Cox–Ingersoll–Ross model ( $d r_{t} = σ r_{t} d B_{t} + (a - b r_{t}) d t$ )

A nonnegative mean-reverting short-rate diffusion with positive parameters. Its mean-reversion level is

a / b

and its diffusion size is proportional to

r

. The Feller positivity condition for the CIR model controls whether zero is reachable. Its stationary law is a gamma distribution and its zero-coupon bond prices are exponential-affine.

Table of contents
- CIR bond pricing Cox–Ingersoll–Ross model
- Feller positivity condition for the CIR model Cox–Ingersoll–Ross model

CIR bond pricing ( $P (t, T) = A (T - t) e^{- B (T - t) r_{t}}$ )

 0  0

For the CIR model under its money-market risk-neutral measure, the Feynman-Kac formula gives

B^{'} = 1 - b B - σ^{2} B^{2} /2

and

A^{'} / A = - a B

, with

B (0) = 0

A (0) = 1

. Solving this Riccati equation gives explicit exponential-affine zero-coupon bond prices. The positive root parameter is

b^{2} + 2 σ^{2}

; the squared diffusion coefficient, not the volatility itself, enters this expression.

Feller positivity condition for the CIR model ( $2 a \geq σ^{2}$ )

 0  0

For a CIR model started at a positive rate, this condition makes the zero boundary inaccessible. Below the threshold, zero can be reached, but the usual nonnegative solution does not cross into negative rates. Thus nonnegativity and strict positivity are different properties.

 Ancestors (8)

 Synonyms (1)

codex/cir-model

 View article source

 Discussion (0)

New discussion

There are no discussions about this article yet.

 Articles by others on the same topic (1)

Cox–Ingersoll–Ross model by

Wikipedia Bot  1

 View more

The Cox-Ingersoll-Ross (CIR) model is a mathematical model used to describe the dynamics of interest rates. It is part of the class of affine term structure models and is particularly known for its ability to capture the behavior of interest rates in a way that ensures non-negative rates. The CIR model was introduced by economists David Cox, Jonathan Ingersoll, and Stephen Ross in the early 1980s.

 Read the full article

  See all articles in the same topic Create my own version

Cox–Ingersoll–Ross model (drt​=σrt​​dBt​+(a−brt​)dt)

CIR bond pricing (P(t,T)=A(T−t)e−B(T−t)rt​)

Feller positivity condition for the CIR model (2a≥σ2)