Mandatory Courses Offered in the Fall Semester

MATH GR 5010 Introduction to the Mathematics of Finance

The mathematics of finance, principally the problem of pricing of derivative securities, developed using only calculus and basic probability. Topics include mathematical models for financial instruments, Brownian motion, normal and lognormal distributions, the Black-Scholes formula, and binomial models.

STAT GR 5263 Statistical Inference / Time-Series Modeling

Modeling and inference for random processes, from natural sciences to finance and economics. ARMA, ARCH, GARCH, and nonlinear models, parameter estimation, prediction, and filtering.

STAT GR 5264 Stochastic Processes – Applications

Basics of continuous-time stochastic processes. Wiener processes. Stochastic integrals. Ito’s formula, stochastic calculus. Stochastic exponentials and Girsanov’s theorem. Gaussian processes. Stochastic differential equations. Additional topics as time permits.

Mandatory Courses Offered in the Spring Semester

STAT GR 5265 Stochastic Methods in Finance

Mathematical theory and probabilistic tools for modeling and analyzing security markets are developed. Pricing options in complete and incomplete markets, equivalent martingale measures, utility maximization, term structure of interest rates.

MATH GR 5030 Numerical Methods in Finance

Review of the basic numerical methods for partial differential equations, variational inequalities, and free-boundary problems. Numerical methods for solving stochastic differential equations; random number generation, Monte Carlo techniques for evaluating path-integrals, numerical techniques for the valuation of American, path-dependent and barrier options.

MATH GR 5050 Practitioners’ Seminar

The seminar consists of presentations and mini-courses by leading industry specialists in quantitative finance. Topics include portfolio optimization, exotic derivatives, high frequency analysis of data and numerical methods. While most talks require knowledge of mathematical methods in finance, some talks are accessible to a general audience.

These courses are not mandatory. MAFN students may choose their electives from across the university, subject to the constraints of the MAFN degree requirements and the constraints imposed by the schools and departments offering the courses.

Math GR 5220 Quantitative Methods in Investment Management

Surveys the field of quantitative investment strategies from a “buy side” perspective through the eyes of portfolio managers, analysts, and investors. Financial modeling there often involves avoiding complexity in favor of simplicity and practical compromise. All necessary material scattered in finance, computer science, and statistics is combined into a project-based curriculum, which gives students hands-on experience to solve real-world problems in portfolio management. Students will work with market and historical data to develop and test trading and risk management strategies. Programming projects are required to complete this course.

Math GR 5280 Capital Markets and Investments

Risk/return tradeoff, diversification and their role in the modern portfolio theory, their consequences for asset allocation, and portfolio optimization. Capital Asset Pricing Model, Modern Portfolio Theory, Factor Models, Equities Valuation, definition and treatment of futures, options, and fixed income securities will be covered. Many business school finance courses have a Capital Markets and Investments prerequisite, and Math GR5280 satisfies this prerequisite. However, even if you satisfy the prerequisite, there is no guarantee that you can cross-register into any particular business school course.

Math GR 5300 Hedge Funds Strategies and Risk

The hedge fund industry has continued to grow after the financial crisis, and hedge funds are increasingly important as an investable asset class for institutional investors as well as wealthy individuals. This course will cover hedge funds from the point of view of portfolio managers and investors. We will analyze a number of hedge fund trading strategies, including fixed income arbitrage, global macro, and various equities strategies, with a strong focus on quantitative strategies. We distinguish hedge fund managers from other asset managers and discuss issues such as fees and incentives, liquidity, performance evaluation, and risk management. We also discuss career development in the hedge fund context.

Math GR 5340 Fixed Income Portfolio Management

Prerequisites: Students should be comfortable with algebra, calculus, probability, statistics, and stochastic calculus. The course covers the fundamentals of fixed income portfolio management. Its goal is to help the students develop concepts and tools for the valuation and hedging of fixed income securities within a fixed set of parameters. There will be an emphasis on understanding how an investment professional manages a portfolio given a budget and a set of limits.

Math GR5400 Non-Linear Option Pricing

Prerequisites: We assume familiarity with Brownian motion, Itô’s formula, stochastic differential equations, and Black-Scholes option pricing.

