These course descriptions provide a general overview of course coverage and credit awarded (in parentheses following the course name). Our faculty routinely update and adapt offerings to address new trends in the subject area and based on student feedback. For information on degree 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 CFRM MS degree or the Computational Finance certificate requirements.

CFRM 405, 410, 415, and 420 comprise the Quantitative Fundamentals of Computational Finance certificate.

**CFRM 405:**Mathematical Methods for Quantitative Finance (3)

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.

**Prerequisites**

University-level calculus courses that include an introduction to multivariable differential calculus (equivalent to UW MATH 124, 125 and 126); additional introductory mathematics and statistics coursework is desirable.

**CFRM 410:**Probability and Statistics for Computational Finance (3)

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.

**Prerequisites**

At least one undergraduate statistics course, e.g. UW STAT 311. Prior or concurrent enrollment in CFRM 405 is also helpful.

**CFRM 415:**Introduction to Financial Markets (3)

**Learning Objectives:**

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

**Prerequisites**

Mathematics and probability/statistics at the level of CFRM 405 and 410.

**CFRM 420:**Introduction to Computational Finance and Financial Econometrics (3)

- 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.

**Prerequisites**

Prior completion or concurrent enrollment in CFRM 405 and 410 or permission of instructor.

**CFRM 425:**R Programming for Quantitative Finance (1)

Formerly CFRM 463. 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.

Topics include:

- 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.

**Prerequisites**

Prior experience in another programming language is desirable.

# Mandatory CFRM MS Courses

The following courses are part of the core Master’s program curriculum and must be completed by all students seeking the degree.

CFRM 501-503 comprise the Computational Finance certificate.

**CFRM 501:**Investment Science (4)

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 (compounding, present value, internal rate of return)
- Fixed Income Securities (bonds, value formulas, yield, duration, convexity, immunization)
- Term Structure of Interest Rates (discount factors, forward rates, short rates)
- Mean-Variance Portfolio Theory (efficient frontiers, quadratic utility, benchmark tracking)
- Factor Models (CAPM, linear regression and prediction, multi-factor models, APT)
- General Principles (expected utility maximization, coherent and tail risk measures)
- Futures and Forwards (futures and forward prices, margin, hedging with futures)
- Options Part 1: (option payoffs, trading strategies, binomial models, risk neutral pricing)
- Options Part 2: (Ito process and lemma, GBM, Black-Scholes, hedging, implied volatility)

**Prerequisites**

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.

**CFRM 502:**Financial Data Modeling and Analysis in R(4)

This course is an in-depth hands-on introduction to the R statistical programming language (www.r-project.org) for computational finance. The course will focus on R code and code writing, R packages, and R software development for statistical analysis of financial data including topics on factor models, time series analysis, and portfolio analytics. Topics include:

- The R Language. Syntax, data types, resources, packages and history
- Graphics in R. Plotting and visualization
- Statistical analysis of returns. Fat-tailed skewed distributions, outliers, serial correlation
- Financial time series modeling. Covariance matrices, AR, VecAR
- Factor models. Linear regression, LS and robust fits, test statistics, model selection
- Multidimensional models. Principal components, clustering, classification
- Optimization methods. QP, LP, general nonlinear
- Portfolio optimization. Mean-variance optimization, out-of-sample back testing
- Bootstrap methods. Non-parametric, parametric, confidence intervals, tests
- Portfolio analytics. Performance and risk measures, style analysis

**Prerequisites**

CFRM 501 or equivalent and competency in R at the level of CFRM 425 or permission of instructor.

**CFRM 503:**Portfolio Optimization and Asset Management (4)

The topics to be covered are listed below. On average one lecture will be devoted to each topic. The content coverage will blend theory and methods with computing using R and R packages.

- Mean-variance optimization (MVO) theoretical foundations
- Numerical MVO with box, group, turnover constraints and concentration penalties
- Covariance matrix estimator methods: shrinkage, robustness
- Portfolio performance analysis and back-testing
- Downside and tail risk measures, expected shortfall, coherent risk measures
- Expected shortfall based portfolio optimization theory and computing
- Expected utility and coherent risk measure portfolio optimization
- Non-convex portfolio optimization: DE Optim and random portfolios
- Active portfolio management and active management characterization
- Information ratio and the fundamental law of active management
- Time series factor model fitting, risk analysis and hedge funds applications
- Fundamental factor model fitting, risk analysis and equity portfolio applications
- Factor mimicking portfolios and factor investing
- Marginal contributions to risk, risk-parity portfolios, equal weight portfolios
- Bayes models and Bayes-Stein shrinkage estimator
- Black-Litterman model and its calibration
- Statistical factor models for portfolio optimization and risk analysis

**Prerequisites**

CFRM 501 and 502 or permission of instructor.

