The Master of Financial Mathematics (MFM) program consists of four course sequences:
| Fall | Spring |
|---|---|
| FM 5091 | FM 5092 |
| FM 5011 | FM 5012 |
| FM 5021 | FM 5022 |
| FM 5031 | FM 5032 |
These sequences may be taken either in parallel or sequentially, following their numerical order (with the exception of FM 5091/5092, -that is recommended to be taken as early as possible.)
The MFM program is structured with a balance between theory and practice in mind. It features rigorous coursework in mathematics and statistics alongside a practicum course that is comprised of a variety of modules taught by industry professionals. In order to provide additional experience, students are invited to participate in modeling workshops in which they work in teams on a project mentored by a financial industry practitioners.
Student admitted in the program often have a variety of backgrounds. Some need to “brush up” on their mathematical foundation in order to be better prepared for the high level of mathematics throughout the MFM courses. The sequence FM 5001/5002 is offered to these students. It is not a core sequence of the MFM program, however, students who do not have the appropriate mathematical background are required to complete it before taking any of the FM 5011/5012, FM 5021/5022 and FM 5031/5032. Together with FM 5091/5092, FM 5001/5002 form the Fundamentals of Quantitative Finance (FQF) Post Baccalaureate Certificate program.
The MFM program welcomes working individuals who are planning to change careers or enhance their understanding of quantitative finance. The courses are offered in the evening and structured to accommodate full time working students.
This sequence is intended to introduce students to the principles and practicalities of programming in the context of finance. Students will first be exposed to programming through MATLAB during the first semester. A large minority of the first semester and the whole of the second semester are then dedicated to learning C#. The class is project-based and students are evaluated on the quality and functionality of their code. All projects aim to solve practical finance problems dealing with financial derivatives, simulation, and optimization.
A theoretical sequence that focuses on graduate level mathematics and statistics that builds a solid foundation for modeling and using financial data.
FM 5011: This course covers the basics of probability and measure theory useful in stochastic calculus. Its purpose is to develop many of the advanced mathematical tools that are necessary for the understanding of stochastic calculus and the derivation of the Black Scholes option pricing formula. Topics will include: Sample spaces, Lebesgue measure and Lebesgue integral, limit theorems, martingales, elements of stochastic processes (example: Brownian motion), stochastic integration and Ito’s lemma, stochastic differential equations, some numerical approximations (example: Euler and Milstein), the derivation of the Black-Scholes option pricing formula.
FM 5012: The objective of this course is to introduce core ideas behind statistical methods and optimization techniques, with a special focus on their application in financial mathematics. Topics include univariate and multivariate random variables, distributions, time series analysis - especially ARMA and GARCH models, univariate and multivariate regressions and optimization – theory and applications.
FM 5021: Linear contracts: forwards, futures and swaps. Arbitrage. Cash and carry arguments. Introduction to the valuation of options, binomial tree approach, delta-hedging argument in discrete time. and Monte Carlo simulation. Basic properties of the Brownian Motion, stochastic integral and Itô’s lemma. Black-Scholes-Merton model. Delta-hedging argument in continuous time. Greek letters. Volatility smiles.
FM 5022: Review of Black-Scholes, Greeks and shortcomings. Value at risk. Principal component analysis. Introduction to time series, applications to volatility estimation: ARCH, GARCH. Exotic Options. Stochastic and local volatility models. Equivalent martingale measure approach. Interest rate derivatives, standard market models, 1-factor and 2-factor models of the short rate. Heath-Jarrow-Morton model. LIBOR market model.
This practicum sequence features six modules that are taught by financial industry practitioners. Each module is an independent mini-course that exposes students to various aspects of financial practice.
Risk and Asset Allocation I & II: The objective of the two modules is to provide students with grounding in theoretical and applied statistics as it relates to investments, with emphases on risk measurement and decision techniques for portfolio design.
