Estimating CIR Model Parameters Beyond Euler Discretization

Summary

The document raises a parameter-estimation question for the Cox-Ingersoll-Ross process, whose diffusion scale depends on the square root of the state. The author describes fitting the model with an Euler discretization that can be rearranged into a linear regression, using ordinary least squares for the mean-reversion parameters and residual variance for the volatility parameter.

The motivation is to consider predictor-corrector and Milstein discretizations, which do not reduce to the same simple regression form because of nonlinear terms or non-normal error structure. The document does not provide a solution, estimator, comparison, or empirical results; it is a request for guidance. Its useful lesson is that discretization choice affects the form of the estimation problem, and that methods suitable for the Euler approximation may not transfer directly to higher-order schemes.

Key ideas

• The CIR process has mean-reverting drift and state-dependent diffusion proportional to the square root of the state.
• Euler discretization can yield a regression-based approach for estimating the drift parameters and diffusion scale.
• Milstein and predictor-corrector schemes introduce estimation complications that do not fit the same simple linear regression setup.
• The document poses the estimation problem but does not establish which alternative method performs best.

Tags

# Calibration for CIR Model Discretization for Predictor Corrector and Milstein method


# Calibration for CIR Model Discretization for Predictor Corrector and Milstein method












I'm new to Quantitative Finance. I've data which I need to fit a CIR model and estimate its parameters.

$ dX_{t+1} = a(b-X_{t})dt + \sigma \sqrt{X_t}dW_{t} $

While I can fit and obtain parameparameterates using Euler-maurayana discretization and then linear regression closed form solution, I'd like to explore better approaches. This method was quite easy as after discretizing it, it became:

$ \frac{S_{t+1}-S{t}}{\sqrt S_{t}} = \frac{ab\Delta t}{\sqrt{S_{t}}} - a\sqrt{S_{t}}\Delta t + \sigma \sqrt{\Delta t} \epsilon_{t} $

where $ \epsilon_{t}$ is Normal(0,1)

I estimated a and b through OLS and got sigma estimate through error variance. I'd like to estimate parameters of this CIR model through better discretizing approaches. I tried discretizing the model using Milstein and Prediction and Error Correction Method. But both aren't in traditional linear regression form (either there are nonlinear terms or error terms aren't just normal). I'm not aware of how to proceed from here. Since there's stochastic component here, I don't know how to estimate these parameters (including sigma) with approaches (gradient descent or something).

Could someone kindly please help?

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.