Our faculty routinely update and adapt offerings to address new trends in the subject area and based on student feedback. Syllabi provided are examples from past courses, and future courses are subject to change at the discretion of the instructor. For information on degree program requirements, please refer to the MS curriculum overview. Certificate program courses are subject to the requirements of the respective certificate.
Undergraduate-Level Preparatory Courses
Undergraduate credit does not count toward the MS-CFRM degree nor the Computational Finance certificate requirements.
CFRM 405, 410, 415, and 420 comprise the Quantitative Fundamentals of Computational Finance certificate.
This course reviews the mathematical methods fundamental for the study of quantitative and computational finance. The areas of focus include calculus and multivariable calculus, constrained and unconstrained optimization, and linear algebra. Topics covered include the following:
- Functions and inverse functions
- Limits, derivatives, partial derivatives, and chain rule
- Integrals and multiple integrals, changing the order of differentiation and integration
- Taylor series approximations
- Newton’s method
- Lagrange multiplier method
- Vector and matrix arithmetic, determinants, eigenvalue-eigenvector decomposition, singular value decomposition
- Numerical methods for optimization
Upon completion of the course students will know the fundamental mathematical concepts needed to effectively study quantitative finance areas such as fixed income, options and derivatives, portfolio optimization, and quantitative risk management.
Either AMATH 352, MATH 136, or MATH 308.
Topics covered include the following:
- Probability theory: set theory, probability spaces, joint probability, conditional probability, Bayes theorem
- Univariate and multivariate random variables: distribution and density functions, moments, normal and fat-tailed skewed distributions, linear and non-linear transformations, conditional expectations
- Limit theorems: random variable convergence types, law of large numbers, central limit theorem
Parameter estimation theory: variance, bias and mean-squared error, maximum likelihood estimation of mean and standard deviation for normal distributions and location and scale for non-normal distributions
Upon completion of the course students will know the basic probability and statistics tools needed to effectively study quantitative finance areas such as fixed income, options and derivatives, portfolio optimization, and quantitative risk management.
This course will introduce students to the fundamentals of financial derivatives. Topics will include the basics of interest rates and present value calculations, term structure of interest rates, the concepts of financial arbitrage, the pricing of futures, forwards, and call/put options, and the binomial lattice.
Upon successful completion of the course the student will be able to:
- Convert between different interest rate compounding conventions
- Price the current value of future cash flows and compute discount factors
- Understand the term structure of interest rates
- Understand the pricing of forwards and futures and the use of futures for hedging risk
- Understand how to use no-arbitrage arguments to value options and financial derivatives contracts
- Write computer programs to compute implied volatility and price an option contract using a binomial lattice
CFRM 405 and 410; may be taken concurrently.
This course is an introduction to computational finance and financial econometrics. The course uses the material contained in CFRM 405 and CFRM 410 to build and analyze statistical models for asset returns.
- Asset return calculations
- Probability and statistics applied to asset returns, including: univariate and multivariate distributions, covariance, descriptive statistics, time series concepts, estimation, hypothesis testing, Monte Carlo simulation, bootstrap standard errors
Asset return calculations
- Optimization methods involving equality and inequality constraints
- Matrix algebra
- Statistical distributions and models for asset returns
- Value-at-risk, expected shortfall and portfolio risk budgeting
- Mean-variance portfolio theory
- Statistical analysis of portfolios
- Capital asset pricing model (CAPM)
- Investment performance measurement and analysis
Upon completion of the course, students will be able to apply the fundamental mathematical and statistical concepts needed to estimate and analyze statistical models for asset returns and to apply these models to portfolio theory and risk analysis.
CFRM 405, CFRM 410, or permission of instructor.
This course is an introduction to the R programming language (r-project.org) for students who have had little to no prior computer programming experience in R. Students will learn the fundamentals of R programming.
