Multilevel Monte Carlo method

My main research on Monte Carlo methods concerns the development of multilevel methods. Based on multigrid ideas for the iterative solution of discretised PDEs, this is a variance reduction approach for infinite dimensional integration which combines stochastic simulations with differing levels of resolution. This results in a significant reduction in the order of complexity, the computational cost required to obtain a user-specified accuracy.

• My first paper on this topic presented numerical results for SDEs using an Euler discretisation, but it also analysed the computational complexity for a general class of methods and applications, assuming that the discretisation satisfies certain conditions.

  The MATLAB code used to produce the figures for the paper is available here.

• Paper #2 presented significantly improved numerical results using the Milstein discretisation. The Milstein method's improved strong convergence leads to most of the computational effort being confined to the coarsest levels.

  The MATLAB code used to produce the figures for the paper is available here.

• Paper #3 was a collaboration with Des Higham and Xuerong Mao from the University of Strathclyde, in which we performed a numerical analysis of the method in the first paper.

• Paper #4 was a collaboration with Ben Waterhouse of the University of New South Wales. This used quasi-Monte Carlo techniques based on a rank-1 lattice rule to further improve the computational efficiency.

• Paper #5 is a numerical verification that the Multilevel Milstein treatment is also applicable to basket options.

• Paper #6 is a collaboration with Rob Scheichl and Aretha Teckentrup at the University of Bath, and Andrew Cliffe at the University of Nottingham. This applies the multilevel approach to elliptic SPDEs which arise in the modelling of nuclear waste repositories, with the permeability of the rock being modelled as a log-Normal stochastic field.

• Paper #7 with my student Yuan Xia tackles Merton-style jump-diffusion models. The key feature of this paper is the use of a change of measure to cope with cases in which the jump rate is path-dependent which would otherwise lead to jumps at different times on coarse and fine paths.

• Paper #8 with my student Sylvestre Burgos deals with the calculation of sensitivities. This involves differentiating the payoff, and the loss of smoothness causes difficulties for the multilevel method.

• Paper #9 is a collaboration with my colleague Christoph Reisinger. This is another SPDE application, but in this case it is an unusual parabolic SPDE which arises in a financial credit modelling application.

• Paper #10 is a collaboration with my colleague Lukas Szpruch. This addresses this use of the Milstein approximation in multiple dimensions. This usually requires the simulation of Lévy areas, but we have developed an antithetic technique which gives a high rate of multilevel convergence without simulating Lévy areas.

• Paper #11 continues the collaboration with Rob Scheichl and Aretha Teckentrup at the University of Bath.

Current research, involving several collaborations, is addressing the following:

• multilevel simulation of stochastic PDEs
• several issues concerned with Milstein discretisation for multi-dimensional SDEs
• Lévy processes
• stopping times
• parallel implementation on NVIDIA GPUs

• American and Bermudan options

Publications

1. M.B. Giles. `Multi-level Monte Carlo path simulation'. Operations Research, 56(3):607-617, 2008. (PDF)
2. M.B. Giles. `Improved multilevel Monte Carlo convergence using the Milstein scheme'. 343-358, in Monte Carlo and Quasi-Monte Carlo Methods 2006, Springer, 2008. (PDF)
3. M.B. Giles, D.J. Higham and X. Mao. 'Analysing multilevel Monte Carlo for options with non-globally Lipschitz payoff'. Finance and Stochastics, 13(3):403-413, 2009. (PDF)
4. M.B. Giles and B.J. Waterhouse. 'Multilevel quasi-Monte Carlo path simulation'. pp.165-181 in Advanced Financial Modelling, in Radon Series on Computational and Applied Mathematics, de Gruyter, 2009. (PDF)
5. M.B. Giles. `Multilevel Monte Carlo for Basket Options'. Winter Simulation Conference '09. (PDF)
6. K.A. Cliffe, M.B. Giles, R. Scheichl, A.L. Teckentrup, 'Multilevel Monte Carlo Methods and Applications to Elliptic PDEs with Random Coefficients', Computing and Visualization in Science, 14(1):3-15, 2011. (PDF)
7. Y. Xia, M.B. Giles. `Multilevel path simulation for jump-diffusion SDEs', in Monte Carlo and Quasi-Monte Carlo Methods 2010, Springer, 2012. (PDF)
8. S. Burgos, M.B. Giles. `Computing Greeks using multilevel path simulation', in Monte Carlo and Quasi-Monte Carlo Methods 2010, Springer, 2012. (PDF)
9. M.B. Giles, C. Reisinger. 'Stochastic finite differences and multilevel Monte Carlo for a class of SPDEs in finance', to appear in SIAM Journal of Financial Mathematics, 2012. (PDF)
10. M.B. Giles, L. Szpruch. 'Antithetic multilevel Monte Carlo estimation for multi-dimensional SDEs without Lévy area simulation', 2012. (PDF)
11. A.L. Teckentrup, R. Scheichl, M.B. Giles, E. Ullmann. 'Further analysis of multilevel Monte Carlo methods for elliptic PDEs with random coefficients', 2012. (PDF)

Acknowledgements

• Oxford-Man Institute of Quantitative Finance
• EPSRC, through a Springboard Fellowship