Explain the treatment of the dynamics of the volatility term for the Cox-Ingersoll-Ross interest rate model

What will be an ideal response?

For the Cox-Ingersoll-Ross specification interest rate model, we look at the case for γ = 1/2 . Substituting this value for γ into σrγdz, we get the following model identified by Cox-Ingersoll-Ross who first proposed it:
γ = 1/2: σ(r,t) = σ (Cox-Ingersoll-Ross specification).

The Cox-Ingersoll-Ross (CIR) specification, referred to as the square-root model, makes the volatility proportional to the square rate of the short rate. Negative interest rates are not possible in this square-root model.

One can combine the dynamics of the drift term and volatility term to create the following commonly used interest rate model:
dr = .

This model specifies a mean reversion process for the drift term and the square root model for volatility and is referred to as the mean-reverting square-root model.

Although there have been many developments in equilibrium models, the best known models are the Vasicek and CIR models. To apply these models, estimates of the parameters of the assumed interest-rate process are needed, including the parameters of the volatility function for interest rates. These estimated parameters are typically obtained using econometric techniques that rely on historical yield curves without regard to how the final model matches any market prices.