Proposal for an MS Program in Financial Mathematics in the School of Humanities and Sciences
Contentssubmitted by George Papanicolaou Department of Mathematics email: papanico@math; telephone -32081 March 6, 1999 SUMMARY PREFACE 1. Name of Program: Master of Science in Financial Mathematics or MS-FM for short. 2. Objectives: To provide an attractive masters level education in applied mathematics, statistics and financial applications to individuals with very strong mathematical skills. We expect three broad classes of candidates for this degree. - PhD students in Economics, Mathematics or Statistics (as well as SCCM, ISL, ...) who want to get a masters degree in an applied area (for mathematics, statistics and economics). - Candidates who are admitted directly into the masters program. The main admission requirement will be a strong background in science and mathematics as spelled out in the detailed description of the program on page 9. - Undergraduates from Mathematics or the Mathematical and Computational Sciences Program (MCS) who will be able to enter this masters program in financial mathematics by enrolling for a fifth year (coterminal program). 3. Brief history: This is a new program. The decision to try to initiate an MS-FM Program here was made in the Fall of 1997 in discussions that Dembo, Lai and Papanicolaou had concerning new financial mathematics programs in mathematics/statistics departments in other universities (Chicago, NYU, Columbia, CMU, etc). Several meetings were held among the Core Faculty (see the list below), the Mathematics Department Faculty, the Statistics Department Faculty, the Steering Committee, the Dean and Associate Dean of H&S, etc, since that time. 4. Program Leadership: The core faculty for the proposed program is as follows. Economics: Amemiya, Kurz, Wolak Mathematics: Dembo, Diaconis, Papanicolaou Statistics: Cover, Dembo, Diaconis, Lai, Owen Graduate School of Business: Duffie, Harrison, Singleton Engineering Economic Systems and Operations Research: Glynn, Iglehart, Luenberger Electrical Engineering: Cover The administration of the program will be handled by the Steering Committee consisting of A. Dembo (mathematics and statistics) P. Glynn (EESOR) T. Lai (statistics) A. Owen (statistics) G. Papanicolaou, (mathematics; director of the Program) K. Singleton (GSB) F. Wolak (economics) The chairmen of the Mathematics and Statistics Departments are invited to attend all meetings of the Steering Committee. The members of the Steering Committee are also the liaisons between the Financial Mathematics Program and the five main Departments involved (Economics, EESOR, GSB, Mathematics, Statistics). If the Program is approved, it will start in the Fall of 1999. We plan to have it reviewed after two years, in the Spring of 2001, by a panel appointed by the Dean of H&S. The Dean will also appoint a new director at that time. 5. Courses and Degree Material: This is a demanding MS program with roughly one third Mathematics, one third Statistics and one third Finance. Of the twelve one-quarter courses (36 units) that are needed for the degree, the following six graduate courses, or their equivalents, are required: 1. Introduction to Stochastic Differential Equations (Math 236) 2. Statistical Methods in Finance (Stat 240; course to start in 1999-00) 3. Partial Differential Equations (Math 220B) 4. Computing and Simulation in Finance (Math 240, same as Stat 245; course to start in 1999-00) 5. Introduction to Financial Mathematics (Math 180; course to start in 1999-00) 6. Mathematical Finance (Math 241, same as Stat 250 and Econ 289; course started in 1998-99) The first two courses are in Stochastic Processes and Statistics, the second two in Applied and Computational Mathematics, and the last two in Finance. Six electives are to be taken from an approved list, two from each of these three groups: Statistics and Stochastic Processes Differential Equations, Simulation and Computing Economics and Finance. Further details on courses and prerequisites are given on pages 9-11 of this proposal. This is an academically challenging and well integrated interdisciplinary program in Probability, Differential Equations, Statistics, Computation and Finance. It is comparable to similar MS programs in Financial Mathematics in leading universities (Carnegie Mellon, Chicago, Columbia, Courant (NYU), Oxford and Toronto). 