Nonlinear Option Pricing is a major and popular theme of research today in quantitative finance, covering a wide variety of topics such as American option pricing, uncertain volatility, uncertain mortality, different rates for borrowing and lending, calibration of models to market smiles, credit valuation adjustment (CVA), transaction costs, illiquid markets, super-replication under delta and gamma constraints, etc.

The objective of this course is twofold: (1) introduce some nonlinear aspects of quantitative finance, and (2) present and compare various numerical methods for solving high-dimensional nonlinear problems arising in option pricing.

This course also exposes the students with a wide variety of Machine Learning techniques, old and new. These techniques allow us to compute some quantities that are key ingredients of the nonlinear Monte Carlo algorithms.

Math GR 5420 Modeling and Trading Derivatives

Prerequisites: Math GR5010 Required: Math GR5010 Intro to the Math of Finance (or equivalent),Recommended: Stat GR5264 Stochastic Processes – Applications I (or equivalent). The objective of this course is to introduce students, from a practitioner’s perspective with formal derivations, to the advanced modeling, pricing and risk management techniques of vanilla and exotic options that are traded on derivatives desks, which goes beyond the classical option pricing courses focusing solely on the theory. It also presents the opportunity to design, implement and backtest vol trading strategies. The course is divided in four parts: Advanced Volatility Modeling; Vanilla and Exotic Options: Structuring, Pricing and Hedging; FX/Rates Components: Discounting, Forward Projection, Quanto and Compo Options; Designing and Backtesting Vol Trading Strategies in Python.

Math GR 5430 Machine Learning for Finance

The application of Machine Learning (ML) algorithms in the Financial industry is now commonplace but still nascent in its potential. This course provides an overview of ML applications for finance use cases, including trading, investment management, and consumer banking. Students will learn how to work with financial data and how to apply ML algorithms using the data. In addition to providing an overview of the most commonly used ML models, we will detail the regression, KNN, NLP, and time-series deep learning ML models using desktop and cloud technologies. The course is taught in Python using Numpy, Pandas, scikit-learn, and other libraries. Basic programming knowledge in any language is required.

Math GR 5440 Price Impact Models and Applications to Quantitative Trading

At the end of the course, students are expected to understand how to design live trading experiments, fit price impact models and apply price impact models to a broad set of quantitative strategies. Special emphasis is placed on acquiring the ability to communicate precise assumptions and actionable results to a general audience within the finance community. The class is divided into three modules: (a) a quick primer on trading, the role of price impact in quantitative finance and the database language kdb+ (b) real-life applications of price impact models within trading teams, including optimal execution, statistical arbitrage, and liquidity risk management (c) the design and study of live trading experiments using causal inference with applications to Transaction Cost Analysis (TCA) and high frequency trading.

Math GR 5510 MAFN Fieldwork

Prerequisites: all 6 MAFN core courses, at least 6 credits of approved electives, and the instructors’ permission. This course provides an opportunity for MAFN students to engage in off-campus internships for academic credit that counts towards the degree. Graded by letter grade. Students need to secure an internship and get it approved by the instructor. This course provides an opportunity for MAFN students to engage in off-campus internships for academic credit that counts towards the degree.This course helps the students understand the job search process and develop the professional skills necessary for career advancement. The students will not only learn the best practices in all aspects of job-seeking but will also have a chance to practice their skills. Each class will be divided into two parts: a lecture and a workshop. In addition, the students will get support from Teaching Assistants who will be available to guide and prepare the students for technical interviews.

Math GR 5520 Career Development for Quantitative Finance

This course helps the students understand the job search process and develop the professional skills necessary for career advancement. The students will not only learn the best practices in all aspects of job-seeking but will also have a chance to practice their skills. Each class will be divided into two parts: a lecture and a workshop. In addition, the students will get support from Teaching Assistants who will be available to guide and prepare the students for technical interviews.

NOTE: Offered to first semester MAFN students only.

Stat GU 4205 Linear Regression Models

Theory and practice of regression analysis, Simple and multiple regression, including testing, estimation, and confidence procedures, modeling, regression diagnostics and plots, polynomial regression, colinearity and confounding, model selection, the geometry of least squares. Extensive use of the computer to analyze data. For students who do not have a strong background in Linear Regression.

