MASTER OF SCIENCE IN FINANCIAL MATHEMATICS

PROPOSED CURRICULUM

Department of Mathematics

University of Dayton

The University of Dayton proposes to offer a professional Master degree program in Financial Mathematics. The Department of Mathematics and the Department of Economics and Finance are collaborating to offer this exciting new program. The program requires:

· exhibition of sufficient background knowledge in methods of applied mathematics or successful completion of introductory course (MTH 511) in methods of applied mathematics;

· exhibition sufficient background knowledge in methods finance or successful completion of MBA 620;

· successful completion of 18 credit hours (6 courses) of required course work at the graduate level;

· successful completion of 9 credit hours (3 courses) of elective course work (does not include MTH 511 or MBA 620) at the graduate level;

· three credit hours devoted to a capstone project requirement in the Mathematics Clinic (MTH 541);

· completion of 10 workshops in topics essential to successful performance in the corporate world. This requirement is referred to as a Business Survival Kit.

Upon admission to the program, the student will be assigned an advisory committee consisting of a primary advisor and two other committee members. The primary advisor is a member of the Department of Mathematics Graduate Faculty. The two additional committee members can be selected from the faculty of the Departments of Mathematics, Economics & Finance, Computer Science, or Engineering Management & Systems, and at least one of these two members can be selected from local or regional business or industry.

Up to 6 credits of courses equivalent to the courses listed for this program may be transferred from another university, provided that they were never used as credit toward any degree at any institution.

The Department of Mathematics will also offer three different post-baccalaureate Certificate programs in Financial Mathematics: a Certificate in Computational Finance, a Certificate in Financial Statistics, and a Certificate in Financial Risk Management. The Certificate programs are designed for students who do not want to commit to a full program of study at the Master level. The requirements related to each of these Certificates are stated below under the title, 6. Certificate Programs.

The proposed M.S. program is designed as a two-year program for students employed as Graduate Teaching Assistants. Students with strong undergraduate degrees in the mathematical sciences will be able to complete the degree requirements in twelve months. Hence, the proposed M.S. program is also designed as a five-year B.S.+M.S. program.

Sophisticated financial tools are now routinely employed in many areas of the corporate world. The proposed curriculum at the University of Dayton focuses on development, adaptation, and application of computational and statistical financial tools. Graduates of this program will competitively find job opportunities in Financial Risk Management, Asset Management Liability, Portfolio Management and other exciting careers in the corporate world.

1. THE INTRODUCTORY COURSES

1. Advanced Mathematical Analysis, MTH 511 (3 credit hours) Students entering with sufficient background in methods of applied mathematics will have the MTH 511 requirement waived.

Course Description: Review of basic results of calculus: power series, directional derivatives and gradient, extrema and Lagrange multipliers, and multiple integration. Advanced topics in calculus: uniform convergence of sequences of functions, uniform convergences of series of functions, implicit function theorems, and functions defined by integrals. Linear Algebra: solution of systems of equations and matrices, vector spaces, and eigenvalue problems. Multivariate calculus: directional derivatives, chain rule, Lagrange multipliers, Taylor’s formula, the mean value theorem, the inverse and implicit mapping theorems, iterated integrals and Fubini’s theorem.

2. Financial Analysis and Markets, MBA 620 (3 credit hours) Students entering with sufficient background in finance will have the MBA 620 requirement waived.

Course Description: An overview of finance to include the analysis of financial statements, valuation concepts, capital budgeting techniques, capital structure analysis, working capital management, and capital market financing instruments.

2. THE SIX REQUIRED COURSES

1. Introduction to Financial Mathematics, MTH 538 (3 credit hours)

Course Description: Markov random processes, Brownian motion in financial markets, stochastic integral, stochastic differential equation, Ito calculus, European style stock options, Single-agent consumption and investment, equilibrium in a complete market.

2. Numerical Analysis I, MTH 555 (3 credit hours)

Course Description: Computational methods for solving general mathematical problems, number representation and errors, numerical linear algebra, approximation theory and methods, linear and nonlinear systems, numerical integration, numerical solution of ordinary differential equations, error analysis and conditioning, data smoothing with applications.

3. Numerical Analysis II, MTH 556 (3 credit hours)

Course Description: Numerical solution of partial differential equations. The course materials include finite difference methods, stability, convergence, error estimate, numerical treatments of the Black-Scholes equation, splitting method, adaptive method, binomial trees and Monte Carlo computational methods.

4. Time Series, MTH 544 (3 credit hours)

Course Description: Stochastic models for discrete time series in the time-domain; smoothing with moving averages; autoregressive processes (ARMA, ARCH, GARCH); vector autocorrelation; Bayesian analysis; Kalman filter; model identification; parameter estimation; forecasting; Markov chains and changes in regimes.

