Cox–Ingersoll–Ross model

• Future distribution

The distribution of future values of a CIR process can be computed in closed form:

${\displaystyle r_{t+T}={\frac {Y}{2c}},}$

where ${\displaystyle c={\frac {2a}{(1-e^{-aT})\sigma ^{2}}}}$, and Y is a non-central chi-squared distribution with ${\displaystyle {\frac {4ab}{\sigma ^{2}}}}$ degrees of freedom and non-centrality parameter ${\displaystyle 2cr_{t}e^{-aT}}$. Formally the probability density function is:

${\displaystyle f(r_{t+T};r_{t},a,b,\sigma )=c\,e^{-u-v}\left({\frac {v}{u}}\right)^{q/2}I_{q}(2{\sqrt {uv}}),}$

where ${\displaystyle q={\frac {2ab}{\sigma ^{2}}}-1}$, ${\displaystyle u=cr_{t}e^{-aT}}$, ${\displaystyle v=cr_{t+T}}$, and ${\displaystyle I_{q}(2{\sqrt {uv}})}$ is a modified Bessel function of the first kind of order ${\displaystyle q}$.

• Asymptotic distribution

Due to mean reversion, as time becomes large, the distribution of ${\displaystyle r_{\infty }}$ will approach a gamma distribution with the probability density of:

${\displaystyle f(r_{\infty };a,b,\sigma )={\frac {\beta ^{\alpha }}{\Gamma (\alpha )}}r_{\infty }^{\alpha -1}e^{-\beta r_{\infty }},}$

where ${\displaystyle \beta =2a/\sigma ^{2}}$ and ${\displaystyle \alpha =2ab/\sigma ^{2}}$.

• Mean reversion,
• Level dependent volatility (${\displaystyle \sigma {\sqrt {r_{t}}}}$),
• For given positive ${\displaystyle r_{0}}$ the process will never touch zero, if ${\displaystyle 2ab\geq \sigma ^{2}}$; otherwise it can occasionally touch the zero point,
• ${\displaystyle \operatorname {E} [r_{t}\mid r_{0}]=r_{0}e^{-at}+b(1-e^{-at})}$, so long term mean is ${\displaystyle b}$,
• ${\displaystyle \operatorname {Var} [r_{t}\mid r_{0}]=r_{0}{\frac {\sigma ^{2}}{a}}(e^{-at}-e^{-2at})+{\frac {b\sigma ^{2}}{2a}}(1-e^{-at})^{2}.}$

• Ordinary least squares

The continuous SDE can be discretized as follows

${\displaystyle r_{t+\Delta t}-r_{t}=a(b-r_{t})\,\Delta t+\sigma \,{\sqrt {r_{t}\Delta t}}\varepsilon _{t},}$

which is equivalent to

${\displaystyle {\frac {r_{t+\Delta t}-r_{t}}{{\sqrt {r}}_{t}}}={\frac {ab\Delta t}{{\sqrt {r}}_{t}}}-a{\sqrt {r}}_{t}\Delta t+\sigma \,{\sqrt {\Delta t}}\varepsilon _{t},}$

provided ${\displaystyle \varepsilon _{t}}$ is n.i.i.d. (0,1). This equation can be used for a linear regression.

• Martingale estimation
• Maximum likelihood

Stochastic simulation of the CIR process can be achieved using two variants:

• Discretization
• Exact