Analysis of multilevel Monte Carlo path simulation using the Milstein discretisation

Michael B. Giles; Kristian Debrabant; Andreas Rössler

doi:10.3934/dcdsb.2018335

Article Contents

2019, Volume 24, Issue 8: 3881-3903. Doi: 10.3934/dcdsb.2018335

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Analysis of multilevel Monte Carlo path simulation using the Milstein discretisation

1.
Mathematical Institute, Oxford University, Oxford OX2 6GG, United Kingdom
2.
Department of Mathematics and Computer Science, University of Southern Denmark, 5230 Odense, Denmark
3.
Institute of Mathematics, University of Lübeck, 23562 Lübeck, Germany

^* Corresponding author: Michael B. Giles
^* Corresponding author: Michael B. Giles

Received: March 2018

Early access: January 4, 2019

Published online: January 4, 2019

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Abstract

The multilevel Monte Carlo path simulation method introduced by Giles (Operations Research, 56(3):607-617, 2008) exploits strong convergence properties to improve the computational complexity by combining simulations with different levels of resolution. In this paper we analyse its efficiency when using the Milstein discretisation; this has an improved order of strong convergence compared to the standard Euler-Maruyama method, and it is proved that this leads to an improved order of convergence of the variance of the multilevel estimator. Numerical results are also given for basket options to illustrate the relevance of the analysis.

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Mathematics Subject Classification: Primary: 60H10, 60H35, 65C05, 65C30.

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Figure 1. Asian option

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Figure 2. Lookback option

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Figure 3. Barrier option

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Figure 4. Digital option

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Table 1. Orders of convergence for $ V_\ell $ as observed numerically and proved analytically for both the Euler-Maruyama and Milstein discretisations; $ \delta $ can be any strictly positive constant

	Euler-Maruyama		Milstein
option	numerical	analysis	numerical	analysis
Lipschitz	$ {O} (h) $	$ {O} (h) $	$ {O} (h^2) $	$ {O} (h^2) $
Asian	$ {O} (h) $	$ {O} (h) $	$ {O} (h^2) $	$ {O} (h^2) $
lookback	$ {O} (h) $	$ {O} (h) $	$ {O} (h^2) $	$ {O} (h^2 (\log h)^2) $
barrier	$ {O} (h^{1/2}) $	$ {o} (h^{1/2-\delta}) $	$ {O} (h^{3/2}) $	$ {o} (h^{3/2-\delta}) $
digital	$ {O} (h^{1/2}) $	$ {O} (h^{1/2}\log h) $	$ {O} (h^{3/2}) $	$ {o} (h^{3/2-\delta}) $

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References

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