Program in Mathematical Finance

Interdisciplinary Program

The following list reflects the 2006/2007 faculty.

Director Andrew Lyasoff

Associate Director Paolo Guasoni

Faculty

Jérôme Detemple Professor of Finance and Economics, School of Management. BA, ESSEC (France); MA, PhD, Wharton School, University of Pennsylvania; Doctorat d’État et Sciences Économiques, Université Louis Pasteur (France)

Paolo Guasoni Assistant Professor of Mathematics and Statistics, College of Arts and Sciences. Laurea in Mathematics, University of Pisa (Italy); MA, Columbia University; PhD, Scuola Normale Superiore (Italy)

Konstantinos Kardaras Assistant Professor of Mathematics and Statistics, College of Arts and Sciences. BSc, MSc, University of Athens (Greece); PhD, Columbia University

Andrew Lyasoff Associate Professor of Mathematics and Statistics, College of Arts and Sciences. BS, MS, PhD, Sofia University St. Kliment Ohridski (Bulgaria)

Affilated Faculty

Adele Barron Norman Professor in Management, School of Management. PhD, Massachusetts Institute of Technology

Zvi Bodie Professor in Management, School of Management. PhD, Massachusetts Institute of Technology

Thomas Hawkins Professor of Mathematics, College of Arts and Sciences. BA, Houghton College; MS, University of Rochester; PhD, University of Wisconsin

Mark Kon Professor of Mathematics, College of Arts and Sciences, BA, Cornell University; PhD, Massachusetts Institute of Technology

Laurence J. Kotlikoff Professor of Economics, College of Arts and Sciences. BA, University of Pennsylvania; MA, PhD, Harvard University

Jianjun Miao Assistant Professor of Economics, College of Arts and Sciences. BS, University of Science and Technology of China (China); MA, Llingnan College, Zhongshan University (China); MA, Queen’s University (Canada); MA, PhD, University of Rochester

Erol A. Pekoz Associate Professor of Operations Management, School of Management. BS, Cornell University; MS, PhD, Univesity of California, Berkeley

H. Eugene Stanley Director, Center for Polymer Studies, Graduate School of Arts and Sciences; University Professor and Professor of Physics, College of Arts and Sciences (condensed matter theory); Professor of Physiology, School of Medicine. BA, Wesleyan University; PhD, Harvard University

Murad Taqqu Professor of Mathematics, College of Arts and Sciences. BA, University of Lausanne (Switzerland); MA, PhD, Columbia University

Pirooz Vakili Associate Professor of Manufacturing Engineering, College of Engineering. BS, Arya-Mehr University (Iran); PhD, Harvard University

Arien Verdelhan Assistant Professor of Economics, College of Arts and Sciences. Engineering deg., Ecole Supérieure d’Electricité (France); BA, Université Paris 1, Sorbonne (France) PhD, University of Chicago

The interdisciplinary Graduate Program in Mathematical Finance (GRMF) offers both the MA and the PhD degrees. Its curriculum combines the study of advanced topics in finance with that of high-end mathematical concepts and computational tools. Participating faculty are from the Departments of Economics, Finance and Economics (School of Management), and Mathematics and Statistics.

From its origins in the late 1970s, the field of mathematical finance has grown to embrace the analysis of virtually any investment activity that involves a certain amount of risk. An underlying premise of mathematical finance is that complete understanding of the most advanced topics in finance requires quantitative skills on a par with those normally acquired through doctoral study in mathematics. Yet, prior to the recent emergence of graduate programs in mathematical finance, much of the research in financial economics was conducted by individuals whose formal academic training was limited, on the one hand, to finance and economics or, on the other hand, to applied mathematics. The GRMF has been conceived to transcend that disciplinary divide, by not only involving the relevant disciplines in a single program, but also integrating them systematically in core courses that foster students’ thinking in financial categories with a high degree of mathematical sophistication.

The Graduate Program in Mathematical Finance is designed for students with a strong affinity toward quantitative reasoning and ability to connect advanced mathematical theories with the real world. It is expected of students who enter the Graduate Program in Mathematical Finance—at both the master’s and doctoral levels—that their background in mathematics will be comparable to that of students entering the Graduate Program in Mathematics at the corresponding (MA or PhD) level.