- R language syntax and control
- R data structures
- Data import and export capabilities
- Functions and scripts
- Graphics and plotting
- R Package system
Upon completion of the course students will know to write R scripts to access data, perform basic analysis, and graphing for data visualization.
Prior experience in another programming language is desirable.
Mandatory MS-CFRM Courses
The following courses are part of the core Master’s program curriculum and must be completed by all students seeking the degree.
CFRM 501, 502 and 503 comprise the Computational Finance certificate.
This course is an introduction to the mathematical, statistical and financial foundations of investment science. Learning of the theoretical concepts will be re-enforced through use of R computing exercises. The material is similar in scope to an MBA level investments course, but at a significantly higher quantitative level. Topics include:
- Basic Theory of Interest Rates
- Term Structure of Interest Rates and Fixed Income Securities
- Mean-Variance Portfolio Theory
- Factor Models and CAPM
- Risk Preferences and Measures
Coursework in multivariate calculus, linear algebra, and one-dimensional optimization at the level of CFRM 405, and probability and statistics at the level of CFRM 410. Familiarity with the material in CFRM 420 is desirable.
Introduces financial modeling and data analysis for computational finance application. Focuses on the statistical analysis, modeling methods, and computational techniques in key quantitative finance areas including factor modeling, financial time series, and portfolio analytics. Topics include:
- Financial data visualization
- Regression methods and factor models
- Principal component analysis
- Financial time series modeling
- Parametric vs. non-parametric methods
CFRM 501, or equivalent and competency in R at the level of CFRM 425, or permission of instructor.
Covers long-only and long-short portfolio optimization with real-world constraints and costs using industrial strength optimization software; classical mean-variance and modern mean-versus downside risk optimization for dealing with fat-tailed skewed asset returns; optimization and risk analysis with factor models; and equity, mixed asset class, and fund-of-hedge portfolios. Topics include:
- Mean-variance optimization (MVO) theoretical foundations
- Portfolio performance analysis and back-testing
- Downside and tail risk measures, expected shortfall, coherent risk measures
- Expected utility and risk sensitive portfolio optimization
- Active portfolio management and active management characterization
- Factor portfolios, strategies, and risk analysis
CFRM 501 and CFRM 502, or permission of instructor.
This course provides basic knowledge of the theory, statistical modeling and computational methods of pricing options and other derivative products. The course blends mathematical and statistical theory with hands-on computing. Topics include:
- Introduction to financial contracts: forwards, futures, and options
- The Binomial Model for Derivatives Pricing
- Arbitrage and the fundamental theorems of asset pricing
- Incomplete markets
- Introduction to stochastic calculus
- Black-Scholes and the Greeks (price sensitivities)
CFRM 425 and CFRM 501, or equivalent. CFRM 502 is recommended.
This course covers a broad range of standard and specialized Monte Carlo methods in finance with applications to derivatives pricing, trading, and risk management. Students will learn the theoretical rationale for the methods as well as practical implementations through programming assignments. Topics include:
- Simulating random variables: inverse transform, acceptance-rejection methods, etc.
- Simulating stochastic processes: Brownian motions, stochastic differential equations, jump diffusions
- Variance reduction methods: antithetic variables, control variates, importance sampling, conditional MC
- Vanilla & exotic derivatives pricing and sensitivity estimation via Monte Carlo
- Risk management applications: calculating VaR and CVaR, portfolio risk estimation
CFRM 504 or equivalent, or permission of instructor.
Addresses ethical theory to recognize and demonstrate an applied understanding of ethical conduct in financial markets, financial management and financial services. Explore assessments of, and responses to, ethical challenges in finance. Includes financial law and regulation. Involves case studies with modern applications related to computational finance and risk management.
Required Course Option
One of the following courses must be completed in order to meet MS-CFRM degree requirements.
