# Cox-Ingersoll-Ross

I am looking at a displaced CIR model and try to calibrate it to market data. I think my results looks reasonable but would like to sense-check with other studies. Does anyone know what "reasonable" parameters (which region they should be in) for a CIR model is? If anyone has some good articles that describe this, I would be glad if you would share it. I can't really seem to find some.

• Could you explain the displaced CIR model? – bcf Jun 3 '16 at 12:40
• Too little information provided. If you can provide a detailed description of your model, in mathematical forms, the parameters you have estimated, and the data that you have used, it will be more helpful for your question. – Gordon Jun 3 '16 at 13:01
• Modified or extended CIR??? – user16651 Jun 3 '16 at 17:01
• By displaced I mean $dr_t = \kappa(\theta-r_t)dt+\sigma \sqrt{r_t} dW_t$, where $r_t=x_t+\alpha$, with $\alpha>0$ being a constant. The displacement makes it possible for the short-rate to be negative. – Bohlke Jun 4 '16 at 9:56
• @BehrouzMaleki please formulate a real answer if you have one, not just links in comments. – SRKX Jul 5 '16 at 1:39

It totally depends on the asset class / precise underlying obviously. You just cannot transpose a parameter set from one asset class to another. In fact you cannot transpose from one underlying to another within the same asset class since the parameters encode pretty rich heterogeneous behaviour that eg one rate or intensity will display relative to another. The key thing is, do the parameters look sensible for the problem you are looking at. Using the specification mentioned in comments:

$$\mathrm{d}r_t = \alpha(\beta-r)\:\mathrm{d}t + \sigma \sqrt{r_t\:} W_t$$ Will assume from the comments we are talking about rates and not credit intensities (for which CIR is also very useful due to analyticity of survival probabilities), I would suggest that the parameters that are easiest to evaluate as being qualitatively sensible or not are $\beta$ and $\alpha$, respectively the mean reversion rate and target. Once you have these within reasonable bounds then a risk neutral square root process vol drops out.

Going into a bit more detail regarding the possible structure you can impose (within any degrees of freedom not controlled by your calibration instruments):

1. Try to make sure that $\beta$ is sensible, are your rates mean reverting to something unrealistic? Is there a valid 'equilibrium' level you think could make sense?
2. Think about the dynamics imposed by the reversion parameter $\alpha$: If this is high, you will find in simulations that high levels of $r_t$ will equate to flatter curves, so the forwards will correlate negatively with outright levels of spot. This is crucial eg for behaviour of CMS pricing. In credit world, this is often observed where participants buy front end protection to indemnify against default risk of distressed credits - inversion is very common and therefore having a low $\alpha$ would not be suitable at all. Low $\alpha$ would be associated with high rates <=> high forwards, is this valid? In rates world, think about bear/bull flattening / steepening, which central bank policy action may influence.

Once you are happy with mean reversion / forward dynamics, I think you can be fairly happy with what you've got. The best way is to simulate and assess the parameters, a good test of #2 being for example what is the correlation your parameter set is creating between spot and forwards?

Anyway that's how I would assess the validity of parameters, there really is no universal set that makes sense in all currencies, regions, asset classes etc etc, but luckily they are quite intuitive I think. Interested in other practitioners' thoughts.

I am not sure what you mean by a displaced CIR model, but for the following CIR model for annualised interest rates $$\textrm{d}r = \alpha(\beta-r)\:\textrm{d}t + \sigma \sqrt{r\:} W_t$$ I have seen papers use similar values to $\alpha = 0.6\:\textrm{year}^{-1}$, $\beta = 0.06$, $\sigma = 0.25\:\textrm{year}^{-\frac{1}{2}}$. An example of such a paper is:

• "A stochastic partial differential equation model for the pricing of mortgage-backed securities", by Ferhana Ahmad, Ben Hambly, and Sean Ledger, 2016.
• See Section 9, albeit they don't give many references to how they picked these values. – oliversm Jun 4 '16 at 11:46
• Shouldn't that be $\sqrt{r_t}$ instead of $\sqrt{t}$? And displaced usually means you subsitute $r_t\rightarrow r_t + c$ for some displacement constant. The model can then handle negative rates up to $r_t \geq -c$. – Olaf Jun 4 '16 at 12:51
• @Olaf, thanks for spotting the typo. I have quoted the above model as it is one of the few where I can find values quoted for the parameters, where the values are used in active models. – oliversm Jun 4 '16 at 13:09