Stat GU 4224 Bayesian Statistics

This course introduces the Bayesian paradigm for statistical inference. Topics covered include prior and posterior distributions: conjugate priors, informative and non-informative priors; one- and two-sample problems; models for normal data, models for binary data, Bayesian linear models; Bayesian computation: MCMC algorithms, the Gibbs sampler; hierarchical models; hypothesis testing, Bayes factors, model selection; use of statistical software. Prerequisites: A course in the theory of statistical inference, such as STAT GU4204, a course in statistical modeling and data analysis, such as STAT GU4205.

Stat GU 4291 Advanced Data Analysis

Prerequisites: STAT GU4205 and at least one statistics course numbered between GU4221 and GU4261. This is a course on getting the most out of data. The emphasis will be on hands-on experience, involving case studies with real data and using common statistical packages. The course covers, at a very high level, exploratory data analysis, model formulation, the goodness of fit testing, and other standard and non-standard statistical procedures, including linear regression, analysis of variance, nonlinear regression, generalized linear models, survival analysis, time series analysis, and modern regression methods. Students will be expected to propose a data set of their choice for use as case study material.

Stat GR 5206 Statistical Computation and Intro Data Science

Corequisites: STAT GR5204 and GR5205 or the equivalent. Introduction to programming in the R statistical package: functions, objects, data structures, flow control, input and output, debugging, logical design, and abstraction. Writing code for numerical and graphical statistical analyses. Writing maintainable code and testing, stochastic simulations, parallelizing data analyses, and working with large data sets. Examples from data science will be used for demonstration.

NOTE: Opens to MAFN students during the second week of classes.

Stat GU 5261 Statistical Methods in Finance

Prerequisites: STAT GU4205 or the equivalent. A fast-paced introduction to statistical methods used in quantitative finance. Financial applications and statistical methodologies are intertwined in all lectures. Topics include regression analysis and applications to the Capital Asset Pricing Model and multifactor pricing models, principal components and multivariate analysis, smoothing techniques and estimation of yield curves statistical methods for financial time series, value at risk, term structure models and fixed income research, and estimation and modeling of volatilities. Hands-on experience with financial data.

Math GR5260 Programming for Quantitative & Computational Finance

This course covers programming with applications to finance. The applications may include such topics as yield curve building and calibration, short rate models, Libor market models, Monte Carlo simulation, valuation of financial instruments such as options, swaptions, and variance swaps, and risk measurement and management, among others. Students will learn about the underlying theory, learn coding techniques, and get hands-on experience in implementing financial models and systems. The Spring version of this course uses Python.

Math GR5320 Financial Risk Management and Regulation

Prerequisites: The student is expected to be mathematically mature and to be familiar with probability and statistics, arbitrage pricing theory, and stochastic processes. The course will introduce the notions of financial risk management, review the structure of the markets and the contracts traded, introduce risk measures such as VaR, PFE, and EE, overview regulation of financial markets, and study a number of risk management failures. After successfully completing the course, the student will understand the basics of computing parametric VaR, historical VaR, Monte Carlo VaR, credit exposures, and CVA and the issues and computations associated with managing market risk and credit risk. The student will be familiar with the different categories of financial risk, current regulatory practices, and the events of financial crises, especially the most recent ones.

Math GR5360 Math Methods in Financial Price Analysis

This course covers modern statistical and physical methods of analysis and prediction of financial price data. Methods from statistics, physics, and econometrics will be presented with the goal to create and analyze different quantitative investment models.

Math GR5380 Multi-Asset Portfolio Management

The course will cover practical issues such as: how to select an investment universe and instruments, derive long-term risk/return forecasts, create tactical models, construct and implement an efficient portfolio, to take into account constraints and transaction costs, measure and manage portfolio risk, and analyze the performance of the total portfolio.

Math GR5430 Machine Learning for Finance

The application of Machine Learning (ML) algorithms in the Financial industry is now commonplace but still nascent in its potential. This course provides an overview of ML applications for finance use cases, including trading, investment management, and consumer banking. Students will learn how to work with financial data and how to apply ML algorithms using the data. In addition to providing an overview of the most commonly used ML models, we will detail the regression, KNN, NLP, and time-series deep learning ML models using desktop and cloud technologies. The course is taught in Python using Numpy, Pandas, scikit-learn, and other libraries. Basic programming knowledge in any language is required.

