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 BESQδ
Add the drift 2βt r(t)) [Girsanov]. Multiply the process by σt
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)~BESQδ
And
Elementary calculations show that one can generate any positive covariances as polynomial of grade 3 from BESQ0 using two factors.
For grade 4 polynomials the same procedure applies but starting with BESQδ instead of BESQ0, 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 BESQ1, more explicitly:
Both being BESQ1 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 + 2BESQ1 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 fvt 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 BESQ0.
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 [