Summary
This post is a summary of a talk I gave at the University of Wrocław about what it’s actually like for a mathematician to work in finance. We went over the different roles you can land, what skills you actually need (beyond just knowing your way around a proof), and how to navigate the industry.
The Financial Ecosystem
When you first look at financial markets, they can seem like a chaotic mess of numbers, shouting (well, maybe less shouting now that everything is electronic), and complex jargon. But at their core, markets are just ecosystems where capital and risk are being traded. It’s a place where people who have money but no projects meet people who have projects but no money.
The characters you meet on the trading floor generally fall into a few buckets. You have the Speculators, who are the ones taking on risk because they think they can predict where the price is going. They provide the liquidity that makes the market move. Then you have Hedgers—these are the “insurance seekers” who already have a risk (like a farmer worried about wheat prices) and want to get rid of it.
In between them, you find the Arbitrageurs. These are the “market cleaners” who look for tiny inconsistencies in prices between different places. If gold is cheaper in London than in New York, they’ll buy in one and sell in the other until the prices align. Finally, you have the Intermediaries (like brokers and market makers) who just want to make the trade happen and pocket a small fee for the service.
For a mathematician, this is a playground. Why? Because every single one of these interactions involves uncertainty, and if there’s one thing mathematicians are good at, it’s quantifying the unknown.
Why Do They Need Us?
You might wonder why a bank would hire someone who spent four years studying non-local unbounded operators. The reason is that modern finance is built on models that are essentially massive math problems.
Take pricing, for example. If you want to sell someone an option—the right to buy a stock at a certain price in six months—you need to know what that right is worth today. That involves stochastic calculus, partial differential equations, and a lot of numerical methods. It’s not just “guessing”; it’s solving the Heat Equation in a financial context.
Then there’s Risk Management. After the 2008 crisis, “winging it” stopped being an option. Banks need to know, with a high degree of mathematical certainty, how much they could lose if the market takes a dive. This requires sophisticated probability theory and statistics to model “fat tails”—those rare but catastrophic events that standard models often miss.
Beyond pricing and risk, we’re heavily involved in Algorithmic Trading, where we use optimization and signal processing to execute trades in milliseconds, and Portfolio Optimization, where we use linear algebra to find the best possible balance between return and risk.
The Many Flavors of Quants
Not all “Quants” do the same thing. Depending on your interests, you might find yourself in very different environments. Some roles are all about the “hard math” of derivatives, while others are more about being a software engineer who happens to know what a Yield Curve is.
Develops and applies quantitative models for pricing, risk management, and trading strategies directly on the trading desk.
Develops and validates models to measure and manage market risk, ensuring the bank stays within its limits and regulatory requirements.
Focuses on the risk of people or companies defaulting on their debts. Heavy on probability and statistical modeling.
The ‘internal police.’ They independently check the models built by other quants to make sure they aren’t broken or dangerous.
The ‘think tank’ role. Conducts cutting-edge research to find new trading signals or better ways to price assets.
Writes the code that actually executes trades. Needs to understand how markets move on a second-by-second basis.
The backbone of the operation. They build the high-performance systems that the math runs on.
The Reality of the Daily Grind
Life as a quant isn’t just solving theorems on a whiteboard. A huge chunk of your day is actually spent coding and testing. You might have a great idea for a model, but if you can’t implement it efficiently or prove that it works on historical data, it’s useless. You need to be comfortable with “messy” data—the kind that has holes, errors, and outliers that would make a pure mathematician cry.
There’s also a big communication aspect. You’ll often work with traders or managers who are brilliant at their jobs but don’t know a Taylor series from a Taylor Swift song. Being able to explain why a model is giving a certain result, without using Greek letters, is a superpower.
Finally, you have to deal with pressure and regulations. In some roles, if your code has a bug, you could lose millions of dollars in minutes. In others, you might spend weeks writing documentation for regulators to prove that your model isn’t going to blow up the economy. It’s a job that requires extreme attention to detail and a thick skin.
Where to Look for Work?
If you’re starting out, you have a few options depending on where you want to live.
In Wrocław, we have a surprisingly strong quant scene with players like UBS, BNY Mellon, and Santander. If you look at Poland as a whole, the list expands to Goldman Sachs, HSBC, and Revolut in Warsaw. Of course, the global hubs remain London, New York, and Hong Kong, but the barrier to entry has lowered significantly with the rise of remote and hybrid work.
Final Thoughts
The transition from academia to finance can be jarring, but it’s incredibly rewarding if you like seeing your math have a direct, tangible impact on the world. If you’re interested in pursuing this path, my best advice is to get comfortable with Python or C++, learn the basics of probability, and start reading up on the “classics” of the field.
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