Strong Order-One-Half Convergence of the Projected Euler–Maruyama Method for the Cox–Ingersoll–Ross Model

Why It Matters

Half‑order strong convergence across wider CIR parameters enhances the accuracy of interest‑rate derivative pricing models, reducing model risk for financial institutions.

Key Takeaways

• Projected EM achieves 0.5 strong convergence order
• Works across broader CIR parameter regimes
• Extends normalized error analysis to projection methods
• Numerical experiments confirm theoretical rates
• Improves reliability for interest‑rate derivative pricing

Pulse Analysis

The Cox–Ingersoll–Ross (CIR) process remains a cornerstone in modeling short‑term interest rates, underpinning a vast array of fixed‑income derivatives. Traditional discretization techniques, such as the full‑truncation Euler–Maruyama method, deliver reliable results but only under restrictive parameter conditions that ensure positivity and stability. Practitioners often grapple with regimes where volatility or mean‑reversion parameters push the model toward the boundary, leading to numerical bias or even negative rates, which are economically implausible. Consequently, extending strong convergence guarantees to these challenging settings is a pressing research priority.

The projected Euler–Maruyama (PEM) method tackles this issue by enforcing a non‑negative constraint through a simple projection after each time step. Coupled with the normalized error analysis pioneered by Cozma and Reisinger, the authors demonstrate that PEM retains the classic Lp‑strong convergence order of ½, even when the Feller condition is violated. This hybrid approach sidesteps the need for ad‑hoc truncations while preserving the stochastic dynamics essential for accurate term‑structure modeling. The rigorous proof expands the theoretical foundation of stochastic numerical methods, offering a clear pathway to implement PEM in high‑frequency calibration routines without sacrificing convergence speed.

For quantitative analysts and risk managers, the practical upshot is significant. A discretization scheme that guarantees half‑order convergence across a broader parameter spectrum translates into tighter error bounds for Monte Carlo simulations of interest‑rate derivatives, such as caps, floors, and swaptions. This reliability reduces the capital allocated for model risk and improves hedging precision. Moreover, the method’s simplicity—requiring only a projection step—facilitates integration into existing pricing libraries, accelerating adoption. Future work may explore adaptive step‑size strategies or extensions to multi‑factor CIR frameworks, further cementing PEM’s role in modern financial engineering.