Computational Finance: Carnegie Mellon’s Place in History
Computational finance has a fascinating pedigree, and Carnegie Mellon is in its bloodline. To large extent, the field relates to game theory, in which players try to maximize returns based on different strategies. Nobel laureate and Carnegie Mellon alumnus John Nash (B.S, M.S. ’48) invented the Nash equilibrium, which describes the balance between strategies and payoffs in a game where no player benefits if he changes his position unilaterally — one without other players changing positions, too.
Historic models developed in computational finance irrevocably alter society. A case in point: Black, Morton and Scholes. Robert Morton, who first introduced stochastic calculus into the study of finance, teamed with Fischer Black and Myron Scholes to develop their celebrated formula, called the Black-Scholes option pricing formula. It revolutionized the finance industry because it initiated the development of a host of new securities that could be used for managing risk and for investment purposes.
“Their formula provided a satisfying solution for an important practical problem — that of finding a fair price for the right to buy one share of a given stock at a specified price and time,” says Steve Shreve, the Orion Hoch Professor of Mathematical Sciences. “These ‘European call options’ are frequently purchased by investors as a risk-hedging device.”
The Black-Scholes option pricing formula, while remarkable, didn’t account for interest rate fluctuations that affect the securities market. In 1992, Dave Heath, working with collaborators outside Carnegie Mellon, tackled this problem and facilitated vastly better modeling that transformed financial operations. Heath retired from Carnegie Mellon in spring 2006 after spending more than a decade educating students and building relationships between the university and industry.
While computational finance is at the end about problem solving, it also is a rich area of pure mathematical research, according to Dmitry Kramkov, professor of mathematical sciences.
“In the theoretical mathematical finance community, we want axioms to say as much as possible. We want to obtain the same results as our more practically oriented colleagues, but with fewer assumptions,” he maintains, all the while noting that the drive to explain more complex activities with fewer equations ultimately yields results with profound practical applications. “For instance, the popular Heath-Jarrow-Morton methodology for interest rate models was built on the basis of pure mathematical results, namely, first and second fundamental theorems of asset pricing.”
At his desk, Kramkov wrestles with mathematics that he hopes will increase the efficiency of derivates markets. This work ultimately would enhance the flow of capital to productive sectors of the economy.
Shreve, Kramkov and other faculty at Carnegie Mellon within Mathematical Sciences and the Tepper School of Business continue to advance computational finance. In fact, Shreve is president-elect of the Bachelier Finance Society, the world’s leading professional society for quantitative finance. Together, Shreve, Kramkov and Heath have been investigators on projects to conduct research in the mathematics of financial risk management. These projects have received more than $1.7 million in funding from the National Science Foundation.
When he came to Carnegie Mellon in 1980, Shreve teamed with John Lehoczky, dean of the College of Humanities and Social Sciences, to apply his work in process engineering to financial problems. Together, they built on the work of Nobel Laureate Merton to show how to form optimal portfolios of stocks in an unstable, unpredictable climate.
Shreve’s work in quantitative finance has established new methods for pricing exotic derivative securities, including knock-out options, a type of derivative security whose payoff can see a large jump as a result of a small change in the underlying asset price. Shreve has contributed to the basic understanding of convertible bond prices and dealt with mathematical models of markets in which financial risk cannot be hedged using available securities.
Shreve and Lehoczky’s collaboration proved fruitful both for computational finance research and for Carnegie Mellon’s educational initiatives, which in the past two decades have grown substantially.
“Steve has guided the development of a complete curriculum in computational finance, including innovative bachelor’s, master’s and doctor’s degrees,” said Roy Nicolaides, Alexander M. Knaster Professor and Head of the Department of Mathematical Sciences head of the Department of Mathematical Sciences. “In addition to this singular achievement, he is arguably among the top 10 leaders in his field.”
The growth of computational finance adds luster to a Department of Mathematical Sciences already ranked in the top 10 in applied mathematics and logic by U.S. News and World Report magazine. And this same growth promises to ensure that Carnegie Mellon will continue to generate luminaries who alter the landscape of computational finance.