    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  August 2019 Early access  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 and Continuous Dynamical Systems - B, 2019, 24 (8) : 3881-3903. doi: 10.3934/dcdsb.2018335
##### References:
  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.   P. Embrechts, C. Klüppelberg and T. Mikosch, Modelling Extremal Events: For Insurance and Finance, Springer, 2008. 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.   M. B. Giles, Multilevel Monte Carlo path simulation, Operations Research, 56 (2008), 607-617.  doi: 10.1287/opre.1070.0496.   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.  M. B. Giles, Multilevel Monte Carlo methods, Acta Numerica, 24 (2015), 259-328.  doi: 10.1017/S096249291500001X.   M. B. Giles, D. 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.   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.   P. Glasserman, Monte Carlo Methods in Financial Engineering, Springer, New York, 2004.  P. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, Springer, Berlin, 1992. doi: 10.1007/978-3-662-12616-5.   show all references

##### References:
  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.   P. Embrechts, C. Klüppelberg and T. Mikosch, Modelling Extremal Events: For Insurance and Finance, Springer, 2008. 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.   M. B. Giles, Multilevel Monte Carlo path simulation, Operations Research, 56 (2008), 607-617.  doi: 10.1287/opre.1070.0496.   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.  M. B. Giles, Multilevel Monte Carlo methods, Acta Numerica, 24 (2015), 259-328.  doi: 10.1017/S096249291500001X.   M. B. Giles, D. 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.   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.   P. Glasserman, Monte Carlo Methods in Financial Engineering, Springer, New York, 2004.  P. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, Springer, Berlin, 1992. doi: 10.1007/978-3-662-12616-5.   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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