Milstein Discretization for the Time-Varying CIR Process

Summary

The note addresses simulation of a time-varying Cox–Ingersoll–Ross (CIR) process, whose diffusion coefficient scales with the square root of the state. It presents the Milstein scheme as a refinement of Euler discretization. For a general one-dimensional stochastic differential equation, the method adds a correction involving the derivative of the diffusion coefficient and the squared normal shock. Substituting the CIR drift and diffusion gives an update with a term proportional to the square of the shock minus one.

The response also names transformed-volatility, IJK, and quadratic-exponential methods as alternatives, but does not explain or compare them. No numerical experiment, error estimate, or assessment of stability is supplied. In particular, the displayed Milstein update is not accompanied by discussion of how to handle negative simulated states, an issue that can matter for square-root diffusions. The note is therefore an introductory discretization recipe, not evidence that Milstein is the most accurate or suitable choice for every CIR simulation task.

Key ideas

• The CIR process has a state-dependent diffusion coefficient proportional to the square root of its state.
• Milstein discretization adds a correction based on the derivative of the diffusion coefficient.
• For CIR, the correction depends on the squared standard-normal shock minus one.
• The note lists transformed-volatility, IJK, and quadratic-exponential schemes without comparing their performance.
• The response does not address negative simulated values or provide numerical accuracy evidence.

Tags

# Extended CIR and discretization


# Extended CIR and discretization












Did someone know how to discretize this process efficiently :

$dX(t) = \kappa [\theta(t)-X(t)]dt + \sigma \sqrt{X(t)}dW(t)$

I am looking for something more sophisticated than the trivial Euler Schema :

$X(t_{k+1}) =X(t_{k}) + \kappa[\theta(t_k)-X(t_{k})]\Delta t + \sigma \sqrt{X(t_{k})}\Delta_k W$

Thanks in advance,

## Answer by user16651 (score 5)

https://quant.stackexchange.com/a/18586

Milstein Scheme

This scheme is described in Glasserman (2003) and in Kloeden and Platen (1992) for general processes.Hence, for simplicity, we can assume that the Stochastic Process is driven by the SDE \begin{align} &dX_t=\Xi(t,X_t)dt+\Sigma(t,X_t)dW_t\\ \end{align} Milstein discretization is, \begin{align} dX_{t+\Delta t}=X_t+\Xi(t,X_t)dt+\Sigma(t,X_t)\sqrt{\Delta t}\,Z+\frac{1}{2}\Sigma(t,X_t)\frac{\partial\Sigma(t,X_t)}{\partial X_t}\Delta t(Z^2-1) \ \end{align} where Z is is a standard normal variable.The coefficients of C.I.R process are \begin{align} \Xi(t,X_t) = \kappa(\theta(t) − X_t) \end{align} and \begin{align} \Sigma(t,X_t) = \sigma\sqrt X_t \end{align} by application of Milstein scheme,we have

\begin{align} dX_{t+\Delta t}=X_t+\kappa(\theta(t) − X_t)\Delta t+\sigma\sqrt{\Delta t \,X_t}\,Z+\frac{1}{4}\sigma^2\Delta t(Z^2-1) \end{align} Other Methods as follow

- Transformed volatility scheme

- IJK Scheme

- Quadratic exponential method

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