6. Teaching Staff: Of the six required courses, four are new ones with two based in the Mathematics (4 and 5 above) and two in the Statistics Department (2 and 6). In addition, course 1, based in the Mathematics Department, will now be given every year instead of every two years. Course 3 is given every year and will continue this way. The Mathematics Department will use one senior faculty FTE (Papanicolaou) and two Szego assistant professors (Mattingly and another one next year) at one-quarter teaching time to cover the new courses. The Statistics Department will use two senior faculty FTE's (Lai and Owen) at one-quarter teaching time for the new courses. In addition, several other members of the core faculty (Dembo, Duffie, Glynn, Igelhart, Luenberger, Singleton and Wolak) will contribute their time in helping to develop the new courses (2,4,5 and 6 above). 7. Enrollment and Degree Statistics: The Program is running this year as a Departmental MS Degree in Mathematics with Field Designation `Financial Mathematics'. We have about 5 students who have registered or will register for this degree, all at Stanford already, admitted to other programs like Statistics and SCCM. About 4-5 MCS students have expressed interest in the MS-FM as a coterminal degree. We have received some 40-50 emails and phone calls requesting information about the program, which is announced in the Stanford Catalog this year and on the web page of the Mathematics Department. The Math 236 (Stochastic Differential Equations) class has 24 registered students, Math 241 (Mathematical Finance) has 35 registered students and Math 220B (Applied Partial Differential Equations) 27 registered students, this winter term. We will admit about ten students from outside Stanford for the MS-FM degree. We expect that at equilibrium the MS-FM Program will have about twenty candidates each year. 8 Finances: The bulk of the MS-MF Program will run on existing resources from the Mathematics and Statistics Departments. We will provide no financial support for students admitted to this program. We will make available one office in the Mathematics Department for use by the students in this program. There will be two or three departmental computers in this office. We are asking from the School of Humanities and Sciences for a half-time Program Administrator.1 Academic Rigor and Intellectual Coherence of the Core Curriculum 1.1 The core faculty 1.2 The required courses 1.3 Admission and advising 1.4 Scheduling of Required Courses 2 Overlap with Programs Offered or Potentially Offered within Departments or Existing IDP's 2.1 Comparison with the MS degree in EESOR 2.2 Comparison with GSB programs 2.3 Comparison with MS-FM programs in other universities 3 Costs and Benefits Unit-Wide in the Appropriate Division 3.1 Cost of the Program to the Mathematics and Statistics Departments 3.2 Cost of the Program to CS, Econ, EESOR and GSB 3.3 Cooperation with GSB, Econ and EESOR 4 The Proportion of Classes Taught by Academic Council Members 5 The Amount of Interest and Potential Involvement of Students 6 The Strength of the IDP's Administrative Structure1 Academic Rigor and Intellectual Coherence of the Core CurriculumThis is a proposed MS Program in the school of Humanities and Science involving the departments of Economics, Mathematics and Statistics, with the close collaboration of the Graduate School of Business and the departments of Engineering Economic Systems-Operations Research and Electrical Engineering, in the School of Engineering. The purpose of this Program is to provide a masters level education in applied mathematics and statistics with emphasis in financial applications to individuals with very strong mathematical skills. 1.1 The core faculty The core faculty is: Economics: Amemiya, Kurz, Wolak Mathematics: Dembo, Diaconis, Papanicolaou Statistics: Cover, Dembo, Diaconis, Lai, Owen Graduate School of Business: Duffie, Harrison, Singleton Engineering Economic Systems and Operations Research: Glynn, Iglehart, Luenberger Electrical Engineering: Cover 1.2 The required courses This is a demanding MS program with roughly one third Mathematics, one third Statistics and one third Finance. Of the twelve one-quarter courses (36 units) that are needed for the degree, the following six graduate courses, or their equivalents, are required. The detailed description of the course content, as it will appear in the Catalog, is given on page 11. 