Probability Density of the CIR Model

8 Pages Posted: 13 Oct 2014

See all articles by Andras Vanyolos

Andras Vanyolos


Maxx Cho

University of Maryland

Scott Alan Glasgow

Brigham Young University

Date Written: October 11, 2014


We present an alternative derivation of the transition density in the Cox-Ingersoll-Ross (CIR) model. Applying methods developed in elementary quantum mechanics we show that the transition density can be determined from the eigenvalue problem of a second order differential operator with continuous spectrum. The operator turns out to be the stochastic analogue of the time-dependent quantum mechanical position operator in one dimension. Finally, to demonstrate the applicability of this technique to other one-dimensional diffusion problems, we re-derive the familiar transition densities of the Vasicek and Black-Scholes models and write down the equations of motion for the relevant operators in the Courtadon interest rate model.

Keywords: Transition density, CIR model, Kolmogorov equation, Quantum Mechanics

JEL Classification: C63, C69

Suggested Citation

Vanyolos, Andras and Cho, Maxx and Glasgow, Scott Alan, Probability Density of the CIR Model (October 11, 2014). Available at SSRN: or

Andras Vanyolos (Contact Author)

Independent ( email )

No Address Available

Maxx Cho

University of Maryland ( email )

College Park
College Park, MD 20742
United States

Scott Alan Glasgow

Brigham Young University ( email )

Provo, UT 84602
United States

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