Math GR5450 Credit Analytics

This course uses a combination of lectures and case studies to introduce students to the modern credit analytics. The objective for the course is to cover major analytic concepts, ideas with a focus on the underlying mathematics used in both credit risk management and credit valuation. We will start from an empirical analysis of default probabilities (or PD), recovery rates and rating transitions. Then we will introduce the essential concepts of survival analysis as a scientific way to study default. For credit portfolio we will study and compare different approaches such as CreditPortfolio View, CreditRisk+ as well as copula function approach. For valuation we will cover both single name and portfolio models.

Math GR 5460 Interest Rates Models

The objective of this course is to introduce students to interest rates models and to build step by step a coherent understanding of the interest rates world, from the stripping of a yield curve to the modern frameworks of option pricing. Adopting a practitioner’s perspective, it will put an emphasis on building a strong intuition on the products and models, and will adopt a balanced approach between formal derivation and concrete applications.

Math GR5470 Generative AI in Finance

Generative AI is projected to have a transformational impact to the finance industry and macro-economy at large, with Goldman Sachs estimating a potential 7% increase in global GDP. For this exact reason investment banks and hedge funds have established highly specialized Generative AI Quant teams dedicated with the strategic roll-out in all facets of the firm’s operations, from the business to control functions. We see an immense shift of focus and potential with applications across banking, asset management, client advisory, risk management, investment research and many more.

The Generative AI in Finance course at Columbia University’s Mathematics of Finance (“MAFN”) program is exactly geared towards equipping students with the skills necessary to drive and lead the AI transformation in the financial services sector.

Math GR5480 Quantitative Balance Sheet Strategy (QBSS) for Commercial Banks

In response to increasing complexity of the operating business environment, commercial banks continue to invest in transformation of their balance sheet management functions which includes strengthening their quantitative balance sheet strategy (QBSS) capabilities.

This transformation improves quantitative analytics rigor around the both existing Asset & Liability Management (ALM) processes (which are traditionally focused on interest rate risk management) and the evolving areas of bank balance sheet strategy such as management of liquidity and capital positions as well as better understanding of the consumer market trends for bank balance sheet products (e.g., loans and deposits).

Quantitative approach improves banks’ abilities to systematically analyze and address some of their major strategic balance sheet challenges such as navigating macroeconomic cycles, managing evolving regulatory requirements and optimally responding to increasing competitive pressures.

Banking industry events of recent years demonstrated the vital importance of a thoughtful quantitative approach in managing bank balance sheets. This continues to drive the industry’s demand for top quant talent proficient in bank balance sheet strategy matters.

The course provides a practitioner’s perspective on applying a quantitative approach to defining and executing on commercial bank balance sheet strategy. The students will receive working knowledge of analyzing, managing and optimizing interest rate risk, liquidity risk and capital risk exposures of commercial bank balance sheets. The course also provides an overview of quantitative approaches to evaluation of business trends within the US commercial banking industry which informs balance sheet strategy of a typical commercial bank.

Throughout the course we will refer to a stylized bank balance sheet model to illustrate strategy concepts discussed in lectures. The students will be invited to use this model as a starting point for their group projects.

In addition to applied quantitative skills, the course focuses on fostering teamwork and presentation skills which contributes towards a well-rounded set of hard and soft skills required in the growing business field of Quantitative Balance Sheet Strategy (QBSS).

Math GR5510 MAFN Fieldwork

Prerequisites: All 6 MAFN core courses and at least 6 credits of approved electives. As a consequence, this course is not open to students in their first two semesters. This course provides an opportunity for MAFN students to engage in off-campus internships for academic credit that counts towards the degree.

For course description, rules, and procedures, please see Fieldwork Course (CPT)

Stat GR5206 Statistical Computation and Intro Data Science

Introduction to programming in the R statistical package: functions, objects, data structures, flow control, input and output, debugging, logical design, and abstraction. Writing code for numerical and graphical statistical analyses. Writing maintainable code and testing, stochastic simulations, parallelizing data analyses, and working with large data sets. Examples from data science will be used for demonstration.

Stat GR5241 Statistical Machine Learning

Prerequisites: STAT GR5206 or the equivalent. The course will provide an introduction to Machine Learning and its core models and algorithms. The aim of the course is to provide students of statistics with detailed knowledge of how Machine Learning methods work and how statistical models can be brought to bear in computer systems – not only to analyze large data sets but to let computers perform tasks that traditional methods of computer science are unable to address. Examples range from speech recognition and text analysis through bioinformatics and medical diagnosis. This course provides a first introduction to the statistical methods and mathematical concepts which make such technologies possible.