1. Introduction to Stochastic Differential Equations (Math 236) This course is given regularly in the mathematics department by Papanicolaou, with this Winter term (98-99) being the third time in the last six years. It will be given every year, beginning next year, and it will be moved to the fall quarter. J. Mattingly, a Szego assistant professor will give it next fall. Dembo, who teaches advanced probability and stochastic processes classes regularly, may also teach this course in the near future, as well as T.P. Liu. 2. Introduction to Financial Mathematics (Math 180; course to start in 1999-00) This is a new course that will be given normally in the fall quarter, except for 1999-2000 when it will be given in the spring quarter. The level is introductory, an undergraduate upper division or basic graduate course. It is meant to be comparable in content to EESOR 242 (Investment Science) but addressesd to students with a strong background and interest in mathematics. The course will be developed by Papanicolaou in close consultation with core faculty members Duffie, Luenberger, Singleton and Wolak. Many faculty members in the mathematics department have expressed interest in teaching this course in subsequent years (G. Carlsson, R. Cohen, A. Dembo, P. Diaconis, Y. Eliashberg, R. Schoen). Courses like the one we want to introduce are already being given by many mathematics departments in American and European universities and they are very successful in attracting students from many disciplines. 3. Partial Differential Equations of Applied Mathematics (Math 220B) This course is given every winter quarter by the Mathematics Department, by T.P. Liu, Szego assistant professors and G. Papanicolaou. 4. Statistical Methods in Finance (Stat 240; course to start in 1999-00) This is a new course that will be given every winter quarter. It will be developed by A.Owen in the Statistics Department, in close consultation with core faculty members Lai, Singleton and Wolak. In subsequent years it may be given by other faculty members in the Statistics Department. 5. Computing and Simulation in Finance (Math 240, same as Stat 245; course to start in 1999-00) This is a new course that will be given every spring term jointly by Mathematics and Statistics. It will be developed and taught by A.Owen and G. Papanicolaou, in close consultation with core faculty members Duffie, Glynn, Iglehart and Singleton. 6. Mathematical Finance (Math 241, same as Stat 250 and Econ 289; course started in 1998-99) This course will be given every spring quarter. It is being given this winter quarter for the first time, by T. Lai. The course material is developed in close consultation with core faculty members Duffie, Luenberger, Singleton and Wolak. This is a carefully thought out sequence of required courses that are rigorous and comprehensive but should be attractive to students with a certain uency and facility in mathematics. It uses the students' strengths while they are introduced systematically to a lively and demanding field of application. The students will round out their program with six electives: two from Statistics and Stochastic Processes, two from Differential Equations, Simulation and Computing and two from Economics and Finance. A list of possible electives is given on page 10. 1.3 Admission and advising The admission of MS-FM students from outside Stanford will be conducted each year by a committee consisting of three core faculty members who will compile a list of finalists for admission. This list will be circulated to the core faculty members before a final decision is made. Undergraduates in MCS, Mathematics or other departments that wish to enter a coterminal MS-FM program will be interviewed by the Program director and will be asked to complete a proposed schedule of courses. The students will be expected to follow this schedule, to the extent possible if con icts do not arise. Any changes that are made have to be approved by their faculty advisor. PhD students who want to get the MS-FM degree will be interviewed by the Program director and will be asked to complete a proposed schedule of courses. They will be expected to follow this schedule, to the extent possible if con icts do not arise. Any changes that are made have to be approved by their MS- FM advisor. The PhD students will also be asked to get the MS-FM course schedule approved by their main advisor in their department. Students admitted into the MS-FM program from outside Stanford will also be asked to complete a proposed schedule of courses. They will be assigned an advisor from one of the core faculty members. We expect to have about twenty students in this Program: about half that are admitted directly from outside Stanford and half that come from MCS or PhD programs at Stanford. 