Fixed Income Markets: In this module the investment-bank fixed-income trading environment and whole-world fixed-income products are reviewed. We discuss how prices are quoted on dealer screens and how mark-to-market values are computed for portfolios of these products. The focus is on the key concept of delta-neutral spread trades – a long position (receiving fixed) in one security vs. a short position (paying fixed) in another, with the sizes on each side chosen to make the position insensitive to small shifts in the rate curve. Dealers use these spread positions to help neutralize the risk on their books during the trading day, and relative-value traders use them to profit from market irregularities. Finally, basic objectives for relative-value portfolio management and basic trading strategies are discussed, and the behavior of a few real-world fixed-income spread positions over time is examined. Material is constantly updated to keep relevancy in this rapidly evolving financial world.
Copula Models and MCMC: The objective of this module is to give students an introduction to applications of copula models in finance. The examples considered include extreme co-movements of international markets, CDO pricing, and mortgage portfolio risk. The module also touches on several techniques of statistical estimation including maximum likelihood and Bayesian approaches. In order to implement the latter, Markov chain Monte Carlo methods (MCMC) are introduced.
Analysis and Valuation of Mortgage Backed Securities: This module is designed to be an introduction to the Valuation of Mortgage Backed Securities. Static and dynamic interest rate models are used to evaluate mortgage prepayment risks using risk neutral projections of interest rates and corresponding mortgage cash flows. Mortgage cash flows are developed from the most basic assumption that prepayments do not occur. The course continues with stylized views of prepayments and a fully dynamic prepayment model that is driven by the current level of mortgage rates relative to the contractual rate. The remaining part of the course is spent analyzing how various securitization structures can be used to transfer risk from one MBS tranche to another.
Volatility Models in Finance: This module is a survey of equity volatility models used in finance. The course objective is to develop a familiarity with equity models that can be used to accurately price and evaluate risks in exotic options. Students will learn to price options under each model with both closed-form solutions and Monte Carlo methods. Models used throughout the course will include Black’s time-dependent model, Dupire’s local volatility model, and Heston’s stochastic volatility model. With each model, the students will learn the dynamics of the underlying asset and how those dynamics affect the pricing and risk of both vanilla and exotic options.
This is a two-semester refresher sequence in the mathematical background needed for the Financial Mathematics Master Degree program at the University of Minnesota, especially for the sequence FM 5011/5012. The main topics for review are calculus (especially multivariable calculus), linear algebra, probability theory, and differential equations (including an introduction to partial differential equations, with most of the emphasis on linear algebra and probability theory. It is assumed that students have had at least a course in multivariable calculus. Throughout the course, connections are made to basic concepts in financial mathematics.
Choosing an emphasis is not required to complete a Master of Financial Mathematics, but some emphasis options are detailed below.
Fina 6321: Portfolio Analysis and Management - Introduces analytical concepts used to manage security portfolios from perspective of an institutional investor. Market microstructure. Margin purchasing, short selling. Portfolio risk management, risk/return tradeoffs, strategic/tactical asset allocation, active versus passive management. Portfolio revision, performance evaluation.
MBA 6230: Financial Management - Tools/concepts of financial management. Emphasizes use by financial/non-financial managers to measure creation of value within an organization. Evaluating businesses/business opportunities, identifying financial requirements/sources.
MBA 6030: Basic principles of financial accounting involving the consecution/interpretation of corporate financial statements
A MFM with an Emphasis in Statistics requires students to take Stat 5302 and two of the other three class choices.
Stat 5302: Applied Regression Analysis - Simple, multiple, and polynomial regression. Estimation, testing, prediction. Use of graphics in regression. Stepwise and other numerical methods. Weighted least squares, nonlinear models, response surfaces. Experimental research/applications.
Stat 5401: Applied Multivariate Methods - Bivariate and multivariate distributions. Multivariate normal distributions. Analysis of multivariate linear models. Repeated measures, growth curve and profile analysis. Canonical correlation analysis. Principle components and factor analysis. Discrimination, classification, and clustering.
Stat 5601: Nonparametric Methods - Order statistics. Classical rank-based procedures (e.g., Wilcoxon, Kruskal-Wallis). Goodness of fit. Topics may include smoothing, bootstrap, and generalized linear models.
Stat 8511: Time Series Analysis - Time series as multivariate samples of size 1. Discrete and continuous parameter time series. Stationarity. Autocovariance and autocorrelation. ARIMA models, identification, estimation, diagnostic checking. Other time domain models. Forecasting, seasonal adjustment, time series regressions. Introduction to frequency domain methods.
