CIR model and calibration

Question

I am new to quantitative finance.

We know that in the CIR model the short rate can't go negative. My question then concerns calibration of CIR to a ZCB yield curve. Is it (and why?) possible to calibrate the CIR model to a yield curve with negative yields in the short end? Why (or why not) is this possible, when the short rate can't go negative?

There is probably something conceptual I have missed about the relationship short rates and yields.

Answer 1

As you say, in the CIR model with usual assumptions the short rate cannot go negative. This means that yields in the model are always poaitive, so it will not be a good fit to a yield curve which is negative for short maturities.

If you really do want the CIR model, there is a weird extension you could try:

$$ dr_t = \kappa (\theta - r_t) dt + \sigma \sqrt{|r_t|} dW_t,$$

where as usual $\kappa>0$ and $\theta>0$ but now we relax and allow $r_0<0.$

In this extended model, the short rate starts negative, but eventually goes positive and thereafter can never go negative again.

I think (but haven't checked!) that the usual bond pricing formulas extended to a negative initial rate are correct for this extended model.

Answer 2

t.f thanks for the answer.

You say that yields can't go negative in CIR. But if r0 (say 1d rate) is negative (which is the case in many govies today), I guess yields can be negative? And you will in this case be able to actually calibrate a CIR, which gives negative yields in the short end? My question might seem a bid odd, but I was just wondering?

But otherwise than that, the thing is i am investigating a displaced version so that $r_t+\alpha$, where $\alpha$ is a positive constant, has a CIR distribution. This models allows $r_t$ to be negative, and I can use the features from CIR. So I think that will work :-)

Answer 3

In the industry the model I have used is the 'shifted Sabr' where:

$dx(t) = \sigma(t) [x(t)-c]^\beta dW(t)$

$d\sigma(t) = \alpha \sigma(t) dZ(t)$

$dW(t)\ dZ(t) = \rho\ dt$

This allows for rates down to the parameter $c$. If you set, for example, $c=-200bp$ then you can have negative rates. You can define a CIR variant in an analogous way.

I have used this model both for pricing and for risk scenario/ path generation. The rule of thumb I used to set $c$ around 'current rate minus 2 normal implied ATM vols'. Obviously I cannot give proprietory details on the implementation here, but you should be able to work them out yourself.

Ps: There is always the option of the CIR++ models in Brigo and Mercurio, where you have a normal CIR and add a deterministic shift $\theta(t)$. It is a relatively simple and well documented alternative.