The goal of the program is to provide graduates with a strong mathematical background and to develop skills necessary to apply their expertise to the solution of real-world finance problems. Students develop modeling skills so that they are able to formulate a well-posed mathematical problem from a description in financial language, carry out relevant mathematical analysis using tools of stochastic analysis and probability theory, implement the results using advanced numerical methods, and interpret and make decisions based on these results.

The Quant Program requires a total of 36 credits of coursework, 24 of which are required core courses and 12 of which are electives. Most students complete the program in three semesters, but occasionally students will extend the program to a fourth term. Students in the Accelerated Master's Degree Program will complete the same coursework in a different order.

### Structure and Required Coursework

Except in special circumstances, students will take the following core required courses in the prescribed order:

### Semester 1

- Math 472: Numerical Analysis with Financial Applications
- Math 526: Discrete State Stochastic Processes
- Math 573: Advanced Financial Mathematics I
- Stats 500: Applied Statistics I

### Semester 2

- Math 506: Stochastic Analysis for Finance
- Math 574: Advanced Financial Mathematics II
- Stats 509: Statistical Analysis of Financial Data
- 3 credits of electives

### Semester 3

- Math 623: Computational Finance
- 9 credits of electives

This course plan is structured around four course sequences that serve as the foundation of the program. Successful completion of the first course in each sequence is necessary in order to move on to the second. These sequences are described below:

**I. Math 573 – Math 574 : **introduces students to the main concepts of Financial Mathematics and Engineering.

**II. Math 526 – Math 506: **analyzes in more detail the mathematical tools used in Math 573 - Math 574. The two sequences of courses discuss similar problems; however, the coursework in Math 526 – Math 506 focuses on the associated mathematical challenges, while the Math 573 - Math 574 sequence emphasizes the application of mathematical methods to the relevant problems in the financial industry.

**III. Math 472-Math 623: ** focuses on the implementation of the models using tools from numerical methods for solving partial differential equations and Monte-Carlo methods. The students will develop computer programs to calculate the prices of financial derivatives and find ways of hedging risk.

**IV. Stats 500 – Stats 509:** introduces the basic statistical tools for financial data, including regression and time series models, as well as various inference techniques.

### Electives

Students choose 12 or more credits of electives from the following list of courses. Other courses may be approved for elective credit at the discretion of the student's advisor and program administration. To obtain approval, current students should submit a recent syllabus to quantfinms@umich.edu.

**Math Courses**

- MATH 507: Mathematical Methods of Algorithmic Trading (3 cr)
- MATH 561/IOE 510: Linear Programming (3 cr)
- MATH 562/IOE 511: Continuous Optimization Methods (3 cr)
- MATH 663/IOE 611: Nonlinear Programming (3 cr)

**Statistics Courses**

- STATS 503: Applied Multivariate Analysis (3 cr)
- STATS 531/Econ 677: Analysis of Time Series (3 cr)
- STATS 535/IOE 562: Reliability (3 cr)

### Computer Science Courses

- EECS 484: Database Management (4 cr)
- EECS 492: Introduction to Artificial Intelligence (4 cr)
- EECS 545: Machine Learning (3 cr)
- EECS 547: Electronic Commerce (3 cr)
- EECS 597: Language and Information (3 cr)

### Economics Courses

- ECON 411: Monetary and Financial Theory (3 cr)
- ECON 441: International Trade Theory (3 cr)
- ECON 442: International Finance (3 cr)
- ECON 501: Microeconomics (3 cr)
- ECON 502: Macroeconomics (3 cr)

### Finance Courses

- FIN 608: Capital Markets & Investment Strategies (2.25 cr)
- FIN 609: Fixed Income Securities and Markets (2.25 cr)
- FIN 640: Financial Trading (1.5 cr)
- FIN 645: Real Options in Valuation (2.25 cr)

### Detailed Descriptions of Required Courses

**Math 573 (3 cr): Advanced Financial Mathematics I****. **

This is an introductory course in Financial Mathematics. This course starts with the basic version of Mathematical Theory of Asset Pricing and Hedging (Fundamental Theorem of Asset Pricing in discrete time and discrete space). This theory is applied to problems of Pricing and Hedging of simple Financial Derivatives. Finally, the continuous time version of the proposed methods is presented, culminating with the Black-Scholes model. A part of the course is devoted to the problems of Optimal Investment in discrete time (including Markowitz Theory and CAPM) and Risk Management (VaR and its extensions). This course shows how one can formulate and solve relevant problems of financial industry via mathematical (in particular, probabilistic) methods. Although Math 526 is not a prerequisite for Math 573, it is strongly recommended that either these courses are taken in parallel, or Math 526 precedes Math 573.