Stat GR6153 Probability II

Prerequisites: MATH GR6151 MATH G4151 Analysis & Probability I. Elements of the general theory of random processes. Martingales, the Dood-Meyer, and optional decomposition theorems. Stochastic integration, the resulting Ito calculus, Girsanov change of measure, local time, integral equations, and diffusions. Entropy production for Markov processes, connections with optimal transport, gradient flows.

Stochastic theory of portfolios. Notions of deflators, numeraires, viability, and arbitrage. The growth-optimal portfolio and its properties. The basic problems of hedging, and of portfolio optimization, in finance.

Courses approved as electives for the MAFN degree but not offered by the MAFN program taken by our students in the last two years.

• (PhD) Foundations Of Optimization: DROMB9118
• (PhD) Foundations Of Stochastic Modeling: DROMB9119
• Accounting For Value: ACCTB8022
• Adv Analytic Techniques: QMSSG5018
• Advanced Corporate Finance: FINCB8307
• Advanced Data Analysis: STATW4291
• Advanced Machine Learning: STATW5242
• Advanced Software Engineering: COMSW4156
• Algorithms For Data Science: CSORW4246
• Analysis & Probability: MATHG6151
• Analysis Of Algorithms: CSORW4231
• Applied Data Science: STATW4243
• Applied Deep Learning: COMSW4995
• Applied Financial Risk Management: IEORE4745
• Applied Machine Learning: COMSW4995
• Artificial Intelligence: COMSW4701
• Asset Management: FINCB7323
• Bayesian Statistics: STATW4224
• Computer Networks: CSEEW4119
• Cost-Benefit Analysis: INAFU6016
• Decision Models & Managem: PUAFU6033
• Deep Learning: COMSW4995
• Defining/Dev Winning Strategies: MRKTB8625
• Earnings Quality & Fundam: ACCTB8008
• Elementary Stochastic Proceses: STATW5207
• Elements Data Sci First C: COMSW4995
• Empirical Corporate Finance: FINCB9329
• Entrepreneurial Business Creation: IEORE4550
• Fin Tech & Data Driven In: STATG5293
• Foundations Of Operations: DROMB9150
• Functions Of A Complex Variable APMAE4204
• Game Theory: ECONW4415
• Honors Complex Variables :MATHW4065
• Intro Modern Analysis: MATHW4061
• Intro Modern Analysis II: MATHW4062
• Intro To Blockchain Tech: ELENE6883
• Intro-Probability & Stati: IEORE4150
• Introduction To Data Science: STATW5206
• Introduction To Databases: COMSW4111
• Linear Regression Models: STATW4205
• Machine Learning: COMSW4771
• Machine Learning Soc Sci: QMSSG5073
• Machine Learning & High- Dimensional Da: IEORE6617
• Multivariate Stat Inference: STATW4223
• Natural Lang Proc Soc Sci: QMSSG5067
• Natural Language Processes: COMSW4705
• Neural Networks & Deep Learning: ECBME4040
• Optimization I: IEORE6613
• Optimization Models And Methods: IEORE4004
• Partial Differential Equations: APMAE4200
• Pricing Models For Financial Engineering: IEORE4620
• Probability II: MATHG6153
• Probability Theory: STATW4203
• Programming Lang & Translators: COMSW4115
• Public Economics: ECONW4465
• Quantitative Corporate Finance: IEORE4403
• Simulation: IEORE4404
• Social Network Analysis: QMSSG5062
• Stat Inf/Time-Series Mode: STATG4263
• Statistical Machine Learn: STATW5241
• Statistical Methods In Fi: STATW4261
• Statistical Methods In Fi: STATW5261
• Structured&Hybrid Product: IEORE4735
• Topology: MATHW4051

The seminar is a required class offered exclusively to the MAFN students in the Spring semester. It consists of presentations and mini-courses by leading industry specialists in quantitative finance. Please read the list of speakers and topics covered in the seminars offered in the last two academic years.

Practitioner’s Seminar 2024

Practitioners’ Seminar 2023

Practitioners’ Seminar 2022