5. Financial Derivatives and Risk Management, FIN 580 (3 credit hours)

Course Description: This course consists of two parts. The first part covers the usage and the pricing of derivatives – subjects include the basis features of futures and options, binomial and trinomial option pricing, the Black-Scholes formula, exotic options, swaptions, interest rate based derivatives, implied binomial trees, volatility measurement, dynamic trading strategies and varieties of exotic options. It also covers arbitrage-based derivatives pricing approaches, emphasizing economic intuition and understanding of quantitative analysis. The second part of the course covers financial risk measurement and management – including market risk, credit risk, liquidity risk, settlement risk, model risk, volatility risk, and kurtosis risk.

6. Computational Finance and Data Mining, FIN 581 (3 credit hours)

Course Description: This will be a team taught course with a faculty member from the Department of Economics and Finance and a faculty member from the Department of mathematics. Students study topics relevant to solving partial differential equations that arise in finance, topics relevant to simulation methods that arise in the analysis of financial derivatives, and topics relevant to optimization methods in finance. Topics related to partial differential equations include finite difference methods with connection to binomial models and interest rate models. Topics also include non-traditional methods such as neural networks, the genetic Algorithm, Fuzzy Logic, etc. Topics related to simulation methods include random variable generation, variance reduction methods, statistical analysis of simulation output, importance sampling, martingale control variables and stratification. With respect to optimization, both linear and nonlinear methods will be introduced.

3. MATHEMATICS CLINIC

Mathematics Clinic (three credit hours) is the capstone requirement of the program. A team of students will work on a project developing, modifying or applying sophisticated financial instruments. Projects will be sought from local and regional corporations and specifically from internship sponsors. Teams can work with faculty advisors from different departments and teams can work with advisors from the corporate world. A Mathematics Clinic project topic is pre-approved by the team members’ respective advisory committees. Teams will present their work in an oral presentation, via a public briefing, and a written report, submitted to the student’s advisory committee.

4. ELECTIVE COURSES

Regression Analysis, MTH 543 (3 credit hours)

Course Description: Least squares, lack of fit and pure error, correlation, matrix methods, F test, weighted least squares, examination of residuals, multiple regression, transformations and dummy variables, model building, ridge regression, stepwise regression, multiple regression applied to analysis of variance problems.

Partial Differential Equations, MTH 535 (3 credit hours)

Course Description: Classifications of partial differential equations; methods of solution of the wave equation, Laplace’s equation, and the heat equation; the Black-Scholes equation; applications of partial differential equations.

Ordinary Differential Equations, MTH 531 (3 credit hours)

Course Description: Existence and uniqueness theorems, linear equations and systems, self-adjoint equations, boundary value problems, and basic nonlinear techniques.

Statistics for Experimenters, MTH 547 (3 credit hours)

Course Description: Covers those areas of design of experiments and analysis of quantitative data that are useful to anyone engaged in experimental work. Designed experiments using replication and blocking. Use of transformations. Applications of full and fractional factorial designs. Experimental design for developing quality into products using Taguchi methods.

Methods of Mathematical Physics, MTH 551 (3 credit hours)

Course Description: Linear transformations and matrix theory, linear integral equations, and calculus of variations, eigenvalue problems.

Methods of Applied Mathematics, MTH 552 (3 credit hours)

Course Description: Dimensional analysis and scaling, regular and singular perturbation methods with boundary layer analysis, the stability and bifurcation of equilibrium solutions, other asymptotic methods.

Linear Algebra, MTH 565 (3 credit hours)

Course Description: Vector spaces, linear transformations and matrices, inner product spaces, invariant direct-sum decomposition and the Jordan canonical form.

Discrete and Continuous Fourier Analysis, MTH 583 (3 credit hours)

Course Description: Fourier representation of Complex valued functions, rules for finding Fourier transforms, mathematical operators associated with Fourier analysis, fast algorithms, wavelet analysis, selected applications.

Difference Equations, MTH 532 (3 credit hours)

Course Description: The calculus of finite differences; first order equations; linear equations and systems; z-transform; stability; boundary value problems for nonlinear equations; Green’s functions; control theory; applications.

Deterministic Operations Research, ENM 521 (3 credit hours)

Course Description: Introduction to deterministic methods for optimization, with a focus on mathematical programming (linear, nonlinear, integer) and network methods.

Probabilistic Operations Research, ENM 522 (3 credit hours)

Course Description: Introduction to probabilistic methods for modeling and analyzing the performance of complex systems. Topics include Markov chains, queuing, forecasting, discrete event simulation, and inventory modeling.

Data Base Management, CPS 542 (3 credit hours)

Physical and logical organization of data files; hierarchical, network, and relational database models; data definition language and data manipulation language of a commercial database management system; query languages.

