Ideas for MSc projects
Here are some outline ideas for projects for the fulltime MSc in Mathematical and Computational Finance:

Comparison of MLMC, ML2R (PagesLemaire) and randomised MLMC (RheeGlynn);

Comparison of Sobol with "simple" digital scrambling versus
MatousekOwen scrambling;

Multilevel QMC
In the past, I did some work with Ben Waterhouse on combining multilevel Monte Carlo with
QMC methods, using rank1 lattices (PDF).
It would be interesting to revisit 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
lowdiscrepancy 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 noone has
yet tried it. It may be the best way to handle multidimensional
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 parttime 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.

Computing Greeks using adjoints.
I'm always interested in new uses of adjoints for computing Greeks. The key is to
identify applications in which lots of first order sensitivities are needed  if you have
that need, then adjoints are usually the most efficient way to compute the sensitivities,
although discontinuous payoffs can pose a problem.