Working financial analytics practitioners regularly need to access data stored in SQL databases. In addition, it is common for the results of an analysis to be summarized and distributed via an Excel spreadsheet. This course provides practical lessons in the retrieval and manipulation of data using SQL, VBA, and Excel. In addition, it shows how to leverage the powerful financial data modeling and analysis capabilities of R in conjunction with the use of SQL, VBA and Excel. Topics include:
- SQL query development
- Relational database and table design, indexes, triggers, constraints, and stored procedures
- SQL database access from R via DBI
- Data analysis with Excel pivot tables and Solver
- Excel VBA object model and VBA procedure development
- Common VBA application coding exercises encountered in quantitative finance and risk management professions
- SQL database access from Excel
- Excel and R interoperability
- Special topics of current interest in finance
CFRM 425, CFRM 501 or equivalent, or instructor permission.
This course provides an introduction to numerical optimization methods in finance. The course will discuss the theory and efficient solution methods for major classes of optimization problems. Theoretical concepts will be paired with example applications and computing exercises. Homework problems will include use of an industrial strength optimizer to solve finance applications. Topics include:
- Linear Programming Theory, Algorithms and Applications: feasible sets, duality, optimality conditions, simplex method, interior point methods, sensitivity analysis, asset/liability cash flow matching
- Quadratic Programming Theory, Algorithms and Applications: constrained and unconstrained programming, optimality conditions, solution methodologies, mean-variance optimization, relationships to statistical regression, Black-Litterman, returns-based style analysis, risk-neutral density estimation
- General Non-Linear Programming Theory, Algorithms and Applications: univariate and multivariate models, convexity, non-smooth optimization, GARCH model fitting, volatility surface estimation
- Integer Programming Theory, Algorithms and Applications: cutting plane methods, index replication
- Combinatorial and Network Programming Theory, Algorithms and Applications: shortest path, min-cost flow, foreign exchange, arbitrage checking
- Cone Programming Theory, Algorithms and Applications: second-order cone programming, tracking error and volatility constraints, estimating covariance matrices
- Dynamic Programming Theory, Algorithms and Applications: Bellman equations, forward and backward recursion, knapsack problem, option pricing, structured products
- Stochastic Programming Theory, Algorithms and Applications: data uncertainty, multi-stage models, recourse, value at risk, conditional value at risk, asset/liability management, CVaR, transaction costs
- Robust Optimization Theory, Algorithms and Applications: parameter uncertainty, robust constraints, robust objectives, single-period and multi-period portfolio selection
- Additional Topics: Decomposition and Column Generation, Genetic Algorithms, Non-gradient me
CFRM 502 or equivalent, or instructor permission. CFRM 503 is recommended.
Elective MS-CFRM Courses
The following courses may be taken as desired in order to meet the 42-credit minimum of the MS-CFRM program, subject to listed prerequisites.
As described in course information in Time Schedule.
This course is a practical introduction to C++ programming for financial applications. The course will focus on developing basic object oriented programming skills in C++ to implement computational finance solutions. Course coverage will also include integrating C++ applications with R, Excel, and the Boost C++ Libraries. Topics include:
- C++ language, syntax, and control
- The C++ Standard library, and new features in C++14 and evolving standards
- Rcpp interface between R and C++
- xlw interface from Excel to C++
- Integration with mathematical functions in the Boost C++ Libraries
- Introduction to C++ templates
- Modern C++ implementation of financial day count conventions, term structures, and bond calculations
- Real-world application of parallelization and random number generation features in C++ to problems in pricing, trading, and risk management
CFRM 502 and CFRM 504, which may be taken concurrently, or permission of instructor.
Introduces the fundamentals of machine learning techniques with applications to finance. Focuses on assessing, organizing, and analyzing financial data, and learning the analytical tools and numerical schemes in machine learning to perform statistical analysis on financial data. Develop practical financial tools such as trading rules and risk indicators. Topics include:
- Data-driven models for decision making
- Supervised and Unsupervised Learning
- Optimization: formulations and algorithms
- Selected topics, including time series analysis
CFRM 502 or equivalent, programming skills in R or MATLAB.
