Strong convergence rates and temporal regularity for Cox-Ingersoll-Ross processes and Bessel processes with accessible boundaries

TL;DR

This paper establishes strong convergence rates and temporal regularity results for Cox-Ingersoll-Ross (CIR) and Bessel processes with accessible boundaries, relevant for financial modeling and numerical approximation methods.

Contribution

It proves positive strong convergence rates for drift-implicit Euler schemes of CIR processes with accessible boundaries, extending previous results to this important case.

Findings

Drift-implicit Euler approximations converge with positive rates in the accessible boundary case.

Bessel processes exhibit temporal 1/2-Hölder continuity in L^p for all p > 0.

Results are relevant for calibrating financial models like the Heston model.

Abstract

Cox-Ingersoll-Ross (CIR) processes are widely used in financial modeling such as in the Heston model for the approximative pricing of financial derivatives. Moreover, CIR processes are mathematically interesting due to the irregular square root function in the diffusion coefficient. In the literature, positive strong convergence rates for numerical approximations of CIR processes have been established in the case of an inaccessible boundary point. Since calibrations of the Heston model frequently result in parameters such that the boundary is accessible, we focus on this interesting case. Our main result shows for every $p \in (0, \infty)$ that the drift-implicit square-root Euler approximations proposed in Alfonsi (2005) converge in the strong $L^p$-distance with a positive rate for half of the parameter regime in which the boundary point is accessible. A key step in our proof is temporal regularity of Bessel processes. More precisely, we prove for every $p \in (0, \infty)$ that Bessel processes are temporally $1/2$-H\"older continuous in $L^p$.