The CIR model describes the instantaneous interest rate r_t with a
Feller square-root process, whose
stochastic differential equation is :dr_t = a(b-r_t)\, dt + \sigma\sqrt{r_t}\, dW_t, where W_t is a
Wiener process (modelling the random market risk factor) and a , b , and \sigma\, are the
parameters. The parameter a corresponds to the speed of adjustment to the mean b , and \sigma\, to volatility. The drift factor, a(b-r_t), is exactly the same as in the Vasicek model. It ensures
mean reversion of the interest rate towards the long run value b, with speed of adjustment governed by the strictly positive parameter a. The
standard deviation factor, \sigma \sqrt{r_t}, avoids the possibility of negative interest rates for all positive values of a and b. An interest rate of zero is also precluded if the condition :2 a b \geq \sigma^2 \, is met. More generally, when the rate (r_t) is close to zero, the standard deviation (\sigma \sqrt{r_t}) also becomes very small, which dampens 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). In the case 4 a b =\sigma^2 \,, the Feller square-root process can be obtained from the square of an
Ornstein–Uhlenbeck process. It is
ergodic and possesses a stationary distribution. It is used in the
Heston model to model stochastic volatility.
Distribution • Future distribution : The distribution of future values of a CIR process can be computed in closed form: :: r_{t+T} = \frac{Y}{2c}, : where c=\frac{2a}{(1 - e^{-aT})\sigma^2}, and
Y is a
non-central chi-squared distribution with \frac{4ab}{\sigma^2} degrees of freedom and non-centrality parameter 2 c r_te^{-aT}. Formally the probability density function is: :: f(r_{t+T};r_t,a,b,\sigma)=c\,e^{-u-v} \left (\frac{v}{u}\right)^{q/2} I_{q}(2\sqrt{uv}), : where q = \frac{2ab}{\sigma^2}-1, u = c r_t e^{-aT}, v = c r_{t+T}, and I_{q}(2\sqrt{uv}) is a modified Bessel function of the first kind of order q. • Asymptotic distribution : Due to mean reversion, as time becomes large, the distribution of r_{\infty} will approach a
gamma distribution with the probability density of: :: f(r_\infty;a,b,\sigma)=\frac{\beta^\alpha}{\Gamma(\alpha)}r_\infty^{\alpha-1}e^{-\beta r_\infty}, : where \beta = 2a/\sigma^2 and \alpha = 2ab/\sigma^2 . To derive the asymptotic distribution p_{\infty} for the CIR model, we must use the
Fokker–Planck equation: :{\partial p\over{\partial t}} + {\partial\over{\partial r}}[a(b-r)p] = {1\over{2}}\sigma^{2}{\partial^{2}\over{\partial r^{2}}}(rp) Our interest is in the particular case when \partial_{t}p \rightarrow 0, which leads to the simplified equation: :a(b-r)p_{\infty} = {1\over{2}}\sigma^{2}\left(p_{\infty} + r{dp_{\infty}\over{dr}} \right) Defining \alpha = 2ab/\sigma^{2} and \beta = 2a/\sigma^{2} and rearranging terms leads to the equation: :{\alpha-1\over{r}} - \beta = {d\over{dr}} \log p_{\infty} Integrating shows us that: :p_{\infty} \propto r^{\alpha-1}e^{-\beta r} Over the range p_{\infty} \in (0,\infty], this density describes a gamma distribution. Therefore, the asymptotic distribution of the CIR model is a gamma distribution.
Properties •
Mean reversion, • Level dependent volatility (\sigma \sqrt{r_t}), • For given positive r_0 the process will never touch zero, if 2 a b \geq\sigma^2; otherwise it can occasionally touch the zero point, • \operatorname E[r_t\mid r_0]=r_0 e^{-at} + b (1-e^{-at}), so long term mean is b, • \operatorname{Var}[r_t\mid r_0]=r_0 \frac{\sigma^2}{a} (e^{- a t}-e^{-2a t}) + \frac{b \sigma^2}{2 a}(1-e^{- a t})^2.
Calibration •
Ordinary least squares : The continuous SDE can be discretized as follows :: r_{t+\Delta t}-r_t = a (b-r_t)\,\Delta t + \sigma\, \sqrt{r_t \Delta t} \varepsilon_t, :which is equivalent to :: \frac{r_{t+\Delta t}-r_t}{\sqrt r_t} =\frac{ab\Delta t}{\sqrt r_t}-a \sqrt r_t\Delta t + \sigma\, \sqrt{\Delta t} \varepsilon_t, :provided \varepsilon_t is n.i.i.d. (0,1). This equation can be used for a linear regression. • Martingale estimation •
Maximum likelihood estimation Simulation Stochastic simulation of the CIR process can be achieved using two variants: •
Discretization • Exact ==Bond pricing==