# Ideas for MSc projects

Here are some outline ideas for projects for the full-time MSc in Mathematical and Computational Finance:
• Comparison of MLMC, ML2R (Pages-Lemaire) and randomised MLMC (Rhee-Glynn);

• Comparison of Sobol with "simple" digital scrambling versus Matousek-Owen scrambling;

• Multilevel QMC

In the past, I did some work with Ben Waterhouse on combining multilevel Monte Carlo with QMC methods, using rank-1 lattices (PDF). It would be interesting to re-visit this work using Sobol sequences, and to develop a MLQMC version of many of the test applications in my Acta paper (link). One goal of this work would be to make it easy to test alternative methods of constructing low-discrepancy points.

• Quantised Latin Hypercube control variate.

This is an idea someone else has suggested to me. Latin Hypercube is a standard variance reduction technique described in Glasserman's book. Quantised Latin Hypercube uses the central point in each "box", which introduces a bias, but avoids the need to simulate random numbers from possibly quite nasty distributions. The bias can then be corrected by using the quantised Latin Hypercube simulation as a control variate. Quantised QMC might also be interesting to investigate.

• Multilevel change of measure for complex digital options.

This change of measure approach is something I suggested in my MCQMC12 review paper on multilevel Monte Carlo methods, but I think no-one has yet tried it. It may be the best way to handle multi-dimensional digital options, and could also lead to an efficient way to compute their Greeks.

• Approximation of inverse CDF for Levy processes.

An inverse CDF approximation can be used to construct samples from a known distribution. This may be an efficient way of generating samples from certain Levy distributions. Given the characteristic function of the distribution, the aim would be to generate an accurate approximation of the inverse CDF, probably using high order polynomial approximations in some suitably chosen transformed coordinate. This might be a project that is more suitable for someone on the Mathematical Modelling and Scientific Computing MSc.

Here are some outline ideas for dissertation topics for the part-time MSc in Mathematical FInance:
• Multilevel Monte Carlo applied to copula?

This is a much more speculative project. I'm interested in new applications of multilevel Monte Carlo, and I wonder whether there are possibilities with copula methods to define a hierarchy of approximations and thereby construct an efficient multilevel Monte Carlo method.