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Preservation of structural properties of the CIR model by θ-Milstein schemes

Samir Llamazares-Elias, Angel Tocino

TL;DR

The paper addresses numerically solving the CIR process while preserving qualitative features such as non-negativity and mean reversion. It analyzes semi-implicit $\theta$-Milstein schemes with $\theta\ge1$ and derives conditions under which these schemes replicate the exact model's long-term behavior and moments; it also establishes weak order 1 and strong order logarithmic convergence, plus analysis of the long-term second moment. Exact preservation of the long-term second moment occurs only in the special case $4\alpha\mu=\sigma^2$ with $\theta=1$, otherwise a small, step-size-dependent bias remains. Numerical experiments compare with CIR-specific schemes and confirm the theoretical findings, showing that $\theta=1$ often yields the best balance for mean accuracy and moment preservation. Overall, the results provide practical guidance for stable, structure-preserving simulations of the CIR process in financial applications, highlighting the advantages of fully implicit or $\theta\ge1$ Milstein schemes.

Abstract

The ability of $θ$-Milstein methods with $θ\ge 1$ 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.

Preservation of structural properties of the CIR model by θ-Milstein schemes

TL;DR

The paper addresses numerically solving the CIR process while preserving qualitative features such as non-negativity and mean reversion. It analyzes semi-implicit -Milstein schemes with and derives conditions under which these schemes replicate the exact model's long-term behavior and moments; it also establishes weak order 1 and strong order logarithmic convergence, plus analysis of the long-term second moment. Exact preservation of the long-term second moment occurs only in the special case with , otherwise a small, step-size-dependent bias remains. Numerical experiments compare with CIR-specific schemes and confirm the theoretical findings, showing that often yields the best balance for mean accuracy and moment preservation. Overall, the results provide practical guidance for stable, structure-preserving simulations of the CIR process in financial applications, highlighting the advantages of fully implicit or Milstein schemes.

Abstract

The ability of -Milstein methods with 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.

Paper Structure

This paper contains 7 sections, 8 theorems, 39 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Preservation of non-negativity by theta-Milstein methods
  3. Strong and weak convergence of theta-Milstein methods
  4. Mean-reverting theta-Milstein schemes
  5. Long-term second moment
  6. Numerical Experiments
  7. Conclusions

Key Result

Proposition 2.1

The $\theta$-Misltein scheme mil2 with $\theta\ge 1$ starting at $X_0\ge0$ preserves the non-negativity of the exact solution $X(t)$ if the parameters of the CIR model fulfill the condition $4\alpha\mu\ge\sigma^2$. In particular, under the Feller condition it preserves the positivity of $X(t)$.

Figures (5)

  • Figure 1: Log-log plot of the weak error $\varepsilon_X^\Delta$ at time $t = 1$ against $\Delta$ for schemes HH, HM, E(0), DI, MS and implicit Milstein ($\theta=1$) with data \ref{['par1']} (top) and \ref{['par2']} (bottom).
  • Figure 2: Log-log plot of the strong error $\mathcal{E}_X^\Delta$ at time $t = 1$ against $\Delta$ for schemes HH, HM, E(0), DI, MS and implicit Milstein ($\theta=1$) with data \ref{['par1']} (top plot) and \ref{['par2']} (bottom plot).
  • Figure 3: Evolution of the first sample moment (top), and its distance to the long-term mean (bottom) for solving \ref{['CIR']} with parameters \ref{['par1']} and schemes \ref{['mil2']}, HM, DI, E(0), and HH.
  • Figure 4: Evolution of the sample second moment (top) and its distance to the long-term second moment (bottom) for solving \ref{['CIR']} with parameters \ref{['par1']} and schemes \ref{['mil2']}, HM, DI, E(0), and HH.
  • Figure 5: First (left) and second (right) moment errors of implicit $\theta-$Milstein schemes applied to the model with parameters \ref{['par1']} (top) and \ref{['par2']} (bottom).

Theorems & Definitions (11)

  • Proposition 2.1
  • Remark 1
  • Proposition 3.1
  • proof
  • Lemma 1
  • Proposition 3.2
  • proof
  • Theorem 4.1
  • Proposition 4.2
  • Theorem 5.1
  • ...and 1 more