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Multilevel Monte Carlo using approximate distributions of the CIR process

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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. 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.

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Notes

  1. We define \(Q_{d}(0,U)=\lim _{g\rightarrow 0}Q_{d}(g,U)=1\), \(U\in (0,1)\). Then \(\lim _{g\rightarrow 0}{\tilde{Q}}_{d}(g,U)=1\), \(U\in (0,1)\), is guaranteed by the definition of the bilinear approximation.

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Acknowledgements

The author is grateful to Prof Mike Giles for suggesting the research problem and fruitful discussions.

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  1. School of Data Sciences, Zhejiang University of Finance and Economics, Hangzhou, China

    Chao Zheng

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  1. Chao Zheng
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Correspondence to Chao Zheng.

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Communicated by David Cohen.

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This research is supported by National Natural Science Foundation of China (No. 11801504).

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Zheng, C. Multilevel Monte Carlo using approximate distributions of the CIR process. Bit Numer Math 63, 38 (2023). https://doi.org/10.1007/s10543-023-00980-0

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