High order approximations of the Cox-Ingersoll-Ross process semigroup using random grids

TL;DR

This paper introduces high order approximation schemes for the Cox-Ingersoll-Ross process using a novel combination of discretization schemes on random grids, achieving higher convergence orders and computational efficiency.

Contribution

It develops a new technique combining discretization schemes on random grids to attain weak approximations of order 2k for the CIR process, improving accuracy and efficiency.

Findings

Achieves weak convergence order 2k for all k in natural numbers.

Numerical experiments confirm convergence and computational time gains.

Applicable to CIR and Heston models with rigorous convergence analysis.

Abstract

We present new high order approximations schemes for the Cox-Ingersoll-Ross (CIR) process that are obtained by using a recent technique developed by Alfonsi and Bally (2021) for the approximation of semigroups. The idea consists in using a suitable combination of discretization schemes calculated on different random grids to increase the order of convergence. This technique coupled with the second order scheme proposed by Alfonsi (2010) for the CIR leads to weak approximations of order $2k$, for all $k\in\mathbb{N}^*$. Despite the singularity of the square-root volatility coefficient, we show rigorously this order of convergence under some restrictions on the volatility parameters. We illustrate numerically the convergence of these approximations for the CIR process and for the Heston stochastic volatility model and show the computational time gain they give.