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The Ph.D. Program in Mathematical Finance
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Rationale for the Program
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.
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Ph.D. Degree in Mathematical Finance
The Ph.D. Program in Mathematical Finance described here is available to students enrolled in the Carnegie Mellon Department of Mathematical Sciences. Students complete all requirements for a Ph.D. degree in mathematics and receive the degree Ph.D. in Mathematics (Mathematical Finance).
Entering students should have a strong undergraduate background in mathematics, although not necessarily an undergraduate degree in mathematics, and should have a desire and aptitude for pursuing mathematics at the graduate level. The curriculum is designed so that no previous experience in economics is necessary, although an undergraduate course in microeconomics is helpful. A student entering this program without mathematical deficiencies can expect to complete the Ph.D. degree in five years. Many students spend at least one summer in an industrial internship position. Although there is no guarantee that such an internship can be arranged, faculty assist with the search and most students seeking internships in recent years have been placed. Some of the companies with which students have interned are Merrill Lynch, UBS, Bank of Nova Scotia, BNP-Paribas, Bank of America, and Lehman Brothers.
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Requirements
The requirements for the degree are detailed below. Full semester graduate courses carry 12 units of credit, and half-semester courses carry 6 units
The Mathematical Sciences Department requires every student to either take or demonstrate competency in the material of a set of core courses. For students in the Mathematical Finance Ph.D. program, these courses are:
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21-620 Real Analysis (6 units) |
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21-621 Introduction to Lebesgue Integration (6 units) |
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21-640 Functional Analysis (12 units) |
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21-660 Introduction to Numerical Analysis (12 units) |
Students generally take the core courses in their first two semesters of residence. Students are also required by the Mathematical Sciences Department to pass a Qualifying Examination, covering major and minor topics, to certify the students' preparedness to begin research. For students in the Mathematical Finance Ph.D. program, the major topic for the qualifying exam will be stochastic processes. The minor topic may be numerical analysis, statistics, or finance/economics. The content of the qualifying examination, which is determined by the student and the student's advisory committee, is typically the content of four Ph.D. level courses, two in the major area and two in the minor area. The qualifying examination is normally taken in the third year of residence. Additional details on these requirements can be found in the graduate catalogue of the Mathematical Sciences Department. Finally, students must write and successfully defend a Ph.D. dissertation.
A formal recitation of requirements does not fully indicate the extent of resources available to a program. The department regularly offers additional Ph.D. courses on topics related to the mathematical finance curriculum. In addition, Ph.D. students frequently take the courses in the professional Master's degree program in Computational Finance, offered jointly by the Department of Mathematical Sciences, the Department of Statistics, the Tepper School of Business and the Heinz School of Management.
Although the Master's courses are not appropriate for Ph.D. qualifying examination preparation, they provide students insight into the practical issues of mathematical finance. Students in the Ph.D. program in Mathematical Finance generally take a number of these coures in their first few years of residence, and serve as teaching assistants in some of these courses in later years.
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Faculty
The following faculty in the Department of Mathematical Sciences, all of whom have research interest in finance, are affiliated with the Ph.D. Program in Mathematical Finance.
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Dmitry Kramkov, Associate Professor of Mathematics |
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Kasper Larsen, Assistant Professor of Mathematics |
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John P. Lehoczky, Thomas Lord Professor of Statistics and Mathematics |
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Roy A. Nicolaides, Alexander Knaster Professor of Mathematics |
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Steven E. Shreve, Orion Hoch Professor of Mathematics |
Current research interests of faculty include the term structure of interest rates, stochastic calculus models of asset prices and portfolio optimization, and determination of derivative security prices.
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Admission and Support
The application procedure for the Ph.D. in Mathematical Finance is the same as for other students entering the Department of Mathematical Sciences. See the Graduate Admissions and the Financial Aid pages for more information.
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