Preservation of structural properties of the CIR model by {\theta}-Milstein schemes

TL;DR

This paper investigates how -Milstein schemes with  can preserve key properties of the CIR model, such as non-negativity and mean-reversion, and analyzes their convergence and variance preservation.

Contribution

It demonstrates that -Milstein schemes with  maintain the CIR model's structural properties and studies their convergence and long-term variance.

Findings

Schemes with  preserve non-negativity and mean-reversion.

The order of convergence of the schemes is established.

Long-term variance is preserved by the schemes.

Abstract

The ability of $\theta$-Milstein methods with $\theta\ge 1$ to capture the non-negativity and the mean-reversion property of the exact solution of the CIR model is shown. In addition, the order of convergence and the preservation of the long-term variance is studied. These theoretical results are illustrated with numerical examples.