**Program Information**

The Accelerated Master's Degree Program (AMDP) has specific requirements and modifications to the general course-plan mentioned below.

**Quantitative Finance and Risk Management Program **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 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 coursework described below, which includes 24 credits of required courses and 12 credits of electives for a total of 36 credits, lays the foundation for a career as a quantitative analyst in a financial or other institution. These skills will be developed through 4 sequences of required core courses. The first sequence is devoted to introducing the main concepts of Financial Mathematics and modeling; the second is devoted to building a strong mathematical background in stochastic analysis; the third focuses on numerical implementation of the mathematical methods presented in other sequences; and the last sequence in statistics prepares the students for working with financial data.

Students will also be required to take 12 credits of electives which are offered by Mathematics, Statistics, Economics, the College of Engineering (IOE and Electrical), or the Ross School of Business. These will allow students to develop skills in particular areas (i.e. deeper understanding of optimization, economics, statistical methods, data management, special features of financial markets) they are pursuing. We host a weekly financial mathematics seminar, which brings in distinguished speakers from peer institutions and research groups of financial firms. Attendance at these seminars is optional for our Master’s students.

Students who have been admitted to the program but with some missing background may need to take one or two courses during the summer preceding the first semester of the program.

The specifics of the core courses and electives, course numbers and descriptions are below.

### Curriculum

The list of required courses consists of 3 sequences of Math classes: Math 573 - Math 574, Math 526 –Math 506, Math 472 – Math 623; and a pair of Stats courses: Stats 500 – Stats 509, for a total of 24 credit hours. These courses, along with the additional elective courses, are distributed over 3 semesters.

The sequence** Math 573 – Math 574 (Advanced Financial Mathematics I and II)** introduces students to the main concepts of Financial Mathematics and Engineering. This sequence is expected to go in parallel with **Math 526 – Math 506 (Discrete State Stochastic Processes and Stochastic Analysis for Finance)**, which analyze 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.

While the courses listed above focus mainly on developing skills of model building, the **Math 472-623**sequence **(Numerical Analysis with Financial Applications and Computational Finance)**, on the other hand, 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.

**Stats 500 – Stats 509 (Applied Statistics I and Statistical Analysis of Financial Data)** introduces the basic statistical tools for financial data, including regression and time series models, as well as various inference techniques.

**Elective Courses**

Students are required to receive at least 12 credits* of the elective courses listed below.

MATH– 561 (3 cr, Linear Programming, cross-listed as IOE 510) – 562 (3 cr, Continuous Optimization Methods, cross-listed as IOE 511) – 663 (3 cr, Nonlinear Programming, cross-listed as IOE 611) |
STATS– 503 (3 cr, Applied Multivariate Analysis) – 531 (3 cr, Analysis of Time Series, cross-listed as ECON 677) – 535 (3 cr, Reliability, cross-listed as IOE 562) |

ECON– 501 (3 cr, Microeconomics) – 502 (3 cr, Macroeconomics) – 411 (3 cr, Monetary and Financial Theory) – 441 (3 cr, International Trade Theory) – 442 (3 cr, International Finance) |
EECS– 484 (4 cr, Database Management) – 492 (4 cr, Introduction to Artificial Intelligence) – 545 (3 cr, Machine Learning) – 547 (3 cr, Electronic Commerce) – 597 (3 cr, Language and Information) |

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

*Those students who have completed the Undergraduate program in Financial or Actuarial Mathematics at the University of Michigan and have taken one of the approved graduate-level elective courses below, are only required to receive 8 additional credits for elective courses. Econ 411, 441, and 442 have been approved by the Registrar for graduate credit.

*Sample course work for freestanding option for students arriving with a bachelor’s degree (Not AMDP)*

**Fall-Semester 1**

Math 526, Math 573, Stats 500, Math 472 (12 cr)

**Winter-Semester 2
**Math 506, Math 574, Stats 509, Elective (12 cr)

**Fall- Semester 3**

Math 623, Elective, Elective, Elective (12 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.