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In mathematical finance, the Cox–Ingersoll–Ross (CIR) model describes the evolution of
interest rate An interest rate is the amount of interest due per period, as a proportion of the amount lent, deposited, or borrowed (called the principal sum). The total interest on an amount lent or borrowed depends on the principal sum, the interest rate, t ...
s. It is a type of "one factor model" ( short-rate model) as it describes interest rate movements as driven by only one source of market risk. The model can be used in the valuation of interest rate derivatives. It was introduced in 1985 by John C. Cox, Jonathan E. Ingersoll and Stephen A. Ross as an extension of the
Vasicek model In finance, the Vasicek model is a mathematical model describing the evolution of interest rates. It is a type of one-factor short-rate model as it describes interest rate movements as driven by only one source of market risk. The model can be use ...
.


The model

The CIR model specifies that the instantaneous interest rate r_t follows the stochastic differential equation, also named the CIR Process: :dr_t = a(b-r_t)\, dt + \sigma\sqrt\, dW_t where W_t is a Wiener process (modelling the random market risk factor) and a , b , and \sigma\, are the
parameter A parameter (), generally, is any characteristic that can help in defining or classifying a particular system (meaning an event, project, object, situation, etc.). That is, a parameter is an element of a system that is useful, or critical, when ...
s. 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, 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) 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). This process can be defined as a sum of squared Ornstein–Uhlenbeck process. The CIR is an ergodic process, and possesses a stationary distribution. The same process 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_ = \frac, : where c=\frac, and ''Y'' is a non-central chi-squared distribution with \frac degrees of freedom and non-centrality parameter 2 c r_te^. Formally the probability density function is: :: f(r_;r_t,a,b,\sigma)=c\,e^ \left (\frac\right)^ I_(2\sqrt), : where q = \frac-1, u = c r_t e^, v = c r_, and I_(2\sqrt) 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_ will approach a
gamma distribution In probability theory and statistics, the gamma distribution is a two- parameter family of continuous probability distributions. The exponential distribution, Erlang distribution, and chi-square distribution are special cases of the gamma di ...
with the probability density of: :: f(r_\infty;a,b,\sigma)=\fracr_\infty^e^, : where \beta = 2a/\sigma^2 and \alpha = 2ab/\sigma^2 . To derive the asymptotic distribution p_ for the CIR model, we must use the Fokker-Planck equation: : + (b-r)p= \sigma^(rp) Our interest is in the particular case when \partial_p \rightarrow 0, which leads to the simplified equation: :a(b-r)p_ = \sigma^\left(p_ + r \right) Defining \alpha = 2ab/\sigma^ and \beta = 2a/\sigma^ and rearranging terms leads to the equation: : - \beta = \log p_ Integrating shows us that: :p_ \propto r^e^ Over the range p_ \in (0,\infty], this density describes a gamma distribution. Therefore, the asymptotic distribution of the CIR model is a gamma distribution.


Properties

* Mean reversion (finance), Mean reversion, *Level dependent volatility (\sigma \sqrt), *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 _t\mid r_0r_0 e^ + b (1-e^), so long term mean is b, *\operatorname _t\mid r_0r_0 \frac (e^-e^) + \frac(1-e^)^2.


Calibration

* Ordinary least squares : The continuous SDE can be discretized as follows :: r_-r_t = a (b-r_t)\,\Delta t + \sigma\, \sqrt \varepsilon_t, :which is equivalent to :: \frac =\frac-a \sqrt r_t\Delta t + \sigma\, \sqrt \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


Simulation

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


Bond pricing

Under the no-arbitrage assumption, a bond may be priced using this interest rate process. The bond price is exponential affine in the interest rate: :P(t,T) = A(t,T) \exp(-B(t,T) r_t)\! where :A(t,T) = \left(\frac\right)^ :B(t,T) = \frac :h = \sqrt


Extensions

A CIR process is a special case of a
basic affine jump diffusion In mathematics probability theory, a basic affine jump diffusion (basic AJD) is a stochastic process Z of the form : dZ_t=\kappa (\theta -Z_t)\,dt+\sigma \sqrt\,dB_t+dJ_t,\qquad t\geq 0, Z_\geq 0, where B is a standard Brownian motion, and ...
, which still permits a
closed-form expression In mathematics, a closed-form expression is a mathematical expression that uses a finite number of standard operations. It may contain constants, variables, certain well-known operations (e.g., + − × ÷), and functions (e.g., ''n''th r ...
for bond prices. Time varying functions replacing coefficients can be introduced in the model in order to make it consistent with a pre-assigned term structure of interest rates and possibly volatilities. The most general approach is in Maghsoodi (1996). A more tractable approach is in Brigo and Mercurio (2001b) where an external time-dependent shift is added to the model for consistency with an input term structure of rates. A significant extension of the CIR model to the case of stochastic mean and stochastic volatility is given by Lin Chen (1996) and is known as
Chen model In finance, the Chen model is a mathematical model describing the evolution of interest rates. It is a type of "three-factor model" (short-rate model) as it describes interest rate movements as driven by three sources of market risk. It was the fir ...
. A more recent extension for handling cluster volatility, negative interest rates and different distributions is the so-called CIR # by Orlando, Mininni and Bufalo (2018, 2019, 2020, 2021) and a simpler extension limited to non-negative interest rates was proposed by Di Francesco and Kamm (2021, unpublished).


See also

* Hull–White model *
Vasicek model In finance, the Vasicek model is a mathematical model describing the evolution of interest rates. It is a type of one-factor short-rate model as it describes interest rate movements as driven by only one source of market risk. The model can be use ...
*
Chen model In finance, the Chen model is a mathematical model describing the evolution of interest rates. It is a type of "three-factor model" (short-rate model) as it describes interest rate movements as driven by three sources of market risk. It was the fir ...


References


Further References

* * * * *
Open Source library implementing the CIR process in python
* {{DEFAULTSORT:Cox-Ingersoll-Ross Model Interest rates Fixed income analysis Stochastic models Short-rate models Financial models de:Wurzel-Diffusionsprozess#Cox-Ingersoll-Ross-Modell