5. BUSINESS SURVIVAL KIT

The Business Survival Kit is an initiative with a decidedly professional focus. It provides an explicit professional component to the financial mathematics program. As such, it will be a requirement in the program and successful completion will be transcriptable for students enrolled in the M.S. in Financial Mathematics program one of the Certificate programs. To satisfy this requirement, the student will participate in a sequence of weekend workshops. Each workshop addresses topics that help prepare the student to interact successfully in the corporate or industrial environment. Students are required to pay a small tuition to cover the cost of delivering the workshop. Topics include: Writing for Clarity, Group Communication, Business Ethics, Legal Environment of Business, Marketing Management, Operational Management, Investment and Financial Markets, Managerial Accounting, Economics. Experts from the University of Dayton (members of the Law School, School of Business Administration, English, Philosophy and Communication Departments, and the Learning Teaching Center) and experts from the local community will be retained to deliver the sequence of workshops.

Clearly, the extent of the Business Survival Kit is limited; in particular, the Business Survival Kit is not intended to serve as a “mini-MBA” and it will be implemented in an honest and ethical manner. For example, the program is transcriptable for only those students enrolled in the M.S. in Financial Mathematics program or one of the Certificate programs.

6. CERTIFICATE PROGRAMS

Certificate programs exist for students who do not want to commit to the full M.S. program. Upon successful completion of five courses focused on a specific set of concepts, a student will earn a post-baccalaureate Certificate in that area. Successful completion means a minimum GPA of 3.0 in those five courses. The Certificate programs and the associated five courses are:

Certificate in Computational Finance: CPS 542, MTH 538, MTH 555, MTH 556, FIN 591, Business Survival Kit or MTH 541

Certificate in Statistical Finance: ENM 522, MTH 538, MTH 543, MTH 544, FIN 591, Business Survival Kit or MTH 541

Certificate in Financial Risk Management: MBA 625, MTH 538, MTH 535, FIN 580, FIN 581, Business Survival Kit or MTH 541

The Certificate programs are designed as mini-programs in focus areas. Thus, each includes a capstone experience of the Business Survival Kit or the Mathematics Clinic, MTH 541. If the Certificate candidate is already an employed professional, or is enrolled as an MBA candidate, and is seeking further development with respect to quantitative methods, a capstone experience of MTH 541 fits that candidate’s program of study. If the candidate is a full-time graduate student (in a program other than the MBA) who is taking full advantage of opportunities at the University of Dayton, the Business Survival Kit fits that candidate’s program of study. A student’s Advisory Committee approves the selection of the Business Survival Kit or MTH 541 to fulfill the Certificate requirements.

There will be a simple application process for admission to a Certificate program. Students cannot simultaneously be admitted to the M.S. program in Financial Mathematics and one of the Certificate programs. Students can be simultaneously enrolled in any other post-baccalaureate program at the University of Dayton and a Certificate program. Students must meet the entrance standards of the M.S. program in Financial Mathematics to gain admission to a Certificate program.

7. CONTINUING CURRICULUM DEVELOPMENT

The program description given above follows considerable consultation between the respective departments, considerable input from the Advisory Board, input from the one-on-one interviews, and consultation with existing comparable programs. We will continually assess the curriculum so that it best serves the local and regional needs.

It is important to the success of the program that experts in finance and methods of computational financial risk management methods play an active role to develop and deliver the curriculum. Discussions are ongoing with a group of Actuaries and Certified Financial Analysts (CFA) in Asset Liability Management to continually develop the curriculum and who are interested to participate in the delivery of the curriculum.

Further opportunities with respect to the curriculum are currently and will continually be explored. For example, can the first or the first two exams in the CFA program, or other certification programs, be incorporated into the M.S. program in Financial Mathematics?

Discussions with the Departments of Computer Science and Engineering Management & Systems will continue so that database methods, information technology methods and optimization methods will be best utilized to serve the local and regional needs.

Internship possibilities exist in the insurance industry and in the banking industry. Opportunities for internships will be continually developed. MTH 541, Mathematics Clinic, will be coordinated with internships as much as possible. Moreover, long distance teaching methods will be developed.

8. LIST OF PROGRAM FACULTY

The faculty listed below demonstrates a spectrum of expertise in the curriculum for the proposed program. A wide range of research expertise will provide invaluable input in the curriculum and the curriculum development for the proposed program.

Department of Mathematics

Dr. Wiebke Diestelkamp, statistician

Dr. Stephanie Edwards, complex analyst

Dr. Paul Eloe, applied mathematician, functional differential equations

Dr. Peter Hovey, statistician

Dr. Muhammad Islam, applied mathematician, integral equations, transform methods

Dr. John Kauflin, applied mathematician, partial differential equations

Dr. Youssef Raffoul, applied mathematician, functional differential equations

Mr. Gerry Shaughnessy, statistician

Dr. Qin Sheng, computational partial differential equations

Department of Computer Science

Dr. James Buckley, data structures and data mining

Department of Engineering Management & Systems

Dr. Edward Mykytka, stochastic optimization methods

Department of Finance

Dr. Carl Chen, mutual funds performance, asset pricing anomalies, corporate governance

Dr. Peter Lung, financial risk management and derivatives

Dr. Nicholas Tay, financial risk management and derivatives