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
follows the
stochastic differential equation, also named the CIR Process:
:
where
is a
Wiener process (modelling the random market risk factor) and
,
, and
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
corresponds to the speed of adjustment to the mean
, and
to volatility. The drift factor,
, is exactly the same as in the Vasicek model. It ensures
mean reversion of the interest rate towards the long run value
, with speed of adjustment governed by the strictly positive parameter
.
The
standard deviation factor,
, avoids the possibility of negative interest rates for all positive values of
and
.
An interest rate of zero is also precluded if the condition
:
is met. More generally, when the rate (
) is close to zero, the standard deviation (
) 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:
::
: where
, and ''Y'' is a
non-central chi-squared distribution with
degrees of freedom and non-centrality parameter
. Formally the probability density function is:
::
: where
,
,
, and
is a modified Bessel function of the first kind of order
.
*Asymptotic distribution
: Due to mean reversion, as time becomes large, the distribution of
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:
::
: where
and
.
To derive the asymptotic distribution
for the CIR model, we must use the
Fokker-Planck equation:
:
Our interest is in the particular case when
, which leads to the simplified equation:
:
Defining
and
and rearranging terms leads to the equation:
:
Integrating shows us that:
:
Over the range
, 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 (
),
*For given positive
the process will never touch zero, if
; otherwise it can occasionally touch the zero point,
*
, so long term mean is
,
*
Calibration
*
Ordinary least squares
: The continuous SDE can be discretized as follows
::
:which is equivalent to
::
:provided
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:
:
where
:
:
:
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