0.1 Model Description
The Cox-Ingersoll-Ross (CIR) Model describes the evolution of interest rates as driven by market risk and was proposed as an extension of the Vasiček Model. The key difference between the models is that Vasiček is based on the [[ornstein-uhlenbeck-process]] and CIR is based on the CIR process, also known as the Feller square root process.
The CIR model describes the instantaneous interest rate \(\{r_{t}\}\) with the Feller square root process with dynamics described by the stochastic differential equation \[ dr_{t}=a(b-r_{t})dt+\sigma \sqrt{ r_{t} }dW_{t}, \] where \(W_{t}\) is a Wiener process.
The parameter \(a\) corresponds to the speed of adjustment to the mean \(b\), and \(\sigma\) the volatility. The drift factor \(a(b-r_{t})\) is the same as in the Vasiček model, ensuring [[mean-reversion]] of the interest rate towards the long run value \(b\), with speed of adjustment governed by the strictly positive \(a\).
The standard deviation factor \(\sigma \sqrt{ r_{t} }\) avoids the possibility of negative interest rates for all positive values of \(a\) and \(b\). Generally, when the rate \(r_{t}\) is close to zero, the standard deviation \(\sigma \sqrt{ r_{t} }\) becomes very small, dampening the effect of the random shock on the rate. Consequently, when the rate gets close to zero, its evolution becomes dominated by the drift factor which pushes the rate upwards towards equilibrium.
The process is used in the Heston Model used to model stochastic volatility.
0.2 Distribution
In this model, the probability density of interest rate at time \(t\) conditional on its value at the current time \(s\) is given by \[ f(r_{t}|r_{s}, s)=ce^{-u-v}\left( \frac{v}{u} \right)^{q/2}I_{q}(2\sqrt{ uv }), \] where \[ \begin{cases} c= \frac{2a}{\sigma^2(1-e^{-a(t-s)})} \\ u=cr_{s}e^{-a(t-s)} \\ v=cr_{t} \\ q= \frac{2ab}{\sigma^2}-1, \end{cases} \] and \(I_{q}(\cdot)\) is the modified Basel function of the first kind of order \(q\) defined as \[ I_{q}(z) = \left( \frac{z}{2} \right)^q \sum_{k=0}^\infty \frac{1}{ \Gamma(k+1)\Gamma(z+k+1)}\left( \frac{z}{2} \right)^2k. \]