to a medium-term factor, the medium-term factor mean reverts to a long-term factor, and the long-term factor is modelled with time-dependent drift and time-dependent volatility.
B
Under the Gauss+ model, the evolution of short-term rate reflects the fact that the Fed keeps the short-term policy rate pegged at a target, and this helps generate a "hump-shaped" term structure of volatility that fits the empirical data quite well.
C
Under the Gauss+ model, the evolution of short-term rate and medium-term factor is modelled with a pre-specified correlation parameter, while the long-term factor is assumed to be independent of the medium-term factor.
D
Under the Gauss+ model, the speed of mean reversion from short-term to medium-term is assumed to be equal to that from medium-term to long-term, while the empirical evidence suggests that the speed of mean reversion among factors is different.