1.4 Scheduling of Required Courses Fall: Mathematics 236: Introduction to Stochastic Differential Equations; Mathematics 180: Introduction to Financial Mathematics. Winter: Statistics 240: Statistical Methods in Finance; Math 220B: Partial differential equations of applied mathematics. Spring: Statistics 245 (Mathematics 240): Computation and Simulation in Finance; Mathematics 241 (Statistics 250 or Economics 289): Mathematical Finance.2 Overlap with Programs Offered or Potentially Offered within Departments or Existing IDP's2.1 Comparison with the MS degree in EESOR The MS degree in EESOR requires 45 course units, 9 more (three courses) than the proposed MS-FM Program. The three extra courses required in EESOR can be roughly identified with prerequisites for the MS-FM Program so that the two degrees demand a comparable amount of course work. EESOR does not have a degree program infinance but informally trains many MS students through a financial analysis track, and has many course offerings in the field. Analogously, Statistics offers many courses related to the field but does not have a degree program in Financial Mathematics. It has an MA and a PhD program in Statistics. EESOR has MS and PhD degrees and an Engineer's degree in EESOR. The proposed MS-FM program is a degree program in Applied Mathematics and Statistics, specialized to Financial Mathematics. No such degree program is currently available at Stanford and surely none at the level of mathematical competence that MS-FM requires. On the other hand, such degree programs in financial mathematics have recently been started out of Mathematics and Statistics Departments at Carnegie Mellon, Columbia, Chicago, NYU, Oxford, Cambridge, Toronto, etc. In the Stanford Bulletin, EESOR lists 8 research areas and their associated faculty: Optimization, Probability and Stochastic Processes, Systems and Simulation, Economics, Finance and Investment, Decisions, Operations and Services, Strategy, and Policy; the area in Finance and Investment is focused on more general issues like investment, portfolio choice and decision making. In contrast, the proposed MS-FM Program is focused primarily on the mathematics and statistics of derivative pricing and the modeling of financial markets. The intellectual center of gravity in the MS-FM Program is in mathematics and statistics, as it would be for a Biostatistics MS degree or a Computational Fluids Dynamics (CFD) MS degree. The possible lack of breadth in these Programs, in comparison with comparable ones in engineering (for CFD for example), is more than compensated by depth and competence in mathematics. 2.2 Comparison with GSB programs There is little overlap between the MBA degree and the MS-FM degree. MBA students study financial mathematics, in addition to many other things, but focus primarily on facts, institutional details and computations. The MS-FM Program focuses on the mathematics and statistics of derivative pricing and the modeling of financial markets. There is also little overlap with the PhD Program in Financial Mathematics in GSB. This is a very advanced program directing its students into research issues rather quickly. Many of the PhD students in the GSB Program already have PhD's in other fields (physics, astrophysics, etc). 2.3 Comparison with MS-FM programs in other universities The MS Programs in Financial Mathematics that other universities have initiated have many common features with the one proposed here. However, they also have weaknesses that we do not expect our program to have. For example, Columbia's program is strong in statistics but relatively weak in differential equations and computational methods, the NYU (Courant) program is strong in differential equations and computing but weak in statistics, etc. The proposed MS-FM Program at Stanford stands out because it is truly interdisciplinary without compromising competence and fluency in mathematics and statistics.3 Costs and Benefits Unit-Wide in the Appropriate Division3.1 Cost of the Program to the Mathematics and Statistics Departments. The Departments of Mathematics and Statistics will carry most of the cost of the MS-FM program, in terms of faculty time devoted to the Program, increased class size, and offering of some courses every year instead of every other year. The development of the new courses, Math 180, Stat 240, Stat 245 (Math 240), Math 241 (Stat 250, Econ 289), will take a lot of time and effort, but they are an essential part of the proposed Program. In addition to the mathematics and statistics department faculty that will be involved (Dembo, Lai, Owen, Papanicolaou), one Szego assistant professor will also be involved (Mattingly) and we are looking to bring another Szego assistant professor in the financial mathematics area. 3.2 Cost of the Program to CS, Econ, EESOR and GSB The cost of the MS-FM program to these departments will be a small increase in class size in some of their courses that the MS-FM students may take as electives. Given that only ten students will be coming from the outside, the other ten being at Stanford already either in MCS or a PhD program, the increase in expected class size in any one elective class will be very small. 