**CFRM 504:**Options and Other Derivatives (4)

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. The first part of the course will emphasize options on stocks, stock indices, currencies and futures, and the latter part will focus on interest rate derivatives. Course work includes assignments in theory and computation, and either a final exam or a project.

- Brief review of forwards, futures, and options basics
- Black-Scholes theory and dynamic hedging with the Greeks
- Volatility estimation, implied volatility, the volatility smile
- Option prices using additive and multiplicative binomial, and use of trinomial trees
- Option pricing under fat-tailed non-normality
- Computational methods for exotic options and complex derivatives
- Brief review of interest rate basics: zero rates, forward rates and term structure Interest rate derivatives: standard market models, short rate and advanced models
- Analytic models and tree models for pricing interest rate derivatives
- Valuation of bonds with embedded options, option adjusted spreads

**Prerequisites**

CFRM 425 and CFRM 501, or equivalent. CFRM 502 is recommended.

**CFRM 505:**Monte Carlo Methods in Finance (4)

This course covers a broad range of standard and specialized Monte Carlo methods in finance with a focus on accurate derivative pricing. Students will learn the theoretical rationale for the methods and will gain applications knowledge through programming assignments using R or Matlab. The course will begin with an overview Monte Carlo methods and a review of basic derivative pricing method. Topics covered will include:

- Derivative pricing methods: replication, no-arbitrage, risk-neutral pricing, change of numeraire
- Random number generators: linear congruential generators, lattice structure, simulation error
- Sampling methods: inverse transform, acceptance-rejection methods
- Mulivariate random numbers: normal distributions, t-distributions, stable distributions
- Simulating sample paths: univariate and multivariate GBM, path-dependent options, short-rate models and bond prices
- Simulating advanced models: square-root diffusions and bond prices, forward rate models and pricing derivatives, jump processes
- Variance reduction methods: antithetic variables, control variates, stratified sampling, Latin hypercube sampling, matching methods, importance sampling
- Discretization methods: Euler method, second-order methods, applications to extremes and barrier crossings
- Estimating the Greeks sensitivity measures: finite-difference approximations, pathwise derivative estimates, likelihood ratio method
- Pricing American options: random tree methods, stochastic mesh methods, regression methods
- Risk management applications: calculating VaR and CVaR, calculating VaR and CVaR portfolio risk decompositions, delta-gamma based variance reduction, methods for fat-tailed distributions

**Prerequisites**

CFRM 504 or equivalent, or permission of instructor.

**CFRM 509:**Ethics in the Finance Profession (2)

This course description is pending.

# Required Course Option

One of the following courses must be completed in order to meet CFRM MS degree requirements.

**CFRM 506:**Financial Data Access & Analysis with SQL, VBA, Excel (4)

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 use of SQL, VBA and Excel. Course topics include:

- SQL query development
- SQL database access from R via DBI and RODBC
- Data analysis with PowerPivot
- Excel VBA object model and VBA procedure development
- SQL database access from VBA
- Web data access from VBA
- Excel and R interoperability
- VBA client and R server computing

Course will include a project that involves an end-to-end implementation of an analytic solution that emulates the type of implementation using SQL, VBA and Excel that will be required by a finance industry organization.

**Prerequisites**

CFRM 425 or instructor permission.

**CFRM 507:**Optimization Methods in Finance (4)

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

**Prerequisites**

CFRM 502 or equivalent, or instructor permission. CFRM 503 is recommended.

# Elective CFRM MS Courses

The following courses may be taken as desired in order to meet the 42-credit minimum of the CFRM MS program, subject to listed prerequisites.

**CFRM 500:**Special Studies in Computational Finance (variable credit)

**Prerequisites**

As described in course information.

**CFRM 520:**Financial Software Development and Integration with C++ (4)

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 and Excel. Course topics include:

- C++ language, syntax, and control
- Object-oriented programming
- The C++ Standard library
- Rcpp interface from R to C++
- Rinside interface from C++ to R
- xlw interface from Excel to C++
- COM interface with C++
**.**

**Prerequisites**

CFRM 502 and CFRM 504, which may be taken concurrently, or permission of instructor.

**CFRM 521:**Machine Learning for Finance

**(4)**

Course description pending.

**CFRM 522:**Introduction to Electronic Trading (4)

Includes an overview of financial markets, instruments, exchanges, and the electronic trading process. Students will then us a paper trading account with Interactive Brokers (http://www.interactivebrokers.com) to explore electronic trading of stocks, futures, and ETFs. After this preliminary material, students will learn to use the R language for statistical computing (http://www.r-project.org) to develop, evaluate, backtest, and optimize quantitative trading strategies using the R packages xts, quantmod, blotter, quantstrat, and PerformanceAnalytics.Topics include:

- Asset classes, financial instruments, and trade orders
- Direct access trading and market microstructure
- Interactive Brokers, Traders Workstation, and the IB Student Trading Lab
- Quantitative trading strategy development with the TradeStation Platform
- Quantitative trading strategy with R and the quantstrat package
- Trading strategy evaluation, optimization, and backtesting

**Prerequisites**

CFRM 420 and CFRM 425, or equivalent.

