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Rationale for Ph.D. Research in Mathematical Finance

By awarding Harry Markowitz, William Sharpe and Merton Miller the 1990 Nobel Prize in Economics, the Nobel Prize Committee brought to world-wide attention the fact that the last forty years have seen the emergence of a new scientific discipline, the "theory of finance." This theory attempts to understand how financial markets work, how to make them more efficient, and how they should be regulated. It explains and enhances the important role these markets play in reducing risk associated with economic activity. Without losing its application to practical aspects of trading and regulation, the theory of finance has become increasingly mathematical, to the point that problems in finance are now driving research in mathematics.

Harry Markowitz's 1952 Ph.D. thesis "Portfolio Selection" laid the groundwork for the mathematical theory of finance. Markowitz developed a notion of mean return and covariances for common stocks which allowed him to quantify the concept of "diversification" in a market. He showed how to compute the mean return and variance for a given portfolio and argued that investors should hold only those portfolios whose variance is minimal among all portfolios with a given mean return. Although the language of finance now involves Ito calculus, minimization of risk in a quantifiable manner underlies much of the modern theory.

In 1969 Robert Merton introduced stochastic calculus into the study of finance. Merton was motivated by the desire to understand how prices are set in financial markets, which is the classical economics question of "equilibrium," and in later papers he used the machinery of stochastic calculus to begin investigation of this issue.

At the same time as Merton's work and with Merton's assistance, Fischer Black and Myron Scholes were developing their celebrated option pricing formula. This work won the 1997 Nobel Prize in Economics. It provided a satisfying solution for an important practical problem, that of finding a fair price for a European call option, i.e., the right to buy one share of a given stock at a specified price and time. Such options are frequently purchased by investors as a risk-hedging device. In 1981, Harrison and Pliska used the general theory of continuous-time stochastic processes to put the Black-Scholes option pricing formula on a solid theoretical basis, and as a result, showed how to price numerous other "derivative" securities.

Many of the theoretical developments in finance have found immediate application in financial markets. To understand how they are applied, we digress for a moment on the role of financial institutions. A principal function of the nation's financial institutions is to act as a risk-reducing intermediary among customers engaged in production. For example, the insurance industry pools premiums of many customers and must pay off only the few who actually incur losses. But risk arises in situations for which pooled-premium insurance is unavailable. For instance, as a hedge against higher fuel costs, airlines want to buy securities whose value will rise if oil prices rise. But who wants to sell such securities? The role of a financial institution is to design such a security, determine a "fair" price for it, and sell it to airlines. The security thus sold is usually "derivative," i.e., its value is based on the value of other, identified securities. "Fair" in this context means that the financial institution earns just enough from selling the security to enable it to trade in other securities whose relation with oil prices is such that, if oil prices do indeed rise, the firm can pay off its increased obligation to the airlines. An "efficient" market is one in which risk-hedging securities are widely available at "fair" prices.

The Black-Scholes option pricing formula provided, for the first time, a theoretical method of fairly pricing a risk-hedging security. If an investment bank offers a derivative security at a price which is higher than "fair," it may be underbid. If it offers the security at less than the "fair" price, it runs the risk of substantial loss. This makes the bank reluctant to offer many of the derivative securities which would contribute to market efficiency. In particular, the bank only wants to offer derivative securities whose "fair" price can be determined in advance. The mathematical theory growing out of the Black-Scholes option pricing formula has permitted the creation and pricing of a host of specialized derivative securities.

Going hand in hand with the mathematical theory of finance is the advent of computer trading, which has accelerated the pace and expanded the information base for decision-making, much of which is now being automated. In such an environment, the trader who bases decisions solely on "feel" for the market is at a decided disadvantage. Even during the retrenchments on Wall Street in the late 1980s, high technology trading operations and their supporting research groups grew. In the 1990s, these operations expanded at an increasing rate.

As these developments proceed, the issues of regulation and risk appraisal become more complex. Banks are trading in exotic derivative securities whose associated risk must be measured. Private corporations are discovering an increased need to do their own financial analysis. The Society of Actuaries reports that "The actuary of the future will be a financial architect and manager of enterprises built on the applications of financial analysis and risk appraisal." Major accounting firms are hiring persons who can build mathematical models to assess the positions of their clients. Despite the Enron fiasco, electric and natural gas utilities continue to build mathematical models to guide trading.

In recent years, Ph.D. candidates in finance programs at leading business schools have found it necessary to learn considerable amounts of graduate-level mathematics. On the other hand, Ph.D. recipients in mathematics, statistics, physics and computer science are finding employment in the finance industry. There is a clear and growing demand for mathematical scientists with knowledge of financial markets. At Carnegie Mellon, research in finance takes place in three different departments: Mathematical Sciences, Statistics, and the Tepper School of Business. Thus, Carnegie Mellon is well positioned to offer an interdisciplinary Ph.D. This program offers talented individuals the opportunity to study mathematically challenging problems driven by compelling applications to the finance industry.