Cox, Ingersoll and Ross

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1 CIR model
2 Value of a zero-coupon bond
3 Transition probability
4 References

CIR model

The Cox, Ingersoll and Ross (CIR) interest-rate model takes the form

$dr=(\eta-\gamma r)dt+\sqrt{\alpha r}\;dX$ .

The spot rate is mean reverting and if $\eta > \alpha/2$ the spot rate stays positive. There are some closed-form formulae for interest rate derivatives, although typically involving integrals of the non-central chi-square distribution.

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Value of a zero-coupon bond

The value of a zero-coupon bond is

$e^{A(t;T)-rB(t;T)}$

where $A$ and $B$ are given by

$\frac{\alpha}{2} A=a\psi_2\log (a-B)+\psi_2b\log ((B+b)/b)-a\psi_2\log a$ ,

and

$B(t;T)=\frac{2(e^{\psi_1(T-t)}-1)}{(\gamma+\psi_1)(e^{\psi_1(T-t)}-1)+2\psi_1}$

where

$\psi_1=\sqrt{\gamma^2+2\alpha}\;\;\;\mbox{and}\;\;\;\psi_2=\frac{\eta}{a+b}$

and

$b,a=\frac{\pm\gamma+\sqrt{\gamma^2+2\alpha}}{\alpha}$ .

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Transition probability

The exact transition probability is given by:

$\rho(r,\delta t|r_0)=c\,e^{-(u+v)}\left({u\over v}\right)^{q/2} I_q(2\sqrt{u v})$

where:

$c={2\gamma \over \alpha (1-\exp (-\gamma \delta t))}$

$q={2\eta\over\alpha}-1\geq 0$

$u=c\,r_0\exp(-a\delta t),\;\;v=c\, r$

and $I_q$ is the modified Bessel function of order $q$ .

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References

Cox, J, Ingersoll, J & Ross, S 1985 A theory of the term structure of interest rates. Econometrica 53 385-467

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Cox, Ingersoll and Ross

From WilmottWiki

Contents

CIR model

Value of a zero-coupon bond

Transition probability

References

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