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:
- "Is it worth it to do a statistics minor if you want to attend pure math grad school"
- "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.