August  2019, 24(8): 3881-3903. doi: 10.3934/dcdsb.2018335

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

Received  March 2018 Published  January 2019

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.

Citation: Michael B. Giles, Kristian Debrabant, Andreas Rössler. Analysis of multilevel Monte Carlo path simulation using the Milstein discretisation. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 3881-3903. doi: 10.3934/dcdsb.2018335
References:
[1]

R. Avikainen, On irregular functionals of SDEs and the Euler scheme, Finance and Stochastics, 13 (2009), 381-401.  doi: 10.1007/s00780-009-0099-7.  Google Scholar

[2]

P. Embrechts, C. Klüppelberg and T. Mikosch, Modelling Extremal Events: For Insurance and Finance, Springer, 2008. Google Scholar

[3]

M. B. Giles, Improved multilevel Monte Carlo convergence using the Milstein scheme, in Monte Carlo and Quasi-Monte Carlo Methods 2006 (eds. A. Keller, S. Heinrich and H. Niederreiter), Springer, 2008,343–358. doi: 10.1007/978-3-540-74496-2_20.  Google Scholar

[4]

M. B. Giles, Multilevel Monte Carlo path simulation, Operations Research, 56 (2008), 607-617.  doi: 10.1287/opre.1070.0496.  Google Scholar

[5]

M. B. Giles, Multilevel Monte Carlo for basket options, in Proceedings of the 2009 Winter Simulation Conference (eds. M. Rossetti, R. Hill, B. Johansson, A. Dunkin and R. Ingalls), IEEE, 2009, 1283–1290. doi: 10.1109/WSC.2009.5429692.  Google Scholar

[6]

M. B. Giles, Multilevel Monte Carlo methods, Acta Numerica, 24 (2015), 259-328.  doi: 10.1017/S096249291500001X.  Google Scholar

[7]

M. B. GilesD. Higham and X. Mao, Analysing multilevel Monte Carlo for options with non-globally Lipschitz payoff, Finance and Stochastics, 13 (2009), 403-413.  doi: 10.1007/s00780-009-0092-1.  Google Scholar

[8]

M. B. Giles and L. Szpruch, Antithetic multilevel Monte Carlo estimation for multi-dimensional SDEs without Lüvy area simulation, Annals of Applied Probability, 24 (2014), 1585-1620.  doi: 10.1214/13-AAP957.  Google Scholar

[9]

P. Glasserman, Monte Carlo Methods in Financial Engineering, Springer, New York, 2004.  Google Scholar

[10]

P. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, Springer, Berlin, 1992. doi: 10.1007/978-3-662-12616-5.  Google Scholar

show all references

References:
[1]

R. Avikainen, On irregular functionals of SDEs and the Euler scheme, Finance and Stochastics, 13 (2009), 381-401.  doi: 10.1007/s00780-009-0099-7.  Google Scholar

[2]

P. Embrechts, C. Klüppelberg and T. Mikosch, Modelling Extremal Events: For Insurance and Finance, Springer, 2008. Google Scholar

[3]

M. B. Giles, Improved multilevel Monte Carlo convergence using the Milstein scheme, in Monte Carlo and Quasi-Monte Carlo Methods 2006 (eds. A. Keller, S. Heinrich and H. Niederreiter), Springer, 2008,343–358. doi: 10.1007/978-3-540-74496-2_20.  Google Scholar

[4]

M. B. Giles, Multilevel Monte Carlo path simulation, Operations Research, 56 (2008), 607-617.  doi: 10.1287/opre.1070.0496.  Google Scholar

[5]

M. B. Giles, Multilevel Monte Carlo for basket options, in Proceedings of the 2009 Winter Simulation Conference (eds. M. Rossetti, R. Hill, B. Johansson, A. Dunkin and R. Ingalls), IEEE, 2009, 1283–1290. doi: 10.1109/WSC.2009.5429692.  Google Scholar

[6]

M. B. Giles, Multilevel Monte Carlo methods, Acta Numerica, 24 (2015), 259-328.  doi: 10.1017/S096249291500001X.  Google Scholar

[7]

M. B. GilesD. Higham and X. Mao, Analysing multilevel Monte Carlo for options with non-globally Lipschitz payoff, Finance and Stochastics, 13 (2009), 403-413.  doi: 10.1007/s00780-009-0092-1.  Google Scholar

[8]

M. B. Giles and L. Szpruch, Antithetic multilevel Monte Carlo estimation for multi-dimensional SDEs without Lüvy area simulation, Annals of Applied Probability, 24 (2014), 1585-1620.  doi: 10.1214/13-AAP957.  Google Scholar

[9]

P. Glasserman, Monte Carlo Methods in Financial Engineering, Springer, New York, 2004.  Google Scholar

[10]

P. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, Springer, Berlin, 1992. doi: 10.1007/978-3-662-12616-5.  Google Scholar

Figure 1.  Asian option
Figure 2.  Lookback option
Figure 3.  Barrier option
Figure 4.  Digital option
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}) $
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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