Hi everyone again,Last time I posted on this forum was over a year ago to get opinions on where should I study an MSc in Mathematical finance in the UK. Well that is over now. I really enjoyed the subject (specially stochastic calculus) and I'm ready to take the next step, which is doing a PhD preferably in London. That narrows it down pretty much to ICL, UCL, Kings College and the LSE.Thus far, I have contacted a couple of supervisors at UCL and ICL showing interest for their areas of research and recent papers and whether it would be possible to join their pools of PhD students, but I hace received no response from either. Since my grades are not a problem, this leads me to believe that potential supervisors do not give attention to students without a clear well-structured research proposal.I am ultimately interested in doing work relevant to the quest of finding a theory of early exercise of american options. However, I find quite daunting the mathematical rigor and complexity of some of the branches in the mathematics involved (advanced aspects in measure theory, ergodic theory, topology...), and so I have decided to also contemplate other options. These could be research in stochastic control, interest rate modelling, stochastic analysis, or advanced methods for derivatives hedging (possibly involving models with jumps, in line with my MSc dissertation). I would like to point out that I do not find very attractive the prospect of devoting a large part of my thesis to developing or improving numerical methods, but I'm not ruling that out completely.When looking at the research interests of faculty members in London, I've found very difficult to make out from their research work what recent developments in the aforementioned areas actually are, and I am starting to feel a bit scared that it will take too much time trying to find out on my own without help from someone in academia. This all brings me to the following question. Could somebody please give me some intuition on where research is going towards in the above mentioned areas, and some literature that I could read to get a better idea of what is worth researching about at the moment?Thanks before hand for all the help and suggestions. I am interested to hear researchers' and industry professionals' take on this.

QuoteI am ultimately interested in doing work relevant to the quest of finding a theory of early exercise of american options. However, I find quite daunting the mathematical rigor and complexity of some of the branches in the mathematics involved (advanced aspects in measure theory, ergodic theory, topology...), and so I have decided to also contemplate other options. These could be research in stochastic control, interest rate modelling, stochastic analysis, or advanced methods for derivatives hedging (possibly involving models with jumps, in line with my MSc dissertation).In general, it is better if you can propose a subject yourself. It shows you have research potential. (for the record, I supervised 10 MScs last year and others in previous years and most proposed their own topics). I suppose an academic needs to be assured that you can work on your own because they will have very little time for hand-holding.BTW measure theory, topology for American options? How do you envisage that? Sounds kinda non-computable. IMO it is better to propose a concrete feasible project and build on it.Quote I would like to point out that I do not find very attractive the prospect of devoting a large part of my thesis to developing or improving numerical methods, but I'm not ruling that out completely.Are you going to it all on paper?

Thanks for your answer Cuchulainn. What I meant is that I do not want my thesis to be just entirely about developing a numerical scheme or finding new ways to improve Monte Carlo. I want the core of my research to be rather theoretical.

QuoteOriginally posted by: mekornilolThanks for your answer Cuchulainn. What I meant is that I do not want my thesis to be just entirely about developing a numerical scheme or finding new ways to improve Monte Carlo. I want the core of my research to be rather theoretical.You're welcome. I see. Why not just do a PhD in Mathematics?IMO buckets more theory is known in this area compared to computational aspects..Last year someone did a project on multi-level MC. Just an idea. arxiv.org/pdf/1304.5472

Numerical methods and Monte Carlo can also be theoretical. When you develop a new improved methodology, you need theory to justify it's valid rather than just present some numerical results.

QuoteOriginally posted by: londonerNumerical methods and Monte Carlo can also be theoretical. When you develop a new improved methodology, you need theory to justify it's valid rather than just present some numerical results.Indeed.And Numerical Analysis != Numerical Methods.

The challenge of modelling interest rates in the current environment seems the obvious choice.

QuoteOriginally posted by: mekornilolHi everyone again,Thus far, I have contacted a couple of supervisors at UCL and ICL showing interest for their areas of research and recent papers and whether it would be possible to join their pools of PhD students, but I hace received no response from either. Since my grades are not a problem, this leads me to believe that potential supervisors do not give attention to students without a clear well-structured research proposal.QuoteA wild guess is that part of the problem was the quality of the remarks that "show[ed] interest". You would need to impress them with your understanding of the topics. You can do this by reading a paper by someone who you would like to work with, and then asking questions that demonstrate your understanding.CommodityQuant

QuoteOriginally posted by: mekornilolHThus far, I have contacted a couple of supervisors at UCL and ICL showing interest for their areas of research and recent papers and whether it would be possible to join their pools of PhD students, but I hace received no response from either. Since my grades are not a problem, this leads me to believe that potential supervisors do not give attention to students without a clear well-structured research proposal.Those people probably receive dozens of emails like yours every year. What makes you stand out?

QuoteOriginally posted by: Marco72QuoteOriginally posted by: mekornilolHThus far, I have contacted a couple of supervisors at UCL and ICL showing interest for their areas of research and recent papers and whether it would be possible to join their pools of PhD students, but I hace received no response from either. Since my grades are not a problem, this leads me to believe that potential supervisors do not give attention to students without a clear well-structured research proposal.Those people probably receive dozens of emails like yours every year. What makes you stand out?As much as I'm interested in everyone's opinion, I have not asked "why haven't they replied to me?", I already know the answer to that question. My question was "what are recent developments in mathematical finance?". Thanks MattF, Cuchulainn and Londoner for their answers.

QuoteOriginally posted by: mekornilolQuoteOriginally posted by: Marco72QuoteOriginally posted by: mekornilolHThus far, I have contacted a couple of supervisors at UCL and ICL showing interest for their areas of research and recent papers and whether it would be possible to join their pools of PhD students, but I hace received no response from either. Since my grades are not a problem, this leads me to believe that potential supervisors do not give attention to students without a clear well-structured research proposal.Those people probably receive dozens of emails like yours every year. What makes you stand out?As much as I'm interested in everyone's opinion, I have not asked "why haven't they replied to me?", I already know the answer to that question. My question was "what are recent developments in mathematical finance?". Thanks MattF, Cuchulainn and Londoner for their answers.I think the reason you haven't received a lot of useful answers is that your question is wildly open-ended. I am involved in editing a math finance journal, I teach something vaguely related at a master's level, and I have been a moderately successful industry quant for 20+ years, and I couldn't even begin formulating a coherent answer. If you ask something like "what are recent developments in the analysis of American options?" you may actually find somebody here who can respond. Depending on their focus and interest, that could range anywhere from sharper Monte Carlo valuation bounds in high dimensions to utility maximization problems that arise when trying to determine an optimal exercise policy for a large option position on an asset whose price is subject to market impact. Or, "what are interesting derivatives pricing applications of BSDEs and G-expectations?", which I think is a relatively hot topic still, at least if you ask Dilip Madan (a recent editor-in-chief of Mathematical Finance). At a deeper level, I think your quest illustrates the difference between the European (or at least UK) doctoral programs and those in the US. It is fair to say that the two main purposes of US PhD programs are to develop the skills and knowledge base to do research and to help the student formulate one or more research ideas suitable for a thesis. Carrying out the actual research and writing up the thesis are obviously also important, but these are often done fairly independently in a field like ours. A typical MSc program in math finance does not, in my experience, expose the students to much in the way of cutting edge research, and it seems awfully optimistic to imagine that one is ready to formulate a research proposal with only this background.

QuoteOriginally posted by: bearishAt a deeper level, I think your quest illustrates the difference between the European (or at least UK) doctoral programs and those in the US. It is fair to say that the two main purposes of US PhD programs are to develop the skills and knowledge base to do research and to help the student formulate one or more research ideas suitable for a thesis. In theory, those are also the aims of a UK PhD. In practice, I have seen people on both sides of the pond who ended up doing nothing but coding for a few years.

mekornilol,A general structure for one type of dissertation is "Applications of [ .....] to Mathematical Finance". Based upon your expressed interests (American options, models with jumps), perhaps the blank might be filled by "Variational Methods/Weak Solutions". I suggest picking up a copy of Evans "Partial Differential Equations" studying the variational/weak solution stuff, andthen considering how to generalize it to boundary value/evolution problems with Levy-type operators, esp. multi-dimensional affine jump-diffusion types. I don't keep up too much with the journals, but one can rely on a general trend toward greater abstraction, operator methods, andmulti-dimensional problems. So, I am confident there are many Ph.D-worthy subtopics within this general area.

Popular research topics include risk measures, conic finance, non-linear finance (illiquid markets, frictions, etc), fast calculation of portfolio Counterparty pricing/risk, and G-Expectations applied to risk/pricing. It's best to look at the professors recent articles and see if they are involved in anything that interests you.