Mike Giles - Multilevel Monte Carlo software

Multilevel Monte Carlo software

This page provides software developed in a MLMC software project for MLMC computations in a variety of languages.

This software differs from my previous MLMC software in separating the computations from the plotting of the results. Each application code has two parts, a high-level part which calls "mlmc_test" to perform the MLMC tests, using the routine "mlmc", and a low-level part which is called by both "mlmc_test" and "mlmc" to compute the MLMC differences for one particular level of correction.

The application codes produce one or more output text files. The MATLAB routine "mlmc_plot" can then be used to generate the standard set of figures which I use in most of my papers.

MATLAB

Common routines used by all applications:

mlmc.m -- main MLMC driver routine
mlmc_test.m -- routine for MLMC tests
mlmc_plot.m -- routine for plotting results
mlmc_test_100.m -- routine performing 100 MLMC calcs
mlmc_plot_100.m -- routine for plotting results

opre -- financial options based on scalar geometric Brownian motion and Heston models, similar to my original 2008 Operations Research paper, using an Euler-Maruyama discretisation

opre.m -- application code
opre.tex -- LaTeX file for output
opre.pdf -- PDF results

mcqmc06 -- financial options based on scalar geometric Brownian motion, similar to my 2008 MCQMC06 paper, using a Milstein discretisation

mcqmc06.m -- application code
mcqmc06_spmd.m -- second version using spmd parallelism
mcqmc06.tex -- LaTeX file for output
mcqmc06.pdf -- PDF results

basket -- basket options based on 5 underlying assets, similar to my 2009 Winter Simulation Conference paper, using a Milstein discretisation

basket.m -- application code
basket.tex -- LaTeX file for output
basket.pdf -- PDF results

ctmp -- continuous time Markov process, giving results for a testcase in a 2012 paper by Anderson and Higham

ctmp.m -- application code
ctmp.tex -- LaTeX file for output
ctmp.pdf -- PDF results

spde -- SPDE arising from modelling of credit derivatives, giving results similar to those in a 2012 paper with Christoph Reisinger

spde.m -- application code
spde.tex -- LaTeX file for output
spde.pdf -- PDF results

antithetic -- Heston SDE example using the Milstein discretisation without the simulation of Levy areas, showing the benefits of the antithetic treatment analysed in a 2014 Annals of Applied Probability paper with Lukasz Szpruch

antithetic.m -- application code
antithetic.tex -- LaTeX file for output
antithetic.pdf -- PDF results

adapted -- non-standard SDE calculation with an adapted timestep, giving results for a testcase in a 2015 MCQMC14 paper

adapted.m -- application code
adapted.tex -- LaTeX file for output
adapted.pdf -- PDF results

reflected1D -- 1D reflected diffusions of particles with and without mass, giving results for a presentation at SciCADE 2015

reflected1D.m -- application code
reflected1D.tex -- LaTeX file for output
reflected1D.pdf -- PDF results

reflected -- 3D reflected diffusions using adaptive timesteps, giving similar results to a presentation at MCQMC16

reflected.m -- application code with reflection, projection or penalisation at boundaries
reflected.tex -- LaTeX file for output
reflected.pdf -- PDF results

nested -- a very simple nested expectation testcase for estimating EVPPI from a 2019 paper with Takashi Goda

nested.m -- application code
nested.tex -- LaTeX file for output
nested.pdf -- PDF results

EVPPI -- a set of 3 medical decision making EVPPI test problems from the same 2019 paper with Takashi Goda

test1.m, test2.m, test3.m -- application codes
EVPPI.tex -- LaTeX file for output
EVPPI.pdf -- PDF results

C/C++

Note: the Makefiles for these testcases are setup to produce two different executables:

a single-threaded executable using the standard C++11 random number generators
a multi-threaded executable using OpenMP and Intel's MKL/VSL random number generators

In each case, they are based on the same C++ code, with the details of the random number generator hidden within mlmc_rng.cpp. Depending on the CPU(s) being used, the OpenMP executable is typically up to 100 times faster.

Common routines used by all applications:

mlmc.cpp -- main MLMC driver routine
mlmc_test.cpp -- routine for MLMC tests
mlmc_test_100.cpp -- routine performing 100 MLMC calcs
mlmc_rng.cpp -- routines for random number generation using either the standard C++11 functions or Intel's MKL/VSL library

mlmc_plot.m -- MATLAB routine for plotting results
mlmc_plot_100.m -- MATLAB routine for plotting results

mcqmc06 -- financial options based on scalar geometric Brownian motion, similar to my 2008 MCQMC06 paper, using a Milstein discretisation

mcqmc06.cpp -- application code
Makefile -- uses g++ or Intel's icpc for compilation
mcqmc06.m -- MATLAB code to plot results using mlmc_plot.m and mlmc_plot_100.m
mcqmc06.tex -- LaTeX file for output
mcqmc06.pdf -- PDF results

ctmp -- continuous time Markov process, giving results for a testcase in a 2012 paper by Anderson and Higham

ctmp.cpp -- single-threaded application code
poissinv.h -- Poisson inverse CDF function
Makefile -- uses g++ for compilation
ctmp.m -- MATLAB code to plot results using mlmc_plot.m
ctmp.tex -- LaTeX file for output
ctmp.pdf -- PDF results

adapted -- non-standard SDE calculation with an adapted timestep, giving results for a testcase in a 2015 MCQMC14 paper

adapted.cpp -- application code
Makefile -- uses either g++ or Intel's icpc for compilation
adapted.m -- MATLAB code to plot results using mlmc_plot.m
adapted.tex -- LaTeX file for output
adapted.pdf -- PDF results

reflected -- 3D reflected diffusions using adaptive timesteps, giving similar results to a presentation at MCQMC16

reflected.cpp -- application code with reflection, projection or penalisation at boundaries
Makefile -- uses either g++ or Intel's icpc for compilation
reflected.m -- MATLAB code to plot results using mlmc_plot.m
reflected.tex -- LaTeX file for output
reflected.pdf -- PDF results

feynman-kac -- stopped diffusions, giving very similar results to a 2018 paper with Francisco Bernal

fk1.cpp -- application code with/without path-splitting
Makefile -- uses either g++ or Intel's icpc for compilation
fk1.m, mlmc_read.m -- MATLAB codes to plot results
fk1.tex -- LaTeX file for output
fk1.pdf -- PDF results

nested -- a very simple nested expectation testcase for estimating EVPPI from a 2019 paper with Takashi Goda

nested.cpp -- application code
Makefile -- uses g++ or Intel's icpc for compilation
nested.m -- MATLAB code to plot results using mlmc_plot.m
nested.tex -- LaTeX file for output
nested.pdf -- PDF results

Python

Please follow this link.

R

Please follow this link.

Licensing and acknowledgements

This software is freely available to all under a GPL license -- anyone requiring a more permissive license for commercial purposes should contact me.

The software is based on the research reported in the papers listed here. If you find it helpful in your research, the papers there can be cited in any publications. I would also be interested to hear about it, particularly if it is used for novel applications.

The underlying research has been supported by

EPSRC, through a Springboard Fellowship and ICONIC programme grant project funding
Oxford-Man Institute of Quantitative Finance