American Institute of Mathematical Sciences

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

Analysis of multilevel Monte Carlo path simulation using the Milstein discretisation

Michael B. Giles 1,, , Kristian Debrabant 2, and  Andreas Rössler 3,

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.

Keywords: Multilevel, Monte Carlo, stochastic differential equations, numerical analysis, computational finance.
Mathematics Subject Classification: Primary: 60H10, 60H35, 65C05, 65C30.
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. 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.  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. 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.  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 Options

Download full-size image

Download as PowerPoint slide

Figure 2.  Lookback option
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 3.  Barrier option
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 4.  Digital option
Figure Options

Download full-size image

Download as PowerPoint slide

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}) $
Table Options

Download as excel

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}) $
[1]

Ajay Jasra, Kody J. H. Law, Yaxian Xu. Markov chain simulation for multilevel Monte Carlo. Foundations of Data Science, 2021, 3 (1) : 27-47. doi: 10.3934/fods.2021004
[2]

Zhiyan Ding, Qin Li. Constrained Ensemble Langevin Monte Carlo. Foundations of Data Science, 2021  doi: 10.3934/fods.2021034
[3]

Giacomo Dimarco. The moment guided Monte Carlo method for the Boltzmann equation. Kinetic & Related Models, 2013, 6 (2) : 291-315. doi: 10.3934/krm.2013.6.291
[4]

Guillaume Bal, Ian Langmore, Youssef Marzouk. Bayesian inverse problems with Monte Carlo forward models. Inverse Problems & Imaging, 2013, 7 (1) : 81-105. doi: 10.3934/ipi.2013.7.81
[5]

Theodore Papamarkou, Alexey Lindo, Eric B. Ford. Geometric adaptive Monte Carlo in random environment. Foundations of Data Science, 2021, 3 (2) : 201-224. doi: 10.3934/fods.2021014
[6]

Jie Xiong, Shuaiqi Zhang, Yi Zhuang. A partially observed non-zero sum differential game of forward-backward stochastic differential equations and its application in finance. Mathematical Control & Related Fields, 2019, 9 (2) : 257-276. doi: 10.3934/mcrf.2019013
[7]

Tetsuya Ishiwata, Young Chol Yang. Numerical and mathematical analysis of blow-up problems for a stochastic differential equation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 909-918. doi: 10.3934/dcdss.2020391
[8]

Nikolaos Halidias. Construction of positivity preserving numerical schemes for some multidimensional stochastic differential equations. Discrete & Continuous Dynamical Systems - B, 2015, 20 (1) : 153-160. doi: 10.3934/dcdsb.2015.20.153
[9]

Minoo Kamrani. Numerical solution of partial differential equations with stochastic Neumann boundary conditions. Discrete & Continuous Dynamical Systems - B, 2019, 24 (10) : 5337-5354. doi: 10.3934/dcdsb.2019061
[10]

Jiakou Wang, Margaret J. Slattery, Meghan Henty Hoskins, Shile Liang, Cheng Dong, Qiang Du. Monte carlo simulation of heterotypic cell aggregation in nonlinear shear flow. Mathematical Biosciences & Engineering, 2006, 3 (4) : 683-696. doi: 10.3934/mbe.2006.3.683
[11]

Zheng Liu, Tianxiao Wang. A class of stochastic Fredholm-algebraic equations and applications in finance. Discrete & Continuous Dynamical Systems - B, 2021, 26 (7) : 3879-3903. doi: 10.3934/dcdsb.2020267
[12]

Dejian Chang, Zhen Wu. Stochastic maximum principle for non-zero sum differential games of FBSDEs with impulse controls and its application to finance. Journal of Industrial & Management Optimization, 2015, 11 (1) : 27-40. doi: 10.3934/jimo.2015.11.27
[13]

Chuchu Chen, Jialin Hong. Mean-square convergence of numerical approximations for a class of backward stochastic differential equations. Discrete & Continuous Dynamical Systems - B, 2013, 18 (8) : 2051-2067. doi: 10.3934/dcdsb.2013.18.2051
[14]

Ilknur Koca. Numerical analysis of coupled fractional differential equations with Atangana-Baleanu fractional derivative. Discrete & Continuous Dynamical Systems - S, 2019, 12 (3) : 475-486. doi: 10.3934/dcdss.2019031
[15]

Chjan C. Lim, Joseph Nebus, Syed M. Assad. Monte-Carlo and polyhedron-based simulations I: extremal states of the logarithmic N-body problem on a sphere. Discrete & Continuous Dynamical Systems - B, 2003, 3 (3) : 313-342. doi: 10.3934/dcdsb.2003.3.313
[16]

Joseph Nebus. The Dirichlet quotient of point vortex interactions on the surface of the sphere examined by Monte Carlo experiments. Discrete & Continuous Dynamical Systems - B, 2005, 5 (1) : 125-136. doi: 10.3934/dcdsb.2005.5.125
[17]

Olli-Pekka Tossavainen, Daniel B. Work. Markov Chain Monte Carlo based inverse modeling of traffic flows using GPS data. Networks & Heterogeneous Media, 2013, 8 (3) : 803-824. doi: 10.3934/nhm.2013.8.803
[18]

Mazyar Zahedi-Seresht, Gholam-Reza Jahanshahloo, Josef Jablonsky, Sedighe Asghariniya. A new Monte Carlo based procedure for complete ranking efficient units in DEA models. Numerical Algebra, Control & Optimization, 2017, 7 (4) : 403-416. doi: 10.3934/naco.2017025
[19]

Burcu Gürbüz. A computational approximation for the solution of retarded functional differential equations and their applications to science and engineering. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021069
[20]

Marco Campo, José R. Fernández, Maria Grazia Naso. A dynamic problem involving a coupled suspension bridge system: Numerical analysis and computational experiments. Evolution Equations & Control Theory, 2019, 8 (3) : 489-502. doi: 10.3934/eect.2019024

2020 Impact Factor: 1.327

Tools

Metrics

  • PDF downloads (291)
  • HTML views (520)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy