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Firstly, can I say that I hope this is the correct part of SE for this question! If it is not, then I will not be at all offended if someone transfers it to the correct place (providing that they are sure the migration is correct).

My question is as the rather long title:

What maths jobs, particularly but not exclusively in banking, use analysis / probability theory, not statistics?

I'm a third year mathematician at Cambridge, and I am most interested in (and am by far best at) analysis. I realise that, although that in the first year (IA) it is not, higher level probability is very much analysis based; in fact, the third year course Probability and Measure that I could take (but classes with another that I want to do) has only one prerequisite: Analysis II (the 2nd year (IB) analysis course). Further, I am not a statistician! (I didn't take IB Stats.)

I quite like the idea of working in banking, but I am not at all interested in doing statistics. I realise that there are jobs in banking that aren't, but I don't really know what they're called, or how to look them up! (I've spent quite a long time looking - I'm not just trying to use SE as a quick fix so that I don't have to research myself.)

I'm looking to do an internship this summer (I'm in the UK), then going on to Part III (if possible; requires a very high II:1 or a First...) and, possibly, finally doing an MPhil in Finance. (If I were not able to get into Part III, then I would do just the MPhil, but at a different university, for example Warwick or Bath.) Any help that could be given with regards to finding such internships would be enormously appreciated!

Best regards, Sam

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  • There's most likely a person attached to the major or the college who's job is arranging internships and job opportunities. Have you tried asking around there? (This might be on topic for Workplace.SE, but you might not get a lot of answers here) Oct 8 '14 at 23:08
  • @raptortech97 Yeah, there are quite a lot of people, but it's actually very difficult to get meetings, since there are a lot of students asking things! Even more difficult to get a specialist banker!
    – Sam OT
    Oct 9 '14 at 7:49
  • I think in most business environments the distinction between "mathematical analysis/probability theory" and "statistics" is like country vs western music. Saying you're "not at all interested" in working a particular closely-intertwined field in an internship position is going to make it hard to place yourself (as you may have noticed). Perhaps exposure to statistics in an internship context is a good thing? There's plenty of time for intense specialization later in your academic and professional career.
    – teego1967
    Oct 9 '14 at 12:33
  • Fair point - maybe I was being a bit overly harsh on stats. However there is a big mathematical difference between statistics and analysis. I realise that a banking job will probably comprise of both, but what I'm after is one that heavy favours analysis/probability. For example, not using Statistical computer packages - eg not credit score/ratings, which is very statistics based, but more hedge fund style (I believe).
    – Sam OT
    Oct 9 '14 at 21:44
  • Might this be better asked at quant.stackexchange.com? Oct 13 '14 at 20:28
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You might want to check out Phillip Guo's excellent post "Unicorn Jobs".

You may also get something from two different Math.SE answers I have written before that touch on this in various ways:

  1. "Is it worth it to do a statistics minor if you want to attend pure math grad school"
  2. "How to study math to really understand it and have a healthy life style with free time"

The gist, which I can offer anecdotal support of based on my working experience in quant finance, is that the kind of job you're describing does not exist outside of academia. There is not much economic value for purely theoretical work in modeling financial instruments, and there are more than enough capable academic mathematicians who can do short term consulting for those limited scenarios where the theoretical work really matters.

Additionally, you seem to have a bias against statisticians. Most of the statisticians I know are every bit as good at abstract mathematics as any of the academic mathematicians I know, some are even better, and most of those statisticians also know how to do reasonable software development in a few modern programming languages and are proficient with database technologies.

In advanced statistics, such as Markov Chain Monte Carlo methods, computational Bayesian methods, or machine learning, there is a significant amount of required theory (think things like Probably Approximately Correct learning, convergence of random variables, proving that a new hybrid monte carlo method will preserve detailed balance, and so on). You have to be pretty good at grad-level analysis, functional analysis, PDEs, and even algebra, to understand these text books and follow the proofs. Stochastic differential equations and stochastic processes are even harder as they build on measure-theoretic probability theory.

Let me pause to drive home the point. I am a pretty decent math person in regards to all of the topics I listed above. I spent 3 years in a PhD program before quitting with a master's degree to work full time. Despite the skill set I have in terms of math knowledge of all of those topics, overwhelmingly my primary employable skill set is knowledge of software architecture principles as they apply to efficient large-scale scientific computing in the Python programming language. And my second-most employable skills are related to a thorough understanding of database systems and performance tradeoffs between different database engines or NoSQL technologies.

After 7+ years of higher education in theoretical math and stats, the thing that investment research teams want to consume from me is just data infrastructure work -- making simple models and simple calculations run exceedingly fast on distributed architectures.

I never would have guessed this is what I would be doing for a job. In terms of my personal talents, I am way more skilled in machine learning methods than I am in Python scientific computing, though I am not bad at that. My comparative advantage is overwhelmingly in statistics, but companies seem to be overjoyed to allocate me inefficiently to work projects that do not use my comparative advantages.