**Math 574 (3 cr): Advanced Financial Mathematics II****. **

This is a continuation of Math 573. This course discusses Mathematical Theory of Continuous-time Finance. The course starts with the general Theory of Asset Pricing and Hedging in continuous time and then proceeds to specific problems of Mathematical Modeling in Continuous-time Finance. These problems include pricing and hedging of (basic and exotic) Derivatives in Equity, Foreign Exchange, Fixed Income and Credit Risk markets. In addition, this course discusses Optimal Investment in Continuous time (Merton’s problem), High-frequency Trading (Optimal Execution), and Risk Management (e.g. Credit Value Adjustment). Although Math 506 is not a prerequisite for Math 574, it is strongly recommended that either these courses are taken in parallel, or Math 506 precedes Math 574.

**Math 526 (3 cr): Discrete State Stochastic Processes****. **

This is an introductory course in the Theory of Stochastic Processes. The topics covered include Markov and Poisson processes, basic Martingale Theory, and introduction to Brownian Motion. The mathematical theory is illustrated with many relevant examples from Economics and Finance, showing how mathematical (probabilistic) methods can be used in these fields. This course is a good complement to Math 573.

**Math 506 (3 cr): Stochastic Analysis for Finance****. **

This is a continuation of Math 526. This course covers such topics as: Stochastic Integration and Stochastic Differential Equations, Change of Measure, advanced Martingale Theory and Brownian Motion, Levy processes, and Stochastic Control. A strong emphasis is made on applications of the developed methods to the problems of Mathematical Modeling in Finance. In particular, it shows how Stochastic Analysis is applied to problems arising in Equity Derivatives, Foreign Exchange, Fixed Income and Credit Risk markets. This course also demonstrates the use of Stochastic Control in the problems of Optimal Investment and Optimal Execution. This is a good complement to Math 574.

**Math 472 (3 cr): Numerical Methods with Financial Applications.**** **

This is a survey course of basic numerical methods used to solve scientific problems. The emphasis is divided between the analysis of the methods, their practical applications, and getting comfortable using a computer language for implementation. Topics intended to be covered are: root finding methods; system of linear equations; interpolation and polynomial approximation; numerical differentiation and integration; numerical methods for ordinary differential equations; basic Monte-Carlo simulations and financial applications. A part of the coursework requires programming in a high-level language.

**Math 623 (3 cr): Computational Finance.**

This is a continuation of Math 472. This course starts with the introduction to numerical methods for solving differential equations of evolution, including the Partial Differential Equations (PDEs) of parabolic type. Convergence and stability of explicit and implicit numerical schemes is analyzed. Examples include the generalized Black- Scholes PDE for pricing European, American and Asian options. Another part of the course is concerned with the Monte Carlo methods. This includes the pseudo random number generators (with applications to option pricing) and numerical methods for solving stochastic differential equations (with applications to Stochastic Volatility models). Finally, the students are introduced to the idea of calibration, which allows one to determine the unknown model parameters from observed quantities (typically, prices of financial products). The calibration is first formulated as a general inverse problem, then, the solution methods are presented in several specific settings. The theory is accompanied by applications of proposed numerical methods in particular models of Stochastic Volatility and Interest Rate models. This includes an in-depth study of numerical methods for pricing, hedging and calibration in the Hull-White and Black-Derman-Toy models. A part of the coursework requires programming in a high-level language.

**Stats 500 (3 cr): Applied Statistics I****. **

This course introduces the essentials of linear models. Topics include linear models, model fitting, identifiability, collinearity, Gauss-Markov theorem, variable selection, transformation, diagnostics, outliers and influential observations, ANOVA and ANCOVA, and common designs. Applications and real data analysis are emphasized, with students using the computer to perform statistical analyses.

**Stats 509 (3 cr): Statistical Analysis of Financial Data****. **

This course will cover basic topics involved in modeling and analysis of financial data. These include linear and non-linear regression, nonparametric and semi-parametric regression, selected topics on the analysis of multivariate data and dimension-reduction, and time series analysis. Examples and data from financial applications will be used to motivate and illustrate the methods.