This paper presents a general framework of how to generate covariances between riskless interest rate r(t) instruments, and financial instruments with intensity of default h (t),in Cox, Ingersoll, Ross (CIR), or in the extended multifactor CIR model.

Se presenta el marco general para generar covarianzas entre instrumentos con tasas de interés libre de riesgo r(t) e instrumentos con intensidad de incumplimiento λ(t), en el modelo Cox, Ingersoll, Ross [CIR] o en el modelo extendido CIR multifactorial.

The problem of generating covariances between intensity of default and riskless interest rates that are consistent with observed data has been treated in several theoretical and practical settings, [

Sections 1 and 2 present a general method for generating covariances for predefined functions, particularly polynomial ones. Section 2 explains howto deal with plausibly observed negative covariances. Sections 3 and 4 analyze the problem for some classes of functions. It is made clear what can or cannot be done in a CIR setting.

Several relevant quotations shall be considered:

r(s) and λ(s) have negative (observed) correlation -20% [

CIR (CSR) correlated square root models are theoretically incapable of generating negative correlations [

The dynamics of r and A are rich enough to allow for a realistic description of the real-world prices [

The usual construction of the multifactor model for interest rate and intensity of default is as follows:

It is not possible to obtain general covariance structure (even positive) if Xt CIR. Using comparisons theorems for diffusions, one can prove easily that

The second constraint is the following.

t5 cannot be uniformly approximated in [0,l] by Cov(r(t), λ(t)). Using well known formulas such as:

An elementary but somewhat tricky proof follows:

Because:

resulting from:

and,

And

Therefore

then, for f(t) any linear combination

and for g(f) = t5, we have

The third constraint is clear: Negative covariances cannot be generated from the usual construction , [

The easiest way to obtain negative correlations seems to be,

Xi ~ driven by Wi

Yi ~ driven by -Wi

But there is no possibility to get explicit results, the only possible approach is process simulation.

"Explicit” formulas are desirable for:

To be able to reproduce a given arbitrary covariance structure, an extended CIR Model (ECIR) with time dependent parameters has to be used.

First a short, user friendly construction of extended CIR with references quoted in the introduction is presented.

Start from BESQ^{δ}

Add the drift 2β_{t} r(t)) [Girsanov].

Multiply the process by σ_{t}

Now,

And

And

(Sturm-Liouville equation),

or equivalently in terms of Riccati equation

Therefore, explicit formulas for bonds prices depend on the solutions of these equations.

Note that the free term of polynomials should be zero.

For grade 3 polynomials set F(t)=0, fix time t, and

For a moment only one factor will be considered.

Choose F(s) = D(s -t), for s<t, D being constant.

and assume that σ(s) > 0.

If X(s)~BESQ^{δ}

And

Elementary calculations show that one can generate any positive covariances as polynomial of grade 3 from BESQ^{0} using two factors.

For grade 4 polynomials the same procedure applies but starting with BESQ^{δ} instead of BESQ^{0}, δ >0.

For grade 5 polynomials F(s) = D(s - t)^{2} , and so on.

This method leads to a general construction for any positive covariance structure, and

Take factors as BESQ^{1}, more explicitly:

Both being BESQ^{1} process driven respectively by:

Now Cov(λ(t),r(t))=2t(t-AB) is negative for t < AB.

(W(t) + A)^{2} + (W(t) - B)^{2} = constant + 2BESQ^{1} starting at

Grade 3 polynomials with some restrictions can be generated as a combination of the results of this section and the previous one.

Grade 6 polynomials can be obtained for example multiplying by σ(s), σ(s) as before (but without ε)

obtained from F(s) = D(s -t).

Set

For B(0,t)

Set

The density fv_{t} is well known

Set

And explicit results can be

for some factor

for some choice of β(s)

However in this case B(0,t) not explicit.

Calculation of covariances:

The problem is easier if based initially on BESQ^{0}.

Y(0)=1 for example

Other general modeling possibility uses Laplace transform for the process Y(t) for general β(s). In this case,

But now F(t)= β(t)+λ for some λ(*) and general solution of this equation can be obtained solving:

One can find A to satisfy * and get explicit formula for

As has been shown, there are many methods to generate given (observed) covariance structure between instantaneous riskless interest rates, and intensity of default. However, to obtain user friendly results in the case of negative correlations, one should not expect substantial extensions of presented use of "degenerated” CIR's- squares of Brownian motions.

As a final comment, the CIR model is very attractive and interesting, being the "Girsanov version "of square of Brownian Motion, but it has generated in the past many erroneous formulas. See for example the excellent textbook by [Jeanblanc et al, 2009] p. 127, where an erroneous application of Ito's formula appears, this mistake is explained extensively in [