Wednesday, February 25, 2015

stochastic processes - CIR model: is the short rate really non-central chi2 distributed?


Probably simple question. Consider the CIR (1985) model for interest rates dr=k(θr)dt+σrdz

Then it is known in closed form the conditional pdf f(r(s),s|r(t),t) (st) f(r(s),s|r(t),t)=ceuv(vu)q/2Iq(2uv)
where c=2kσ2(1ek(st))u=cr(t)ek(st)v=cr(s)q=2kθσ21
and Iq() is a modified Bessel function of the first kind of order q.


Then authors state:



<< The distribution function is the non central chi-square χ2[2cr(s);2q+2,2u], with 2q+2 degrees of freedom and parameter of non centrality 2u proportional to the current spot rate. >>



Then my questions:



1) Is it correct to say that what is (conditionally on r(t)) non-central χ2 distributed is the variable 2cr(s)?


I can answer by my own to this question: Since the conditional expectation E(r(s)|r(t)) and variance Var(r(s)|r(t)) are provided in the paper (Eq. 19), it'easy to check the validity of 1) verifying that: (2q+2)+(2u)=E(2cr(s)|r(t))=2cE(r(s)|r(t))2[(2q+2)+2(2u)]=Var(2cr(s)|r(t))=4c2Var(r(s)|r(t))

where l.h.s. of both equations are expressions for the first two moments of a non-central χ2 variable with 2q+2 and parameter of non-centrality 2u (you may want to check Wikipedia).


2) If 1), which is the conditional distribution of r(s) alone? Is it still non-central χ2?


I want to be crystal clear: we know that 2cr(s)|r(t)χ2(2q+2,2u). Moreover, we know in closed form the (conditional on r(t)) pdf of r(s) (the f(r(s),s|r(t),t) above)... but then, is r(s) a KNOWN random variable (|r(t))? In particular, is it still non-cenral χ2 distributed? (*)


Thanks for your attention


(*) I'm afraid r(s) cannot still be non-central χ2 since this would imply that the non-central χ2 would be close w.r.t. scaling of the variable, and - I'm not sure - this should not be the case.



Answer



To answer this I sum up a paragraph of "Interest rate models - An Introduction" by A.Cairns: For i=1,,d consider the OU-processes dXit=12αXitdt+αdWit.

Looking at the squared radius Rt=di=1(Xit)2 (in Rd) of this process we get by Ito: dRt=di=1(2XitdXit)+dαdt.
Using the definition of Rt introducing a new Brownian motion Bt we get in distribution that that dRt=α(dRt)dt+4αRtdBt.
Defining rt=Rt/θ with θ=4α/σ2 this is the CIR model. This gives a nice geometric interpretation. I am aware that not all details are covered here.


Recall the definition of the non central chi-squared distribution. Let R=di=1(Wi+δi)2

and λ=di=1δ2i, then R has a non-central chi-squared distribution with d degrees of freedom and non-centrality parameter λ.


Since the Xti above are all normally distributed with variance 1eαt we see that Rt/(1eαt) has non-central chi-squared distribution. Finally we have that for d=4αμ/σ2 that 4αrt/(σ2(1eαt)) has a non-central chi-squared distribution with d degrees of freedom and non-centrality parameter λ=4αr0/(σ2(1eαt)).



Conditionally on rt replace r0 by rt.


The answers then are: i) Yes, the variable that has non-central chi-squared distribution is the complicated expression that you mention.


ii) Only this complicated expression is non-central chi-squared distributed - rs itself is not. As you see in the link the non-central chi-squared distribution relates to standardized Gaussians (variance equals 1). Maybe the Generalized chi-squared distribution could be of help. But I don't know this.


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