- Asset classes, financial instruments, and trade orders
- Direct access trading and market microstructure
- Simulated quantitative trade execution and management with trading platform
- Quantitative trading strategy development, optimization, and backtesting with R and the quantstrat package stack
CFRM 420 and CFRM 425, or equivalent.
This course will provide a detailed research process and tools for replicating, assessing, conceptualizing, and developing systematic trading strategies. Students will apply their knowledge in hands-on projects to replicate and evaluate existing research and to create and evaluate a new strategy model.
Development of systematic trading strategies should follow a highly scientific and repeatable process. This course will start by reviewing categories of systematic strategies, drawing out patterns followed throughout the industry.
We will demonstrate a repeatable process for evaluating ideas, constructing hypotheses, building each of the strategy components, and evaluating and improving the strategy at each step. Students will use the R Language for Statistical Computing and Graphics to replicate academic research and evaluate the claims made in papers. Students will also construct a non-trivial strategy from scratch, evaluate the power of each of its components, and examine the likelihood of overfitting. The strategy will be documented and presented in lieu of a final exam.
CFRM 522 or permission of instructor.
Course description coming soon.
Covers financial technology (FinTech) innovations and development, and the associated computational finance and risk management methods and perspectives. Includes real-world applications, including robo-advising, AI and Machine Learning for trading, etc. Also covers blockchain technology with focus on its applications to finance, especially cryptocurrencies. Course topics coming soon.
CFRM 501, and CFRM 506 or CFRM 507, or equivalent, or instructor permission. Ability to program and compile in R and/or Python.
This course provides an introduction to fixed income markets and securities, as well as solid foundation in fixed income analytics and associated portfolio management, along with real-world fixed income applications. Students will have hands-on experience with fixed income data and computational methods. Topics include:
- Fixed income markets and instruments
- Term structure of interest rates and yield curve construction
- Binomial tree model for pricing of interest rate derivatives
- Continuous-time stochastic fixed income models
- Pricing & hedging of interest rate forwards, swaps, futures and options
Solid understanding of multivariable calculus, linear algebra, probability, and statistics at least at the level of CFRM 405 and CFRM 410.
This course covers fundamental principles of portfolio performance measurement and benchmarking. Topics include:
- The role of performance evaluation in portfolio management
- Rate of return calculations for individual assets and for portfolios
- Manipulating returns: linking, averaging, annualizing
- Adjustments for inflation, currency, taxes, fees
- Cash flow methods: time-weighted returns, money-weighted returns, standard approximations
- Excess returns, arithmetic and geometric
- Sector-based performance attribution
- Volatility and asset pricing-based risk measures
- Risk-adjusted return measures
- Factor-based performance attribution
- Uses of indexes: benchmarking, asset allocation, and the basis for investment vehicles
- Benchmark construction principles and practical issues
- Index calculations, weighting, rebalancing, and maintenance
- Equity style indexes
- GIPS: Global Investment Performance Standards
CFRM 420 and CFRM 501, or equivalent.
The course will focus on the endowment management process and specific challenges facing institutional fund managers. These include evaluating the role of an endowment, portfolio construction, risk management, manager selection, and alternative asset class investing. The course utilizes concepts from finance and investments, macroeconomics, and mathematical optimization. Topics include:
- Endowment policy background and philosophy
- Risk and asset allocation
- Emerging market investing
- Fixed incomes role in endowment
- Liquidity and investing in private equity.
- Reading assignments will form the basis for class discussion and students are expected to be prepared for case discussions.
CFRM 501 or equivalent. A general understanding of economics and a good background in core finance and portfolio optimization, e.g., CFRM 503 is preferred.
