Cox–Ingersoll–Ross model
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| Cox–Ingersoll–Ross model | |
|---|---|
 Thomas Steiner · CC BY-SA 3.0 · source | |
| Name | Cox–Ingersoll–Ross model |
| Type | Stochastic differential equation |
| First described | 1985 |
| Authors | John C. Cox; Jonathan E. Ingersoll, Jr.; Stephen A. Ross |
| Field | Mathematical finance |
| Applications | Interest rate modeling; option pricing; credit risk |
Cox–Ingersoll–Ross model The Cox–Ingersoll–Ross model is a stochastic process introduced by John C. Cox, Jonathan E. Ingersoll, Jr., and Stephen A. Ross as a one-factor model for the short-term interest rate. It specifies mean-reverting dynamics with state-dependent volatility and has been widely used in mathematical finance, actuarial science, and econometrics. The model links to classical work in quantitative analysis and has influenced models used by central banks, investment banks, pension funds, and academic researchers.
Introduction
The model was proposed by John C. Cox, Jonathan E. Ingersoll, Jr., and Stephen A. Ross building on prior work by Paul Samuelson, Robert C. Merton, and Fischer Black and Myron Scholes; it appeared amid developments in the same era as the Black–Scholes model and the Heath–Jarrow–Morton framework. It provided an alternative to the Vasicek model with nonnegative state space and square-root diffusion like processes studied by Feller and in connection with the Bessel process and squared Bessel process. The model's tractability made it attractive for practitioners at institutions such as Goldman Sachs, Morgan Stanley, and J.P. Morgan and for researchers at universities including Princeton University, Massachusetts Institute of Technology, and Harvard University.
Model Definition and Properties
The model is defined by the stochastic differential equation dr_t = κ(θ − r_t) dt + σ√{r_t} dW_t, where parameters κ, θ, σ are positive constants and W_t is a Wiener process introduced in the literature on stochastic calculus by Andrey Kolmogorov and Norbert Wiener. Important properties derive from boundary behavior studied by William Feller and link to the theory of diffusion processes developed by Itô and Kiyoshi Itô. Positivity of r_t is ensured when the Feller condition 2κθ ≥ σ^2 holds, a criterion related to the work of Feller and applied in settings examined by Eugene F. Fama and Kenneth R. French. The generator of the process connects to Kolmogorov forward and backward equations used by Andrei Kolmogorov and Serguei N. Bernstein in probability theory, and sample-path properties have been studied using techniques from Itô calculus and martingale theory from Joseph L. Doob.
Solution and Transition Density
The transition density for the model has a closed-form expression in terms of the noncentral chi-square distribution and uses mathematical results associated with C. R. Rao and Harold Jeffreys on distributions; derivations rely on techniques developed by Itô and Eugene Dynkin and spectral methods related to work by Mark Kac and Israel Gelfand. The conditional characteristic function and Laplace transform can be expressed using confluent hypergeometric functions studied by Ernst Schröder and classical treatments in the tradition of Erdélyi and Harold Jeffreys; explicit formulae are used in computational implementations at firms like Barclays and in research at London School of Economics and Columbia University. Time-homogeneous transition kernels enable pricing via expectation semigroups in the tradition of Andrey Kolmogorov and Paul Lévy.
Parameter Estimation and Calibration
Parameter estimation combines maximum likelihood, generalized method of moments, and Bayesian approaches influenced by methods of Ronald Fisher, Karl Pearson, and Sir David Cox. Exact likelihood exploits the noncentral chi-square transition density; approximate techniques rely on discretizations introduced by Peter E. Kloeden and Eckhard Platen and on quasi-maximum-likelihood methods used in empirical finance by Clive Granger and Robert Engle. Calibration to market data such as yield curves and cap/floor prices is performed by practitioners at Deutsche Bank, UBS, and State Street, and academic comparisons reference work from National Bureau of Economic Research and World Bank datasets. Regularization and filtering methods draw on ideas from Gregory P. King and state-space methods influenced by Rudolf E. Kálmán.
Applications in Finance
The model is applied to short-rate modeling for bond pricing, generating yield curves consistent with empirical shapes observed in studies by John Hull and Alan J. Greenspan; it underpins pricing of interest-rate derivatives such as caps, floors, swaptions, and mortgage-backed securities handled by traders at Citigroup and Wells Fargo. It has been incorporated into multi-factor frameworks and risk management systems at central banks including the Federal Reserve System and the European Central Bank and informs stress-testing practices by International Monetary Fund and European Banking Authority. The model also serves in credit risk models for default intensity inspired by work at Moody's and Standard & Poor's and in asset-liability management used by CalPERS and large pension schemes.
Extensions and Related Models
Extensions include multi-factor versions linked to the Hull–White model and the Black–Derman–Toy model, time-dependent parameters as in work from Darrell Duffie, and jump-diffusion hybrids influenced by Robert Merton and Stephen Ross. Related processes include the Vasicek model, Affine term structure models championed by Duffie and Darrell D. Duffie, and the class of continuous-state branching processes connected to the work of Jean-François Le Gall and John Lamperti. Numerical and simulation improvements draw on advances by Peter Glasserman and computational finance research at Carnegie Mellon University and Stanford University. Contemporary research explores connections to stochastic volatility models developed by Tomaso A. Di Matteo and further calibrations using machine learning approaches from Geoffrey Hinton and Yann LeCun.