# Curriculum

## Coursework Details

### Core Courses

The following 8 core courses required for all Quant students in both the AMDP and freestanding master’s program.

**Semester 1:**

**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 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.

**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.

**Stats 500 (3 cr): Applied Statistics I****/ Statistical Learning I: Regression**

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.

**Semester 2: **

**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 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.

**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.

**Semester 3:**

**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.

### Zeyu Zhang, M.S. '18, Ph.D. Student at Carnegie Mellon University

What appealed to me most about the Quant program was the career-oriented academic environment, which gives you both knowledge in statistical data driven trading and traditional stochastic differential equation knowledge for pricing and optimization. The program director, Professor Erhan Bayraktar, is one of the most prestigious researchers in quantitative finance, so it was an amazing opportunity to have him as the instructor in one of our first courses in the program. Quant Program staff are also committed to helping students get interviews and gain skills to succeed in them.

### Electives

Quant students choose 12 or more credits of electives (3 – 5 courses) from across the university, allowing students to tailor their degree toward their area of interest, whether it’s programming, data science, finance, or a deeper understanding of mathematics. In addition to those listed below, other courses may be selected and used toward the master’s degree with advisor approval.

**Math Courses**

- MATH 561/IOE 510: Linear Programming (3 cr, F/W)
- MATH 562/IOE 511: Continuous Optimization Methods (3 cr, W)
- MATH 597: Analysis II (3 cr, W)
- MATH 602: Real Analysis II (3 cr, F)
- MATH 663/IOE 611: Nonlinear Programming (3 cr)

**Finance Courses**

- FIN 466: Real Estate investment (3 cr, W)
- FIN 551: Financial Management and Policy (3 cr, F/Su)
- FIN 575: Financial Modeling (1.5 cr)
- FIN 580: Financial Derivatives in Corporate Finance (2.25 cr, F/W)
- FIN 608: Capital Markets & Investment Strategies (2.25 cr, F)
- FIN 609: Fixed Income Securities and Markets (2.25 cr, F/W)
- FIN 612: International Finance Management I (1.5 cr, F)
- FIN 614: International Finance Management II (1.5 cr, F)
- FIN 631: Risk Management in Banks and Financial Institutions (2.25 cr)
- FIN 640: Financial Trading (1.5 cr)
- FIN 645: Real Options in Valuation (2.25 cr)
- FIN 725: Maize and Blue Fund (1.5 cr, F)
- FIN 726: Maize and Blue Fund (1.5 cr, W)

**Statistics Courses**

- STATS 415: Data Mining (4 cr, W)
- STATS 503: Statistical Learning II: Multivariate Analysis (3 cr, W)
- STATS 504: Statistical Consulting (3 cr, F)
- STATS 507: Data Science and Analytics Using Python (3 cr, W)
- STATS 531: Analysis of Time Series (3 cr, W)
- STATS 535: Reliability (3 cr)
- STATS 600: Linear Models (3 cr, F)
- STATS 607: Programming and Numerial Methods in Statistics (1.5 cr, F/W)

**Computer Science Courses**

- EECS 402: Programming for Scientists and Engineers (4 cr, F/W)
- EECS 477: Intro to Algorithms (4 cr)
- EECS 484: Database Management (4 cr, F/W)
- EECS 498: Special Topics (select sections)
- EECS 492: Introduction to Artificial Intelligence (4 cr)
- EECS 545: Machine Learning (3 cr, F/W)
- EECS 547: Electronic Commerce (3 cr)
- EECS 586: Design and Analysis of Algorithms (4 cr)
- EECS 592: Foundations of Artificial Intelligence (4 cr)
- EECS 595: Natural Processing Language (3 cr, W)
- 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)

**Other Elective Courses**

- BIOSTAT 615: Statistical Computing (3 cr, F)
- BIOSTAT 650: Applied Statics I: Linear Regression (4 cr, F)
- ENGR 599: Special Topics in Engineering – Multidisciplinary Design Projects (1-4 cr, F/W)
- HS 650: Predictive Analytics (4 cr, F)
- SI 507 : Intermediate Programming (3 cr, F/W)
- SI 618: Data Manipulation and Analysis (3 cr)
- TO 513: Spreadsheet Modeling and Applications (1.5 cr, F/W)
- TO 618: Applied Business Analytics and Decisions (3 cr, W)