Constrained Maximum Likelihood Estimation for the CIR Model
Summary
The document frames parameter calibration for the Cox–Ingersoll–Ross interest rate model as a maximum likelihood problem. Its parameters govern mean reversion, the long-run rate, and volatility; the proposed baseline optimization requires each parameter to be positive. It then raises whether calibration should additionally impose the Feller condition, which is intended to ensure the modeled rate stays positive rather than reaching zero.
The central issue is whether the condition should be treated as a hard constraint during estimation or left unenforced. The document gives no calibration data, likelihood derivation, numerical results, or resolution to that question. It is therefore a problem statement about model specification and constrained optimization, not an empirical comparison of constrained and unconstrained estimates. The condition concerns positivity behavior of the CIR process, while the best estimation choice depends on the modeling objective and assumptions.
Key ideas
- CIR parameters can be estimated by maximizing the model likelihood over observed rates.
- The basic parameter domain requires positive mean reversion, long-run rate, and volatility values.
- The Feller condition is presented as a constraint associated with keeping the rate positive.
- Imposing that condition changes the calibration problem to constrained maximum likelihood.
- The document poses the constraint question but does not provide an answer or empirical evidence.
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Full text
# Nonlinear Constrained optimization for a CIR model
# Nonlinear Constrained optimization for a CIR model
I want to calibrate a CIR model which is commonly used to model the evolution of interest rates. Briefly speaking, we know that its dynamics is of the form
\begin{equation} dr_t = \kappa (\theta - r_t) dt + \sigma \sqrt{r_t} dW_t \end{equation} where $\kappa$, $\theta$, and $\sigma$ are positive parameters, and $W_t$ is a standard Brownian motion process. All parameters can be estimated using maximum likelihood estimation (MLE) method. In practice, we consider the following problem, which can be calculated through a numerical optimization, for example, optim() function in R: \begin{equation} (\hat{\theta}, \hat{\kappa}, \hat{\sigma})=\max_{\theta >0, \kappa >0, \sigma >0} log(L(\theta, \kappa, \sigma; \boldsymbol{r})) \end{equation}
where $L(\theta, \kappa, \sigma; \boldsymbol{r})$ stands for the likelihood function of the CIR model and $\boldsymbol{r}$ represents a sample of size $n$.
In my opinion, there is one thing that needs to be considered when estimating parameters. We know that there is a condition under which we make sure that the interest rate governed by the above SDE will not reach zero and remain positive. That is, we also have this constrain $2\theta \kappa > \sigma ^2$. I wonder if we do not need to take into account this constraint. More precisely, one can have a look at the following optimization instead: \begin{equation} (\hat{\theta}, \hat{\kappa}, \hat{\sigma})=\max_{\theta >0, \kappa >0, \sigma >0} log(L(\theta, \kappa, \sigma; \boldsymbol{r}))\\ \text{s.t} \quad 2\theta \kappa > \sigma^2 \end{equation}
This means that we should find the optimal values under these four constraints. I see that most majority of people overlook this matter. Please let me know what you think in this regard.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.