# MFM Curriculum

## MFM Courses

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.**

## Courses Descriptions

### FM 5091/5092 Computation, Algorithms and Coding in Finance:

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.

### FM 5011/5012 Mathematical Background for Finance:

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 5021/5022 Mathematical Theory Applied in Finance:

**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 Itō'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.

### FM 5031/5032 Practitioners Course:

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.

### FM 5001/5002 Preparation for Financial Mathematics:

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.

### FM 5990 Topics in Financial Mathematics

In this course we will learn modern tools of data science, such as classification and clustering methods, feedforward neural networks, convolutional neural networks, and recurrent neural networks, with the goal of applying them to analyze financial data. The course is structured into two parts:

- Training period (about 7 weeks) where we learn about the techniques, work on small projects to become familiar with the tools. Students may work in teams of up to 3 members on the small projects. Teams will be asked to present their work on these projects and must submit written reports on them.

- Team projects (rest of semester) where students will work in teams (team sizes of 4 to 6 people) on problems originating from industry. There will be weekly team meetings with the instructor during this period where the team will give progress reports. Each team is expected to give a final presentation and report.

### English Language Communication Workshop

The Communication Workshop is intended to help students learn and practice the principles of speaking and writing in MFM throughout the first semester. This is an ungraded, practical workshop to help students strengthen their general English language skills as well as their professional writing, presentation, and interviewing skills. Each session is primarily practice and participation based so students can develop confidence and strong communication skills to succeed in the program and in their future careers.

Minors and Additional Course Work

As an MFM student, you have the option of taking additional courses and/or getting a Minor, but it they are not required to receive your MFM. You do not need to decide at the beginning of your MFM if you will be taking additional courses or getting a minor. However, you must make that decision and include it in your degree completion plan at least nine months before you graduate.

The list below outlines the departments and subject areas where MFM students have been approved to take additional courses and/or talk with that department about pursuing a minor.

**IMPORTANT:** To receive a minor degree in any of the areas listed below, you need to get the approval of the department/program that offers the minor, to make sure that the courses you have chosen satisfy the specific minor's requirement.