3.3 Cooperation with GSB, Econ and EESOR The center of gravity of this Program is in the Mathematics and Statistics Departments, both regarding the intellectual orientation of the requiered courses and the commitment of resources. To maintain the broader interdisciplinary balance that is essential for the success of the Program, we rely on the electives and on the strong and effective cooperation of the core faculty in developing and maintaining the content of the required courses. The way that this will be done is explained in section 1.2 (page 5 mostly). Cooperation is not, however, limited to the electives and to consulting. We could, in the near future, give courses with team-teaching, involving several core faculty, we could invite visiting faculty with a joint appointment to teach some courses, etc. We could also consider joint faculty appointments at some point in the future. In assessing the level of cooperation in this Program between the five Departments that are involved (Math, Stat, GSB, Econ and EESOR), it should be kept in mind that this is a Masters Program so its objectives must by well defined and transparent. Tuition-paying students who enter it must have a clear idea of what they are getting.4 The Proportion of Classes Taught by Academic Council MembersAll required classes are taught by Academic Council members. The electives are standard courses from the Mathematics, Statistics and Economics Departments, as well as EESOR, CS and GSB, so they are almost always given by Academic Council members.5 The Amount of Interest and Potential Involvement of StudentsThe MS-FM Program is running this year as a Departmental MS Degree in Mathematics with Field Designation `Financial Mathematics'. We have about 5 students who have registered or will register for this degree, all at Stanford already, admitted to other programs like Statistics and SCCM. About 4-5 MCS students have expressed interest in the MS-FM as a coterminal degree. We have received some 50-60 emails and phone calls requesting information about the program, which is announced in the Stanford Catalog this year and on the web page of the Mathematics Department. We expect that when the program is fully in operation we will have more than ten applications for each of the ten or so students that will be admitted.6 The Strength of the IDP's Administrative StructureThis is a Masters Program, so once the basic courses are set up and are coordinated with each other and with the overall objectives there is not that much to do. There is enough depth in the teaching faculty, in the steering committee and in the core faculty so that even if some members contribute less than what we expect now, a very unlikely event, the Program will hardly feel it. The commitment of the two principal departments, Mathematics and Statistics, to doing all that is needed to promote this Program is strong and long term. We are planning to have the MS-FM Program reviewed in the Spring of 2001, by a committee appointed by the Dean of H&S. At that time, the Dean will also appoint a new director of the Program. ______________________________________________________________________________Financial Mathematics MS Program Graduate Programs Master of Science Effective Academic Year 1999{2000 The Program requires that the candidate take 36 units of work, or 12 courses of 3 units each, from the list of offerings provided below. Ordinarily, three or four quarters are needed to complete all requirements. To be eligible for admission, candidates are expected to have taken the following courses or their equivalents: 1. Linear algebra at the level of Mathematics 103 and Real Analysis (Advanced Calculus) at the level of Mathematics 115. 2. Basic Ordinary and Partial Differential Equations at the level of Mathematics 130, 131 and 132. 3. Probability and Statistics at the level of Statistics 116, 200 and preferably 217 (Introduction to Stochastic Processes; Discrete and Continuous Markov Chains). 4. Computer programming at the level of CS106A. Most of these courses are offered as Summer courses and can be taken by candidates lacking the required background. For the MS degree each candidate must fulfill the following six required courses: 1. In Stochastic Processes and Statistics: Mathematics 236 (Introduction to Stochastic Differential Equations) and Statistics 240 (Statistical Methods in Finance) or Economics 275 (Time Series). 