**CFRM 523:**Advanced Trading Systems (4)

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.

**Instructor:** Brian Peterson

**Prerequisites**

CFRM 522 or permission of instructor.

**CFRM 530:**Fixed Income Analytics and Portfolio Management (4)

This required course in the MS CFRM program provides a solid foundation in fixed income analytics and portfolio management. Course will include some lectures on real-world fixed income applications by finance industry professional guest lecturers. Computing exercises with R will be used throughout to re-enforce understanding of the theory and methods. Topics covered will include:

- Fixed income instrument types including MBS’s and municipal bonds
- Fixed income data sources, access and manipulation
- Term structure of interest rates and yield curve construction
- Interest rate risk management
- Interest rate forwards, swaps, futures and options
- Introduction to binomial tree pricing of interest rate derivatives
- Case studies

Upon successful completion, students will have a firm understanding of fixed income markets, data and analytics, and be able to apply this knowledge to fixed income portfolio construction, performance analysis and risk management.

**Prerequisites**

Good understanding of multivariable calculus, linear algebra, probability, and statistics at least at the level of CFRM 405 and CFRM 410.

**CFRM 531:**Portfolio Performance Analysis & Benchmarking (4)

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

**Prerequisites**

CFRM 420 and CFRM 501, or equivalent.

**CFRM 532:**Endowment and Institutional Investment Management

**(2)**

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. As such, the course utilizes concepts from finance and investments, macroeconomics, and mathematical optimization. Specific topics include:

- Endowment policy background and philosophy
- Spending
- 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.

**Prerequisites**

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.

**CFRM 540:**Financial Risk Management (4)

This introductory risk management course covers the methodologies used to manage financial risk. Emphasis is given to fixed income and foreign exchange derivatives. The topics covered include:

- An overview of fixed income products
- Duration and convexity and risk management of fixed income portfolios
- Black and Scholes model. Hedging and trading parameters
- Pricing options and swaps
- Introduction to term structure models
- Introduction to credit derivatives
- Introduction to mortgage-backed securities and asset-backed securities
- Introduction to hedge fund strategies and risk management

**Prerequisites**

CFRM 501 or equivalent, or permission of instructor.

**CFRM 541:**Quantitative Risk Management (4)

This is a course in quantitative risk management and financial econometrics. The focus will be on the statistical modeling of financial time series (asset prices and returns) with an emphasis on modeling volatility and correlation for quantitative risk management. The learning goals/objectives of the course are to (1) survey the relevant theoretical and practical literature; (2) introduce state-of-the-art techniques for modeling financial time series and managing financial risk; (3) use the open source R statistical software to get hands-on experience with real world data. Topics to be covered include:

- Empirical properties and stylized facts of asset returns
- Probability distributions and statistical models for asset returns
- Risk concepts
- Volatility modeling
- Extreme value theory
- Multivariate dependence using copulas
- Introduction to credit risk models and management

**Prerequisites**

CFRM 502 or equivalent.

**CFRM 542:**Credit Risk Management (4)

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:

- Credit risk drivers and portfolio diversification (idiosyncratic and systemic risks)
- Applied logistic regression (credit scoring models)
- Credit rating products for individuals and corporations (FICO, S&P, Moodys, Experian)
- Merton model for default risk
- Credit risk economic capital
- Basel II credit capital framework for banks
- Modeling loss frequency (PD) and severity (LGD)
- Credit risks in structured asset backed securities
- Credit default swaps, models for valuation and risk measurement

**Prerequisites**

CFRM 501 and 541 or equivalents, or permission of instrutor.

**CFRM 550:**Stochastic Calculus for Derivatives (4)

**Prerequisites**

CFRM 405 or equivalent. CFRM 504 is strongly recommended.

**CFRM 580:**Energy Markets Analytics and Derivatives (2)

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, etcetera)

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

**Prerequisites**

Basic familiarity with R programming.

**CFRM 586:**Financial Time Series Forecasting Methods

**(4)**

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

**Prerequisites**

CFRM 502 or equivalent.

## Special CFRM Electives

**CFRM 600:**Independent Research or Study (variable credit)

**CFRM 601:**Internship or CPT (variable credit)

**CFRM 700:**Master’s Thesis (variable, but a minimum of 9 credits required)

# 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.