Now, after pursuing various jobs and projects, I wanted to find something to do which was engaging, paid well, and had at least some mathematical side to it.

What I found was the field of arbitrage trading. Arbitrage, pronounced "AR-bi-trazh," means executing more than one trade such that 1) there is little or no risk, and 2) there is some positive value.

For example, suppose that at a farmer's market, three apples could be swapped for two bananas, three bananas could be swapped for two peaches, and three peaches could be swapped for seven apples. Then I could take twenty-seven apples, turn them into eighteen bananas, turn the bananas into twelve peaches, and turn the peaches into twenty-eight apples.

After all is said and done, I've earned an extra apple for my efforts, by taking advantage of the fact that the swap rates between the different fruits aren't exactly right---the market has a slight anomaly. Until the farmers adjust their swap rates, I can execute my series of trades over and over again, netting an extra apple each time.

Of course, in the real markets, anomalies as simple as this rarely occur, and are quickly eradicated when they do. Intense competition from arbitrage traders forces all securities to stay very close to fair valuations: the market is what economists would call efficient.

Trading takes a lot of different forms. At first, I was standing in a "pit" on the floor of the Chicago Board Options Exchange, wearing a brightly colored jacket and waving my arms to get a broker's attention. Later, I sat in front of a bank of computer monitors, directing split-second trades over the phone. In both cases, however, the underlying idea was he same: to exploit the mathematical relationships between different stocks or bonds or options or stock indexes.

That isn't to say it's all about the math. In many trading situations, the mathematical relationships are well-understood, and what matters are things like speed, attention to detail, and gamesmanship. From moment to moment, this type of trading closer resembles a race-to-the-buzzer game show than a math test.

Other types of trading are more cerebral, focused on extensive analysis to extract the relationships underlying security prices. The pricing of options---securities which grant the holder the right, if so desired, to buy a stock at a specific price at a specific price---has inspired a wealth of mathematical research. The 1997 Nobel Prize went to two economists who pioneered a key options pricing model based on partial differential equations. And the field has expanded enormously since their 1973 work, with new models continuing to proliferate in answer to ever-deeper questions. Further afield, areas like bond analysis rest on a deep mathematical foundation.

Traders aren't mathematicians, however. Rather, they are skilled users of mathematical models, often developed by quantitative researchers at the trader's firm. However elaborate, models always fail to capture the human side of the equation - much as poker-playing software can calculate probabilities, but can't infer an opponent's weak hand from her demeanor.

For most types of trading, a solid math background at the college level is crucial. Coursework in finance and economics is also useful. For those more inclined to quantitative research rather than trading, graduate study in math or science may be required.

Trading is in many cases a very competitive, intense enterprise. Those who like games and don't mind stress will often excel. However, building relationships is also a key component, and people skills are a must.

There are a lot of books that can help you get a sense of the culture of trading and finance. A few I'd recommend are Peter Bernstein's Against the Gods, Roger Lowenstein's When Genius Failed, and Burton Malkiel's A Random Walk Down Wall Street.

If you're seriously interested in trading - or finance more broadly - talk to as many people close to the industry as you can. Opportunities range from large investment banks (most of which have large trading operations) to small "boutique" operations with fewer than 100 employees. Use of headhunters is also an important way to gain exposure to Wall Street firms.