At any rate, my conjecture is that you will not find any industry-facing job where your primary set of duties is to undertake formal mathematical modeling of financial concepts, and to output proofs of theorems or structural descriptions of models and their consequences. The people whose strengths are in applied statistics can already do those things, but are never asked to, and even if they were, their more compelling skill sets would be anything related to what actually touches the data.

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  • I've actually used analysis in theoretical modeling of a physical instrument which takes measurements with light. I've done this in both an academic and an industrial setting, so there are industry jobs available which can use analysis. Specifically, analysis applied to stochastic estimation and machine learning are probably the largest areas...
    – daaxix
    Oct 11 '14 at 3:35
  • Thank you for your comprehensive answer. Just to clarify, I have nothing whatsoever again statisticians, I just don't like the topic statistics. I'm taking a second Numerical Analysis course, which has quite a lot on making calculations run faster. Very interesting to see what you said about what employers "use you for", and that it isn't what you are actually best at. Thank you for sharing.
    – Sam OT
    Oct 11 '14 at 7:31
  • @daaxix - could you perhaps expand on your comment, specifically about which industry jobs use analysis? Perhaps do so in an answer, so that it's not liked to EMS's question? Thanks!
    – Sam OT
    Oct 11 '14 at 7:35
  • Actually @daaxix makes a decent point. I used to work in a government defense research lab, and especially with regard to quantum optics, we did use theoretical modeling some. However, even then, it usually represented maybe the first 5% or 10% of a project, and once we were done with the modeling, the rest was implementation and data issues. We might occasionally revisit the theory part if the data showed us something counterintuitive. In this way, it was a lot like a graduate lab. You might find similar things with cyber-security and cryptography. The theory was still just a small part.
    – ely
    Oct 11 '14 at 14:03
  • EMS, I agree that the theory portion was smaller than the implementation details, probably 30-40% of my time. @SmileySam, I work in imaging science, imaging operators, and polarization of light. The use of analysis is mostly applying known results, or tweaking known results slightly, so actually doing new mathematics is unlikely in industry...
    – daaxix
    Oct 11 '14 at 15:14
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I am not in the industry so I could be mistaken, however I have spoken to a few investment banks such as Goldman. When I have asked about roles which lean most heavily on anaylsis they have always just called them quants (quantitative analyst). The issue is that someone who does advanced statistics is still called a quant.

As a side note, whilst functional analysis, measure theory and spectral theory does give you insights into solving PDE's no bank is going to ask you to reproduce a proof of Hahn–Banach theorem. You may love pure mathematics but if you want to hit the ground running at a bank I would suggest you at least practice some applied mathematics. Many of the quants I have spoken to have said they just want smart mathematicians who can program, as many of their models are too complex to solve analytically.

More broadly I think you will have a hard time finding any industry job which wants you to work on pure mathematics, if this is your love perhaps consider academia. Or something like GCHQ if your interested in algebra, not really analysis. If you willing to do applied mathematics, actually solving PDEs, computationally model systems, statistics then really there are tons of industries that would want you, banks, oil and gas, even any major company will have a business intelligence team.

I would strongly recommend you take an intern-ship, and it would be a good idea to go talk to Cambridge careers centre as early as possible. They should be able to give you advice on how and who to apply to. For example if you want to intern at an investment bank it's probably a good idea to join some kind of Cambridge financial club to prove you have an active interest. As well as reading about the latest mergers and financial news on top of learning the industry jargon.

http://www.careers.cam.ac.uk/

Getting a job as a mathematician isn't as straight forward as it is for graduates of professional degrees. On the plus side the range of industries and opportunities is much wider, and a mathematics degree from Cambridge is well respected.

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  • Thanks for that comprehensive, excellent response! :) I appreciate that doing some of the more pure stuff isn't of any actual value in an industry directly. (I'm not saying it's not useful, but not in that way.) I do some applied maths, such as PDE solving (but again in a reasonably rigorous mixed pure/applied way) and am on my second Numerical Analysis course, along with computing projects. The thing is that my programming isn't excellent - it's ok, completely passable, but not excellent.
    – Sam OT
    Oct 10 '14 at 19:45
  • I have considered academia, but I would also like to do something practical. With regards to the careers service, I have been to them a couple of times this term, but they're very busy and it's difficult to get appointments! (I have only been able to get 10 minutes on this topic with them so far!).
    – Sam OT
    Oct 10 '14 at 19:46
  • I'm interested in modelling stuff and actually solving it, but not in just getting loads of data and shoving it into a stats program (as well as doing my own statistical analysis of course!). What do you mean by a "business intelligence team"?
    – Sam OT
    Oct 10 '14 at 19:47
  • I echo your sentiments about not wanting to feed data into a black box. As for BI, different companies may give it a different name, basicly they are the companies resident mathematician, although in many cases it just means statistician (depends on industry, company ect). So they analyse data, whether that be internal performance metrics, or consumer data, make sense of it make predictions and report to management to help guide their decisions. My very limited perspective is that you would need to choice very very carefully to end up doing anything interesting.
    – User
    Oct 10 '14 at 20:06
  • @SmileySam, If for what ever reason you don't do Part III, you could always consider Oxford's Industrial Mathematics graduate program.
    – User
    Oct 10 '14 at 20:14

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