Explanation:
The mean of the short-term rate, as a function of the time horizon, would increase gradually from its current value to its limiting value of θ. This is because the mean reversion process implies that the short-term rate tends to revert to a long-term mean value, denoted by θ. The speed of this reversion is determined by the mean-reverting parameter, k. When k is relatively small, the mean of the short-term rate rises very slowly towards its limiting value of θ. This gradual increase is reflected in the term structure, which shows the relationship between interest rates (or yields) and different time horizons. Therefore, the mean of the short-term rate is directly proportional to the horizon, increasing gradually from its current value towards its limiting value of θ.
Choice A is incorrect. The mean of the short-term rate, as a function of the time horizon, does not remain constant on the term structure. Instead, it increases gradually from its current value to its limiting value of θ, reflecting the influence of mean reversion.
Choice C is incorrect. Similar to choice A, this statement is also inaccurate because it suggests that the mean of the short-term rate will remain constant from its current value to its limiting value of θ. This contradicts with how mean reversion influences term structures; it causes an increase in the mean over time.
Choice D is incorrect. While market volatility can impact short-term rates, it does not directly cause an increase in their means. Rather, any changes in these means are primarily driven by factors such as interest rates and economic conditions rather than market volatility alone.
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Q.1663 Suppose we drew a graph showing the impact of the mean reversion on the terminal distribution of short term rates. Risk-neutral distributions at different time horizons for the short-term rate would show the impact of the mean reversion on the term structures. Which of the following observations would we most likely make?
A
The mean of the short-term rate, as a function of the time horizon, would remain constant or relatively constant on the term structure.
B
The mean of the short-term rate, as a function of the time horizon, would increase gradually from its current value to its limiting value of .
C
The mean of the short-term rate, as a function of the time horizon, will remain constant from the current value to its limiting value of θ.
D
The graph would show an increase in the mean of the short-term rate to match the market volatility.