A MFM with an Emphasis in Applied Economics requires students to take APEC 5151, APEC 8202 or APEC 8206, and one of these course sequences: APEC 5481, APEC 8802 and APCE 8803, or APEC 8601.
APEC 5151: Applied Microeconomics: Firm and Household (3 credits) - Quantitative techniques for analysis of economic problems of firms and households. Links between quantitative tools and economic analysis Regression analysis, mathematical programming, and present value analysis.
APEC 5481: Futures and options Markets (3 credits) - Economic concepts related to futures/options trading. Hedging, speculation.
APEC 8202: Mathematical Optimization in Applied Economics (3 credits) - Economic foundations and applications of mathematical and dynamic programming and optimal control. Mathematical optimization concepts; structures and economic interpretations of various models of the firm, consumer, household, sector, and economy. Model building and solution techniques.
APEC 8206: Dynamic Optimization: Applications in Economics and Management (3 credits) - Formulation/solution of dynamic optimization problems using optimal control theory and dynamic programming. Analytical/numerical solution methods to solve deterministic/stochastic problems for various economic applications.
APEC 8601: Natural Resource Economics (3 credits) - Economic analysis of resource use/management. Capital theory, dynamic resource allocation. Applications to renewable/nonrenewable resources. Empirical studies, policy issues.
Students must receive permission to pursue a MFM with Emphasis in Economics from the MFM Program Director and Economics DGS
A MFM with an Emphasis in Economics requires students to take ECON 8105, ECON 8106, ECON 8107, ECON 8108, and ECON 8001, ECON 8002 or ECON 8101, ECON 8102.
ECON 8001: Microeconomic Analysis (2 credits) - Theories of consumer demand, producer supply, and market equilibrium; general equilibrium and welfare. Sample topics: externalities, economics of information and uncertainty, and game theory. This seven-week course meets with 4161.
ECON 8002: Microeconomic Analysis (2 credits) - Theories of consumer demand, producer supply, and market equilibrium; general equilibrium and welfare. Sample topics: externalities, economics of information and uncertainty, and game theory. This seven-week course meets with 4162.
ECON 8101: Microeconomic Theory (2 credits) - Decision problems faced by the household and firm; theories of choice under conditions of certainty and uncertainty. Partial equilibrium analysis of competition and monopoly. General equilibrium analysis. Welfare economics: economic efficiency of alternative market structures, social welfare functions. Dynamics: stability of markets, capital theory. Seven-week course.
ECON 8102: Microeconomic Theory (2 credits) - Decision problems faced by the household and firm; theories of choice under conditions of certainty and uncertainty. Partial equilibrium analysis of competition and monopoly. General equilibrium analysis. Welfare economics: economic efficiency of alternative market structures, social welfare functions. Dynamics: stability of markets, capital theory. Seven-week course.
ECON 8105: Macroeconomic Theory (2 credits) - Dynamic general equilibrium models: solving for paths of interest rates, consumption, investment, prices. Models with uncertainty, search, matching, indivisibilities, private information. Implications for measurement and data reporting. Overlapping generations and dynasty models. Variational and recursive methods. This seven-week course meets with 4165.
ECON 8106: Macroeconomic Theory (2 credits) - Dynamic general equilibrium models: solving for paths of interest rates, consumption, investment, prices. Models with uncertainty, search, matching, indivisibilities, private information. Implications for measurement and data reporting. Overlapping generations and dynasty models. Variational and recursive methods. This seven-week course meets with 4166.
ECON 8107: Macroeconomic Theory (2 credits) - Dynamic general equilibrium models: solving for paths of interest rates, consumption, investment, prices. Models with uncertainty, search, matching, indivisibilities, private information. Implications for measurement and data reporting. Overlapping generations and dynasty models. Variational and recursive methods. This seven-week course meets with 4167.
ECON 8108: Macroeconomic Theory (2 credits) - Dynamic general equilibrium models: solving for paths of interest rates, consumption, investment, prices. Models with uncertainty, search, matching, indivisibilities, private information. Implications for measurement and data reporting. Overlapping generations and dynasty models. Variational and recursive methods. This seven-week course meets with 4168.