Question: While I love research and academia and would prefer to continue there, I've found myself in industry and haven't felt it's a good match for my interests. Moreover, I'm constantly frustrated by the lack of opportunity to do math research. I'd like to find something I'm genuinely excited about (not to mention using all the mathematics I've spent years and years learning), but I'm not sure what exactly is available in industry for people with backgrounds and interests in pure math. Math research, for example, is not really something that industry does at all. (In fairness, there are a couple of small exceptions, like Microsoft Research and probably something over at Google, but the era of large, well-funded industry labs is over.) As far as I can tell, there are three options:

1. Work for a finance or insurance company doing some sort of mathematical modeling.
2. Work for some sort of government lab, probably in cryptography or a closely related field.
3. Work in something completely unrelated to mathematics, like software engineering.

Those options exist (and, in fact, I've had offers from each of them), but I'm not enthusiastic about them. Is there anything that's more compelling from a theoretical-mathematical standpoint?

Answer: 
There is an abundance of research at Aerospace firms, Trading Houses and Hedge Funds, Private Labs (such as Bell Labs), and lots of others that have R&D divisions that use mathematicians, physicists, and engineers, all of whom focus on applied mathematics research. 

Trading Houses and Hedge Funds absolutely do have research groups. Goldman Sachs (suggested read on [Emanuel Derman](https://en.wikipedia.org/wiki/Emanuel_Derman), the prominent particle physicist who became famous at GS and in the financial derivatives industry) and Morgan Stanley are only two examples of successful I-Banks that have huge research budgets in the areas of quantitative research for derivative products and algorithmic trading.

And before turning away from "applied math," the money can be phenomenally good - would you be more interested working at Goldman if your annual bonus was $\$500$K or more? I've known fresh PhD's to enter with a base salary at $\$100$K+, plus a first year bonus that is equal to a 1 or 2 multiple of the base. And that is 1st year!

Finally, while a career in R&D might not be ideal, it can offer both exciting challenges different from those of a career in pure mathematics, along with financial compensation that would be difficult to match as a university professor. 

On a personal note, I was a professor who left academics a long time ago. I have worked at both leading financial institutions and trading houses and did compromise the intellectual freedom for the financial security. It is a decision one makes and should be without future regret. 

I wish you the sincere best in whatever path you choose. And remember, after you make your first $\$10$ million or so, you can always return to academics. And in the interim, use your spare time to stay current in the research literature and have as much fun with math as you can!
You can always become a professor: get good pay, get grants from industry, and continue your focused research that you say keeps you excited in hopes of advancing your field, at least that's what your application will say. 

Companies are intersted in one thing: making money. Pure theoratical math rarely, if ever, can be used in a away to attain money. Thats why most jobs you find are in applied math (modelling, optimization, applied statistics or machine learning, applied number theory in security, etc). A company can use applied math and apply it to a product, then sell that for money. What do you imagine a company can sell with pure theory? And who will buy it?

Even big companies like Google, Intel, and Microsoft only suppory the kind of research that's likely to advance one of their products so they can make more money. I can assure you their executive couldn't be less enthusiastic about abstractness.

That's why it's important, when applying for a job, to think of what both sides want. A university, on the other hand, will be delighted to have someone doing pure theoratical work with no immediate or obvious way to derive money from it. And you get to be around like minded individuals who care as deeply about theory as you, whereas in industry, people might ridicule you for being too abstract. So being a full time professor is an option to seriously consider before leaving academia.