Probably simple question. Consider the CIR (1985) model for interest rates dr=k(θ−r)dt+σ√rdz
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))
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.
Recall the definition of the non central chi-squared distribution. Let R=d∑i=1(Wi+δi)2
Since the Xti above are all normally distributed with variance 1−e−αt we see that Rt/(1−e−αt) has non-central chi-squared distribution. Finally we have that for d=4αμ/σ2 that 4αrt/(σ2(1−e−αt)) has a non-central chi-squared distribution with d degrees of freedom and non-centrality parameter λ=4αr0/(σ2(1−e−αt)).
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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