Multilevel Monte Carlo Using Approximate Distributions of the CIR Process

Abstract

The Cox-Ingersoll-Ross (CIR) process has important applications in finance. However, it is challenging to develop a multilevel Monte Carlo (MLMC) method with an approximate CIR process such that the relevant MLMC variance has a constant convergence rate for all parameter regimes. In this article, we provide a solution to this problem. Our approach is based on a nested MLMC with approximate normal random variables proposed recently by Giles and Sheridan-Methven (SIAM/ASA Journal on Uncertainty Quantification: 10(1), 200-226, 2022). Specifically, we develop this method by embedding a class of approximations of the CIR process using the quantiles of noncentral chi-squared distributions. Under mild assumptions, we show that the MLMC variance is O(h) for the full parameter range of the CIR process, where h is the step size of the discretization of the CIR process. Furthermore, we extend the approach to a time-discrete scheme for the Heston model. The efficiency of this approach is illustrated by numerical experiments.

Keywords: Multilevel Monte Carlo, approximate distribution, CIR process