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The quants of Renaissance Technolo­gies are unusual in that many might have enjoyed significant careers in academia. But quants of a less exalted sort are becoming ubiquitous at financial institutions. There are quants at investment banks, developing new loan structures. There are quants at hedge funds, crunching years of market data to develop trading algorithms that computers execute in milliseconds. And there are more and more quants at pension funds, trying to understand and value the tools created by the banking quants, and trying to evaluate the methods of the investing quants.

“We used to send our graduates mainly to the big banks,” says Andrew Lo, the director of MIT’s Laboratory for Financial Engineering, where many quants are trained. “Now they’re going everywhere, to pension funds, insurance companies, and companies that aren’t finance companies at all.” MIT’s lab was founded in 1992, one of a host of academic programs in the discipline that have sprung up on campuses around the United States and abroad; a new institute at the University of Oxford is one of the most recent additions. “Financial markets and investment processes are becoming more quant across the board,” says Lo.

To understand who they were and what they were doing, I spoke with current and former quants, on and off the record. Many would speak happily and at length. Others spoke guardedly or anonymously–especially those using proprietary analysis and algorithms to conduct trades. I read memoirs of quants–a recently expanding genre–and dipped into an introductory textbook for quants, Paul ­Wilmott Introduces Quantitative Finance, a 722-page condensation of the author’s 1,500-page, three-volume anvil of a book, Paul Wilmott on Quantitative Finance. And I went to a quant drinking party, which convened in the basement of a pub next to Grand Central Station. The name of that event proves, as much as anything, that the quants have geek in their veins: it was the August meeting of the New York chapter of the Quantitative Work Alliance for Applied Finance, Education, and Wisdom, or QWAFAFEW.

Though derivatives were simpler once, they were never very simple. The breakthrough in the valuation of derivatives in general, and options in particular, was the model and formula know as Black-Scholes, first proposed by Fischer Black and Myron Scholes in the 1970s and formalized by Robert Merton in 1973. (Merton, like so many of the best quants, came not out of Wall Street but out of aca­demia, earning a PhD in economics from MIT in 1970.)

In quantitative finance, the formal expression of Black-Scholes by Robert Merton is so important that everything that followed has been called a “footnote.” The Black-­Scholes model assumes that a stock’s price changes partly for predictable reasons and partly because of random events; the random element is called the stock’s “volatility.” The idea can be represented mathematically by a simple equation:

St is the value of the stock, and dSt is the change in stock price. The symbol µStdt represents the stock’s predictable change and its volatility. (View the results of Black-Scholes model using this interactive calculator.) That final, kabbalistic combination of letters, dWt, is the mathematical expression for randomness, known as either Brownian motion or the ­Wiener process. (Chemically, Brownian motion is the random movement of particles in solution, identified by the botanist Robert Brown in 1828 and mathematically described by the great MIT mathematician Norbert Wiener. Black-Scholes shares some qualities with heat and diffusion equations, which describe everyday events like the flow of heat and the dispersion of populations. That some physical processes seem relevant to finance has inspired all kinds of far-out work, such as efforts to bend general relativity to a theory of finance.) Black-Scholes prices an option according to the amount of randomness in a stock’s price; the greater the randomness, the higher the stock could climb, and thus the more expensive the option.

Quants have since refined Black-Scholes, and with the increasing power of computers, they have developed other, more processing-intensive methods of valuing derivatives. In Monte Carlo simulations, for instance, powerful computers model the performance of a stock millions of times and then average the results. Where Black-Scholes, as a mathematical shortcut, assigns a constant value to a stock’s volatility, Monte Carlo simulations vary the volatility itself. In theory, this provides a better approximation of price fluctuations in the real world. And quants have devised yet more arcane methods of derivatives pricing. Some particularly complicated models track other economic factors–like the stock market as a whole, or even larger macroeconomic factors–in addition to a stock’s price.

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