In the context of the CIR model, yield volatility is constant, but the basis-point volatility equals σr and rises with the level of the rate. This is because the CIR model assumes that the volatility of interest rate changes is proportional to the level of the interest rate. This assumption leads to the conclusion that as the interest rate increases, the basis-point volatility (which is the standard deviation of the change in interest rates) also increases. This is consistent with the empirical observation that higher interest rates tend to be associated with larger changes in interest rates. On the other hand, yield volatility, which is often referred to as σ in the CIR model, is assumed to be constant. This is because it is a parameter of the model that is determined by the market and does not change with the level of the interest rate.

Choice A is incorrect. The basis-point volatility is not constant in the CIR model. Instead, it equals σr and rises with the level of the rate. This is a key feature of the CIR model that distinguishes it from other models.

Choice C is incorrect. While it's true that both yield volatility and basis-point volatility rise with the level of interest rates in some models, this isn't accurate for either Courtdon or Lognormal models derived from CIR model. In these models, only basis-point volatility equals σr and rises with the level of rate while yield volatility remains constant.

Choice D is incorrect. Neither yield nor basis-point volatilities decrease with an increase in interest rates under any circumstances in these models derived from CIR model. As explained above, only basis-point volatility increases proportionally to interest rates while yield volatility remains constant.

Things to Remember

• The Cox-Ingersoll-Ross (CIR) model is a popular interest rate model that assumes interest rate changes are proportional to the level of interest rates.

• In the CIR model, the basis-point volatility (standard deviation of interest rate changes) is not constant but equals σr and rises with the level of the rate.

• Yield volatility, denoted by σ, is assumed to be constant in the CIR model.

• The Courtadon model and the Lognormal model are two models derived from the CIR model, each with its own characteristics regarding yield and basis-point volatility.

• Understanding the relationship between yield volatility and basis-point volatility is crucial in interest rate modeling and risk management.