📂Stochastic Differential Equations

Cox-Ingersoll-Ross Model, CIR Model

Model ¹

$$d X_{t} = \left( \alpha - \beta X_{t} \right) dt + \sigma \sqrt{X_{t}} d W_{t} \qquad , X_{0} > 0$$ Assume $\alpha, \beta, \sigma > 0$ satisfies $2 \alpha > \sigma^{2}$. The stochastic differential equation mentioned above is called the CIR model. $$X_{t} = {{ \alpha } \over { \beta }} + e^{-\beta t} \left( X_{0} - {{ \alpha } \over { \beta }} \right) + \sigma e^{-\beta t} \int_{0}^{t} e^{\beta u} \sqrt{X_{u}} d W_{u}$$

Variables

• $X_{t}$: Represents either the Interest Rate or the Gene frequency.

Parameters

• $\alpha / \beta$: Known as the Mean Reversion Level, to which $X_{t}$ tends to return over the long term.
• $\alpha > 0$: The Speed of Adjustment, where a higher value means a faster return to the mean.
• $\sigma > 0$: Represents Volatility.

Explanation

The CIR model, first introduced by Feller in 1951 as an equation describing population growth, became widely known as a model explaining short-term interest rate movements through Cox, Ingersol, and Ross in 1985. It can also be referred to simply as the Stochastic Mean-reverting Square-root Growth Equation, but it is commonly now called the CIR (Growth) model.

Its derivation is similar to the Ornstein-Uhlenbeck process, and it also has the property of reverting to the mean over the long term. In the short term, the adjustment speed $\alpha$ influences it, but over the long term, it is guaranteed to revert in the direction of $\alpha / \beta$. The diffusion coefficient $\sigma \sqrt{X_{t}}$ describes volatility proportional not to $X_{t}$ itself but to $\sqrt{X_{t}}$. This value is singular at $0$, so if the initial value is $X_{0} > 0$, $X_{t}$ will never be negative for any value of $t$.

These characteristics describe that interest rates will not become negative and that over the long term, variations will be seen around the mean level. Of course, in actual economic phenomena, negative interest rates can exist, but considering just the nominal figures, it can be seen as a reasonably rational assumption.