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MA in Mathematical Finance

Admission Candidates for admission must have met the requirements for a bachelor’s degree and must have taken all courses needed as prerequisites (see below). GRE General Test results are required. GRE Subject Test results in Mathematics, Computer Science, or Economics are recommended, but not required. All applicants for admission from non-English speaking countries must submit results from the Test of English as a Foreign Language (TOEFL) examination.

Prerequisites All levels of Calculus (CAS MA 123, 124, 225 or equivalent), Linear Algebra (CAS MA 242 or equivalent), Differential Equations (CAS MA 226 or equivalent), basic computer programming skills (CAS CS 111 or equivalent, or CS 330 or equivalent).

Course Requirements To graduate from the program, students must earn 40 course credits by completing the following courses within one year (3 semesters) in the full-time option or within two years (5 semesters) in the part-time option. Students must receive a grade of B– or higher in order to accumulate credit from any given course.

Fall Semester: CAS MA 569, CAS MF 572, CAS MF 593, GRS MF 795

Spring Semester: CAS MF 594, GRS MF 796, MET CS 563, GSM FE 828.

Part-time (2 years) Option Part-time students will meet with their academic advisor to set up a plan that allows for the program to be completed within two years.

Certificate in Mathematical Finance

A Certificate in Mathematical Finance may be earned by students who pass at the graduate level the following five courses from the MA program in Mathematical Finance:

• Fundamentals of Finance (MF 502) or Introduction to Mathematical Finance (MF 572)
• Introduction to Probability Theory (MA 590)
• Statistical Analysis of Financial Data (MF 593)
• Stochastic Methods of Mathematical Finance (MF 795)
• Computational Methods of Mathematical Finance (MF 796)

The certificate program is appropriate for professionals who do not need a degree in mathematical finance, but would like to acquire some core knowledge in this field or to enhance their knowledge base.

Qualifying Examination

A Preliminary Qualifying Exam will be given at the end of the Summer 2 term. It will cover the material in CAS MF 502 and MA 590, and will focus on students’ ability to synthesize the material from these basic courses. Students must receive a passing grade in this exam in order to continue in the Master’s program.

A Final Graduate Exam will be given in the last week of the spring semester. It will again test students’ ability to apply their theoretical knowledge and technical skills in specific practical situations.

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PhD in Mathematical Finance

Admissions Candidates must have met the requirements for a bachelor’s degree and must have taken all courses needed as prerequisites (see below). GRE General Test results are required. GRE Subject Test results in Mathematics, Computer Science, or Economics are recommended but not required. All applicants for admission from non-English speaking countries must submit results from the TOEFL examination.

Prerequisites All levels of Calculus (CAS MA 123, 124, 225 or equivalent), Linear Algebra (CAS MA 242 or equivalent), Differential Equations (CAS MA 226 or equivalent), two semesters of Mathematical Analysis at the 500 level or higher (CAS MA 511 and CAS MA 512, or equivalent), at least one semester of Probability Theory at the 500 level or higher (CAS MA 590 or equivalent), at least one semester Finance or Financial Economics at the 500 level or higher (CAS MF 502 or equivalent).

Course Requirements Students must complete and receive a grade of B – or higher in the following 16 courses (64 credits):

Students with an MA or MS degree in a relevant field who wish to pursue a post-MA doctoral program in Mathematical Finance must take at least 8 (eight) graduate courses from the above list and must, through a combination of previous graduate course work and additional courses, satisfy the post-BA doctoral program course requirements described above.

Qualifying Examination Students must pass a comprehensive qualifying exam in the areas of Financial Economics, Probability Theory, and Stochastic Processes within two years of the start of their doctoral study. More detailed information will be provided upon admission to the PhD program.

Residency Requirement Please see “General Requirements for the PhD” in this bulletin.

Prospectus and Dissertation Candidates for the PhD must complete, and defend, a dissertation containing original and significant contributions to research in the area of Mathematical Finance. The student’s proposed dissertation topic must be approved by a faculty member. Within 6 months after passing the comprehensive qualifying exam, students must submit a dissertation prospectus, in accordance with policies and procedures detailed on page 7 of this bulletin.