2. In Differential Equations, Simulation and Computing: Mathematics 220B (Applied Partial Differential Equations B) and Mathematics 240 (same as Statistics 245) (Computation and Simulation in Finance) 3. In Finance and Economics: Mathematics 180 (Introduction to Financial Mathematics) or EESOR 242 (Investment Science) or GSB F620 (Introduction to Financial Economics) and Mathematics 241 (same as Statistics 250 and Economics 289) (Mathematical Finance). These courses must be taken for letter grades, and an overall 2.75 letter grade indicator is required. Courses that are equivalent to the above and have been taken previously may bewaived by the adviser, in which case they must be replaced by elective courses in the same subject area. In addition, each candidate must take at least six approved elective courses from the following. 1. At least two electives in Probability, Stochastic Processes or Statistics from: Statistics: 206 (Applied Multivariate Analysis), 310B,C, 317 (Theory of Probability, Stochastic Processes), 324 (Multivariate Analysis), 326 (Sequential Experimentation), 343 (Time Series Analysis), 376A (Information Theory), Mathematics: 205A,B, 230A (Real Variables, Probability Theory same as Stat 310A), 237 (Asymptotic Analysis of Stochastic Equations). Economics: 276 (Special Topics in Econometrics), GSB: E604 (Advanced Econometrics). 2. At least two electives in Differential Equations, Optimization, Simulation or Computing from: Mathematics: 220C (Asymptotic and Perturbation Methods for PDE), 254A,B (Ordinary Differential Equations), 226, 256A, 266, 267A,B (Topics in Applied Mathematics, PDE, Introduction to Wavelets, Harmonic Analysis), Statistics: 227 (Statistical Computing), 353 (Monte Carlo Methods), CS: 229 (Machine Learning), 237A,B,C (Numerical Linear Algebra, Numerical Solution of Boundary Value Problems, Numerical Solution of Initial Value Problems), 260 (Concrete Mathematics), 261 (Optimization and Algorithmic Paradigms), 336, 337, 339 (Advanced Methods in Matrix Computation, Numerical Methods for Initial-Boundary Value Problems, Topics in Numerical Analysis), 365 (Randomized Algorithms). EES&OR: 311, 313, 314, 332, 351 (Optimization, Vector Space Optimization, Optimization Algorithms, Simulation Theory, Dynamic Programming and Stochastic Control). 3. At least two electives in Economics or Finance: Economics: 265 (International Finance), 281 (Economics of Uncertainty), 284 (Topics in Dynamic Economics), EES&OR: 347 (Capital Investment and Financial Decisions). GSB: F621 (Empirical Finance), F622 (Dynamic Asset Pricing Theory). Other elective courses may be authorized by the adviser if they provide skills relevant to financial mathematics and do not overlap with courses in the candidate's program. DESCRIPTION OF REQUIRED COURSES Mathematics 180. Introduction to Financial Mathematics. Basic theory of interest and fixed-income securities. Preferences and risk aversion, stochastic dominance. Mathematics of efficient portfolio frontier. Capital Asset Pricing Model, arbitrage pricing theory. Utility-based optimization. Mathematics 241 (Economics 289 and Statistics 250). Mathematical Finance. Stochastic models of financial markets. Forward and futures contracts. European options and equivalent martingale measures. Hedging strategies and management of risk. Term structure models and interest rate derivatives. Optimal stopping and American options. Mathematics 236. Introduction to Stochastic Differential Equations. Brownian motion, stochastic integrals, and diffusions as solutions of stochastic differential equations. Functionals of diffusions and their connection with partial differential equations. Random walk approximation of diffusions. Prerequisite: basic probability and differential equations. Mathematics 220B. Partial Differential Equations of Applied Mathematics. Green's functions, integral transforms, variational and distribution theoretic methods for the analysis of differential and integral equations, with illustrative examples. Prerequisite: some familiarity with differential equations and functions of a complex variable. Statistics 240. Statistical Methods in Finance. Regression analysis and applications to the Capital Asset Pricing Model and multifactor pricing models. Smoothing techniques and estimation of yield curves. Classification and credit risk. Statistical analysis and econometric modeling of financial time series. Problem sets will include hands-on experience with real data. Statistics 245 (Mathematics 240). Computation and simulation in Finance. Finite difference methods for the numerical solution of partial differential equations in finance. Binomial and trinomial tree methods. Classical numerical integration. Random variable generation, variance reduction, statistical analysis of simulation output. Applications to scenario analysis and interest rate modeling. Introduction to high-dimensional integration, Quasi-Monte Carlo and mortgage-backed securities.