The CIR model specifies that the instantaneous interest rate follows the stochastic differential equation, also named the CIR Process: where is a Wiener process and,, and are the parameters. 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. 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
Asymptotic distribution
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 asympotic distribution of the CIR model is a gamma distribution.
Properties
Mean reversion,
Level dependent volatility,
For given positive the process will never touch zero, if ; otherwise it can occasionally touch the zero point,
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
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. A more tractable approach is in Brigo and Mercurio 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 and is known as Chen model. A CIR process is a special case of a basic affine jump diffusion, which still permits a closed-form expression for bond prices.
