## Financial Mathematics---From the Author's Last Feature

**April 16, 2000**

"Although computer networks have had an obvious effect on how many exchanges operate---allowing the physical pits to be replaced by a cheaper virtual trading floor---the brute mathematical strength of computers has allowed a second, less obvious, revolution to sweep the world of finance.

"In much the same way that engineers simulate the flow of air over a jet's wing before building the airplane, traders now simulate the behavior of financial instruments before buying or selling them, in particular the class of instruments called derivatives.

" 'The whole industry really exists because of mathematics. The idea that you can model derivatives, not only to value them, but also to synthesize them, means that you can hedge your risks,' said Claude Greengard, manager of IBM Corp.'s worldwide insurance research center [and SIAM's vice president for industry]. . . .

"The [Black/Scholes/Merton] theory has since been extended to develop mathematical formulas, or models, for an enormous variety of derivatives.

"Today, those models are not used solely to price derivatives. Said Greengard at IBM, 'It's not just the valuation, but the hedging that is the key.'

"Because the models allow traders to predict how the value of derivatives will respond to market fluctuations, it becomes possible to hedge one derivative by purchasing a different derivative that will respond in the opposite manner to a market change. In theory, when one derivative goes up in value, the other goes down.

"It is even possible to sell derivatives with very complicated payment conditions over the counter and yet, using the appropriate model, hedge the risk with a collection of exchange-traded vanilla derivatives.

"This strategy is widely employed by investment banks, which profit by selling the complex derivative at a premium price, but mitigate any risks by careful hedging with vanillas, according to Robert Almgren, a senior lecturer in the department of mathematics at the University of Chicago. 'They're sort of hedging apples with oranges.' . . .

"Furthermore, despite advances in pricing and hedging theory, the models are unlikely to ever predict the value of derivatives with the same accuracy that physics can predict the flight of an aircraft.

"Said Almgren, 'You're working in a world where, as soon as you figure something out, people realize that you've figured it out and its behavior changes.'"---*From "Trading on an Idea," which appeared under Ian Mitchell's byline on October 4, 1999. © 1999 by* The Chicago Tribune*.*