Cox-Ingersoll-Ross model
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The Cox-Ingersoll-Ross model in finance is a mathematical model describing the evolution of interest rates. It is a type of "one factor model" (Short rate model) as 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 specifies that the instantaneous interest rate follows the stochastic differential equation, also named the CIR process:

where W_t is a Wiener process modelling the random market risk factor.

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, , corrects the main drawback of Vasicek's model, ensuring that the interest rate cannot become negative. Thus, at low values of the interest rate, the standard deviation becomes close to zero, cancelling the effect of the random shock on the interest rate. Consequently, when the interest rate gets close to zero, its evolution becomes dominated by the drift factor, which pushes the rate upwards (towards equilibrium).

Bond Pricing

An arbitrage-free bond may be priced using this interest rate process. The bond price is exponential affine in the interest rate:

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 (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.

See also

• Hull-White model
• Vasicek model

References

• Hull, John C. (2003). Options, Futures and Other Derivatives. Upper Saddle River, NJ: Prentice Hall. ISBN 0-13-009056-5. 
• Cox, J.C., J.E. Ingersoll and S.A. Ross (1985). "A Theory of the Term Structure of Interest Rates". Econometrica 53: 385–407. doi:10.2307/1911242. 
• Maghsoodi, Y. (1996). "Solution of the extended CIR Term Structure and Bond Option Valuation". Mathematical Finance (6): 89-109. 
• Damiano Brigo, Fabio Mercurio (2001). Interest Rate Models - Theory and Practice with Smile, Inflation and Credit, 2nd ed. 2006, Springer Verlag. ISBN 978-3-540-22149-4. 
• Brigo, Damiano and Fabio Mercurio (2001b). "A deterministic-shift extension of analytically tractable and time-homogeneous short rate models". Finance & Stochastics 5 (3): 369-388.