Recently some universities have opened “quant schools,” programs that educate M.B.A. or other master’s students in the higher applied mathematics of finance, the subtleties of Ito’s lemma and other cornerstones of stochastic calculus. Or else they may train physicists, engineers and mathematicians before moving on to Wall Street. “Market pressures are directing physicists to get more education to try to understand the motivation and intuition underlying financial problems,” says Andrew W. Lo, who heads the track in financial engineering at the Massachusetts Institute of Technology’s Sloan School of Management.
As part of their studies, financial engineers in training learn about the progression of mathematical modeling beyond the original work of Black, Scholes and Merton. The basic Black-Scholes formula made unrealistic assumptions about how the market operates. It takes a fixed interest rate as an input, but of course interest rates change, and that influences the value of an option—particularly an option on a bond. The formula also assumes that changes in the growth rate of stock prices fall into a normal statistical distribution, a bell curve in which events cluster around the mean. Thus, it fails to take into account extraordinary events such as the 1929 or 1987 stock market crashes. Black, Scholes and Merton—and legions of quants—have spent the ensuing years refining many of the original ideas.
Emanuel Derman, head of the quantitative strategies group at Goldman Sachs, is a physicist-turned-quant whose job over the past 13 years has been to tackle the imperfections of the Black- Scholes equation. Derman, a native of Cape Town, South Africa, received his doctorate from Columbia University in 1973 for a thesis on the weak interaction among subatomic particles. He went on to postdoctoral positions, including study of neutrino scattering at the University of Pennsylvania and charmed quark production at the University of Oxford’s department of theoretical physics. In the late 1970s Derman decided to leave academia: “Physics is lonely work. It’s a real meritocracy. In physics, you sometimes feel like you’re either [Richard] Feynman or you’re nobody. I liked physics, but maybe I wasn’t as good as I might have been.”
So in 1980 he went to Bell Laboratories in New Jersey, where he worked on a computer language tailored for finance. In 1985 Goldman Sachs hired him to develop methods of modeling interest rates. He has worked there since, except for a year spent at Salomon Brothers. At Goldman, he met the recently recruited Fischer Black, and the two began working with another colleague, William W. Toy, on a method of valuing bond options. Derman remembers Black as a bluntly truthful man with punctilious writing habits who wore a Casio Data Bank watch. “Black was less powerful mathematically than he was intuitively,” Derman says. “But he always had an idea of what the right answer was.”
Physics Versus Finance
Much of Derman’s recent work on the expected volatility of stock prices continues to refine the original 1973 paper. The Black-Scholes equation was to finance what Newtonian mechanics was to physics, Derman asserts. “Black-Scholes is sort of the foundation on which the field rests. Nobody knows what to do next except extend it.” But the field, he fears, may never succeed in producing its own Einstein—or some unified financial theory of everything. Finance differs from physics in that no mathematical model can capture the multitude of ever mutating economic factors that cause major market perturbations— the recent Asian collapse, for instance. “In physics, you’re playing against God; in finance, you’re playing against people,” Derman declares.