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Courses

CAS MA 569 Optimization Methods of Operations Research

Prereq: CAS MA 225 or MA 230, and MA 242 or MA 442. General nonlinear optimization methods: Kuhn-Tucker conditions and Lagrange’s method. Applications to utility maximization and mean-variance analysis. Linear programming, the simplex method and duality theory. Quadratic programming and portfolio optimization. Assignment problems, dynamic programming and the simulated annealing method. Lyasoff. 4 cr, 1st sem.

CAS MA 590 Introduction to Probability Theory

Prereq: CAS MA 225 or CAS MA 230. Combinatorics; conditional probability; independence; discrete and continuous random variables; sigma algebras; joint, marginal, and conditional distributions. Conditional and unconditional expectations and variance; derived distributions; characteristic functions; convergence of random variables; limit theorems; unbiased estimates of mean and variance; and Cochran’s Theorem. (Cannot be taken for credit in addition to CAS MA 381 or MA 581.) This course is part of the Mathematical Finance Program, but is open to all students with permission. Guasoni. 4 cr, Summer 2

CAS MF 502 Fundamentals of Finance

Prereq: CAS MA 123, MA 124, MA 225 or MA 230, and MA 242 or MA 442. Financial systems, financial statements, time value of money, interest rates, return sales, spot rates, forward rates, inflation rates, cost of capital rates, tax rates, bonds, cash-flow models, risk management. This course is part of the Mathematical Finance Program but is open to all students. Kardaras. 4 cr, Summer 2

CAS MF 572 Introduction to Mathematical Finance

Prereq: CAS MA 225, CAS MA 226 or MA 231, and MA 590. An introduction to mathematical finance, emphasizing the rigors of the analytical methods used in finance. Mathematics of Modern Portfolio Theory; Capital Asset Pricing Model (CAPM); Arbitrage Pricing Theory (APT); options, futures, and swaps; and risk management are discussed in depth and at a rigorous and analytical level. Valuation and hedging models are derived based on concepts from Itô Calculus. Guasoni. 4 cr, 1st sem.

CAS MF 593 Statistical Analysis of Financial Data

Prereq: CAS MA 225, CAS MA 226 or MA 231, and MA 590. Statistical/probabilistic techniques commonly used in financial engineering and risk management. Topics include estimation of diffusion processes, elements of linear (ARMA) and nonlineary ((G)ARCH) time series-analysis, multivariate data analysis, extreme value theory and Monte Carlo methods. Guasoni. 4 cr, 1st sem.

CAS MF 594 Stochastic Optimal Control and Investment

Prereq: CAS MA 225 or MA 230, MA 242 or MA 442, and MA 590. Dynamic hedging, option valuation, Brownian motion and stochastic dynamic programming, Bellmann’s equation, contingent claims analysis, optimal stopping rules, dynamic equilibrium, sequential investment. Lyasoff. 4 cr, 2nd sem.

GRS MF 772 Mathematical Finance Theory

Prereq: GRS MA 711, MA 717, and MF 795, or consent of instructor. Martingale and local martingale measures. Notions of arbitrage. Fundamental Theorem of Asset Pricing in the semimartingale setting. Super-hedging and optional decomposition. Snell Envelopes and American options. Predictable representation and complete markets. Convex duality and utility maximization. Risk measures. Lyasoff. 4 cr, 2nd sem.

GRS MF 795 Stochastic Methods of Mathematical Finance

Prereq: CAS MA 225 or MA 230, and MA 590. Stochastic calculus: Brownian motion, martingales, stochastic integrals, Itô’s lemma, Girsanov’s formula, diffusion processes. Mathematical finance theory: financial markets; arbitrage pricing; derivative instruments; exotic options; change of numeraire; LIBOR and Swap market models. Lyasoff. 4 cr, 1st sem.

GRS MF 796 Computational Methods of Mathematical Finance

Prereq: CAS MA 590 and GRS MA 795. The most essential computational methods, including various Monte Carlo methods, used to price derivatives instruments, including the most common exotic and path-dependent options, options with transaction costs, and some fixed-income derivative instruments. Kardaras. 4 cr, 2nd sem.

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31 October 2007
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