Cox–Ingersoll–Ross model

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Three trajectories of CIR Processes

In mathematical finance, the Cox–Ingersoll–Ross model (or CIR model) describes the evolution of interest rates. 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.

The model[edit]

CIR process

The CIR model specifies that the instantaneous interest rate follows the stochastic differential equation, also named the CIR Process:

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,  b to the mean 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 is at a low level (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[edit]

  • Future distribution

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

r_{t+T} = cY,

where c=\frac{(1 - e^{-aT})\sigma^2}{4a}, and Y is a non-central Chi-Squared distribution with \frac{4ab}{\sigma^2} degrees of freedom and non-centrality parameter \frac{r_te^{-aT}}{c}.

  • Conditional distribution

Given r_0 and defining c_t=\frac{4 a}{\sigma^2(1-e^{-a t})}, v=\frac{4 a b}{\sigma^2} and \lambda_t=c_t r_0 e^{-a t}, it can be shown that  c_t r_t follows a noncentral chi-squared distribution with degree of freedom v and non-centrality parameter \lambda_t. Note that v is a constant independent of r_0.

Properties[edit]

  • 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,
  • E[r_t|r_0]=r_0 e^{- a t} + b (1-e^{- a t}), so long term mean is b,
  • Var[r_t|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[edit]

The continuous SDE can be discretized as follows

 r_{t+\Delta t}-r_t = a (b-r_t)\,\Delta t + \sigma\, \sqrt r_t \epsilon_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\, \epsilon_t . This equation can be used for a linear regression.

Simulation[edit]

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

Bond pricing[edit]

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{2h \exp((a+h)(T-t)/2)}{2h + (a+h)(\exp((T-t)h) -1)}\right)^{2ab/\sigma^2}
B(t,T) = \frac{2(\exp((T-t)h)-1)}{2h+(a+h)(\exp((T-t)h)-1)}
h = \sqrt{a^2+2\sigma^2}

Extensions[edit]

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. A CIR process is a special case of a basic affine jump diffusion, which still permits a closed-form expression for bond prices.

See also[edit]

References[edit]