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Analysis of multilevel Monte Carlo path simulation using the Milstein discretisation

  • * Corresponding author: Michael B. Giles

    * Corresponding author: Michael B. Giles 
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  • 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.

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

    Citation:

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