This course covers the methodologies used to manage financial risk. Emphasis is given to fixed income and foreign exchange derivatives. Topics include:
- Overview of fixed income products and players in the market
- Introduction to Market Risk (duration, convexity, VaR, expected shortfall)
- Introduction to Credit Risk (secured vs. unsecured, probability of default, loss given default, credit VaR)
- Introduction to Operational, Model and Liquidity Risk
- Enterprise Risk Management and Governance (Economic Capital, RAROC, Basel)
CFRM 501 or equivalent, or permission of instructor.
This course provides a comprehensive treatment of the theoretical concepts and modeling techniques, and statistical methods for quantitative risk management with practical applications and hand-on computational experience. It covers methods for market, credit, and operational risk modeling, and includes modeling volatility and correlation for quantitative risk management. Topics include:
- Empirical properties of asset returns and associated statistical models
- Volatility estimation and modeling
- Risk concepts
- Multivariate dependence of risk factors
- Credit risk: models and risk management
CFRM 502 or equivalent.
This course is an introduction to the mathematical, statistical and financial foundations of models for analyzing, predicting, and mitigating credit risks. Students will learn the theoretical basis for widely-used modeling methods for credit risk assessment and implement those methods through programming assignments using R. The course will focus on both obligor-level and portfolio-level credit risks, as well as valuation and risk analysis of assets and derivatives with credit risk. Topics include:·
- Applied logistic regression (credit scoring models)
- Merton model for default risk
- Mortgage credit risks and cash flows
- Credit risks in structured asset backed securities
- Credit default swaps, models for valuation and risk analysis
CFRM 501, CFRM 541 or equivalents, or permission of instructor.
This is an introductory course on stochastic calculus for computational finance and risk management. The course covers concepts in probability theory and stochastic processes, and discusses a number of fundamental theorems and results in stochastic calculus, along with their applications to finance. Topics include:
- Martingales and stopping times
- Brownian motion and Brownian bridge
- Stochastic differential equations (SDE’s) and Ito’s formula
- Solutions to SDEs and associated stochastic models
- Black-Scholes option pricing model and partial differential equations
- Feynman-Kac Formula and their financial applications
CFRM 405 or equivalent. CFRM 504 is strongly recommended.
Upon completion of the course, students will be able to:
- Implement appropriate methodologies for modeling, implementing and estimating energy price processes (Geometric Brownian motion, mean reversion, jump diffusion)
Implement various numerical techniques for valuing options, modeling physical assets and measuring risk
- How to treat generation assets (thermal, wind and hydro power plants) as ‘real options’
- Determine the effects of operational constraints and emissions on the value of generation assets
- Understand the different techniques for valuing a portfolio of gas storage contracts
- Analyze a portfolio of gas swing contracts including key features such as minimum bill make up
- Understand what constitutes a ‘best-in-class’ risk management within an energy company
- Understand the types and relevance of available market risk metrics for energy portfolios
- How to include physical assets and complex contracts into market risk calculations such as value-at-risk, cash-flow-at-risk, profit-at-risk, gross-margin-at-risk, revenue-at-risk and earnings-at-risk
- Understand different credit risk measures including potential future exposure and credit VaR as applied to energy portfolios
Basic familiarity with R programming.
This course is an introduction to the role that forecasts can play in investment decisions, especially investing that involves views on short-term opportunities that are implemented through informed rebalancing or explicit asset class tilts away from benchmark. Learning of the theoretical concepts will be re-enforced through use of computing exercises. Topics include:
- Types of forecasts, dynamic forecasts, direct forecasts
- Forecasts by simulation for nonlinear models
- The role of macroeconomic forecasts in investing
- An approach to macroeconomic forecasting
- Asset class returns forecasts
- Ways to combine forecasts using dynamically updated weights
- Ways to account for nonlinearity
- Foreign exchange (FX) forecasts: carry trade motive, momentum strategies, incorporating long-run valuation correction
CFRM 502 or equivalent.
Special CFRM Electives
Elective AMATH Courses
AMATH 582 and 583 are acceptable elective courses in the CFRM MS program. Please review the current course information on the Applied Math site.