Suppose that the short rate is currently 4 % and its...

Suppose that the short rate is currently $4 \%$ and its standard deviation in a short period of time $\Delta t$ is $0.01 \sqrt{\Delta t}$. What happens to this standard deviation when the short rate increases to $8 \%$ in (a) Vasicek's model, (b) Rendleman and Bartter's model, and (c) the Cox, Ingersoll, and Ross model?

Key Concepts

Cox-Ingersoll-Ross (CIR) Model

The CIR model incorporates a diffusion term that is proportional to the square root of the short rate. This implies that as the short rate increases, the standard deviation increases too, but only by the square root of the rate's change. Hence, doubling the rate results in the volatility increasing by a factor corresponding to the square root of the rate ratio.

Vasicek Model

The Vasicek model uses an Ornstein-Uhlenbeck process where the diffusion coefficient is constant. This means that the standard deviation of the short rate is independent of the current rate level; even if the short rate increases, the volatility (or standard deviation) remains unchanged.

Multiplicative Noise Models (e.g., Rendleman-Bartter Model)

In models with multiplicative noise, the volatility is directly proportional to the level of the process. Thus, when the short rate is higher, the standard deviation increases proportionally. An increase from a lower rate to a higher rate results in a commensurate scaling up of volatility.

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