No given job is stable, and that applies from the head of Merrill Lynch to the person who who cleans his floor.
To misquote Woody Allen, "dying doesn't scare me, it's staying dead, that gives me the shits".
Static stability is not only unattainable, it is often unpleasant in the long term, since you may feel trapped and/or stale. The single biggest clump I know of such people is Britain's witless semi-monopoly Telco, British Telecom. It pays these jokers more than they are worth, so they can't get proper jobs since it would hurt financially.
Imagine a truly huge firm staffed 100% by people with no interest other than keeping their jobs, it's not pretty. Those unfortunate enough to fly British Airways will see the same thing, where their naked contempt for customers who fly with them is caused by their own frustration. Every BT/BA employee I have encountered has struck me as in borderline clinical depression. Not always dumb, but always wasted.
Whay you need is dynamic stability. The ability to get another job when this one dies.
This not only will improve your risk adjusted return, but will actually make you happier. I rarely say "X will make you happier", because my job as a headhunter is to find out what you want, and try to get it. It's not my place to tell you what to like.
But in this case I observe that optionality has profound value in your morale, even if you don't really want to leave. A good analogy is that you you pick up a very attractive person of a compatible sex and go back to their room. As you are getting undressed, you hear the door being locked. Your utility may droop noticeably at this point.
One way you can extract utility from the pestering calls of headhunters is to ask them about what variants on your skills would make you more marketable. This is a tragically noisy signal, but is usually honest since HHs want people they can sell. You should not panic the first time you hear "there's not so much call for pricing Nigerian options in Delphi these days", but if the signal repeats, time to think about diversification.
In the Guide, I make a big thing of using your financial maths skills to manage your career. There is an inherent trade off between risk and return. Specialisation helps you get a better return, but when that niche goes bad, it can hurt. I know this from personal experience, indeed one major reason I feel qualified to offer career advice is that I can help people avoid making the mistakes I have made. Some are so bad, I don't put them on this forum so I can use them in after dinner speeches, and my forthcoming novel.
However, risk is not a monotonic function of specialisation, at any point in time for a given person there is an optimum. This is because someone who is generally average across a broad front risks coming 2nd in lots of interviews, but first in none. You need a few sexy specialist skills to get any given decent job.
Networking is also useful, but at more levels than one might at first think.
Yes, some people have good good jobs through the people they know, but more critically you can model how the market for your skills is evolving. You can identify "low hanging fruit", skills you don't have that are congruent with those you already have.
It's often necessary to incur debt to buy into this line of work, and that's usually rational. I counsel you however to keep your debts down, because it reduces the set of options you can choose between.
It was a huge jump for me to become a headhunter, and I could take that risk because I had no debt at all. That's bigger than is rational for most people most of the timed, but recall your Real option theory when looking at the effect of debt you incur.
There are also "insurance skills". These are things people will at least pay you something if it all goes tits up. These are of course the equivalent of low risk/low return assets in your personal portfolio. Excel/VBA is one of these, but you should ensure you have a couple. There is always work somewhere for an Excel jockey, and it may be your way back into banking if you fall out.