# Program

## Fall First Semester Core Courses (Click Course Title for Syllabus)

FSRM 16:958:581 Probability and Statistical Inference with Financial Applications. Prerequisites: One year of Calculus.  Probability spaces, distributions (with an emphasis on distributions that are important for financial applications, e.g. lognormal and heavy-tailed distributions), random variables, VaR, multivariate distributions (including multivariate normal), copulas, expectation, conditional probability and expectation, binomial options pricing model, law of large numbers, central limit theorem.

Theory of point and interval estimation and their relevance for estimating returns, volatility, and correlation. Topics covered include method of moments, maximum likelihood, unbiasedness, mean-squared error, sufficiency, and Cramer-Rao lower bound. Hypothesis testing. Factor models and principal components analysis in finance. Introduction to nonparametric regression.

FSRM 16:958:563 Regression Analysis in Finance. Prerequisites: Level IV Statistics. Basic concepts in Probability and Statistics and matrix algebra; Correlation and Portfolio management; Simple linear regression and capital asset pricing model; Multiple linear regression and foreign exchange models, multi-factor pricing models, Nelson-Siegel yield curves, and some hedging strategies; Nonlinear regression models and asymmetric CAPM models, nonlinear interest rate models, and regime switching models; Nonparametric regression models and nonparametric interest rate models, nonparametric implied volatility models and automatic technical analysis; Regression with ultra-high dimensional data; Principle and factor analysis.

FSRM 16:958:590 Foundations of Financial Statistics and Risk Management. The emphasis of this course will be on (1) basic banking, financial market and risk management concepts and (2) on the use of discrete stochastic models and optimization for portfolio management, derivatives pricing and risk management. Our examples will draw from many asset classes including equities, fixed income, foreign exchange, credit, mortgage backed securities and structured products.

## Spring Second Semester Advanced Courses (Click Course Title for Syllabus)

FSRM 16:958:535 Advanced Statistical Methods in Finance.  Prerequisites: 16:958:563. Conditional expectation and martingales, return and yield curve, portfolio
theory, derivatives, risk neutral measure and complete market in discrete models, forward-futures spread, Brownian motion and stochastic calculus, Girsanov’s theorem, the Black-Scholes-Merton model, Greeks, implied volatility, financial risks, value at risk, back test and stress test, estimation of volatilities and correlations, principle component analysis and factor models, credit risk, estimation of default rate, copulas, interest rate derivatives, short rate models, more if time permit.

FSRM 16:958:565 Financial Time Series Analysis. Prerequisites: 16:958:563 or permission of instructor. Features of financial time series. Model-based forecasting methods, autoregressive and moving average models, ARIMA, ARMAX, ARCH, GARCH, stochastic volatility model, regime switching models, state-space models and nonparametric time series models. Model building, estimation, forecasting and model validation, missing data, parametric and nonparametric bootstrap methods for time series. High frequency financial data, Value-at-risk and extreme value theory.

FSRM 16:958:589 Advanced programming for financial applications.  The course covers the basic concepts of object oriented programming and the syntax of the Python language.  The course objectives include learning how to go from the different stages of designing a program (algorithm) to its actual implementation. This class lays the foundation for applying Python for interactive financial analytics and financial application building.

FSRM 16:958:534 Advanced Methods for Risk Management Practice: This course added for first time in Spring 2017 as a required course for the Risk Management track option  bridges the gap  between 16:958:590 given in the first Fall semester and which establishes quantitative foundational concepts common to both the financial statistics and financial risk management tracks  and 16:958:536 given  in the final Fall semester. 16:958:536 focuses primarily on case studies, research and projects using redacted financial firm and commercial vendor risk management data and case studies. 16:958:534 given in the Spring bridges this gap by covering the analytics involved in risk management practice areas including market, liquidity, credit, operational risk management and risk models.

## Fall Third Semester Advanced Courses (Click Course Title for Syllabus)

FSRM 16:958:588 Financial Data Mining. Prerequisites: 16:958:563. Supervised and unsupervised learning; shrinkage and regularization in regression; splines and kernel smoothing; linear discriminant analysis, logistic regression, supper vector machines and regularization in classification;  model assessment, model selection and cross-validation; tree based methods and boosting; introduction to neural networks; introduction to text mining. Emphasis on the use of data mining techniques in finance.

FSRM 16:958:587 Advanced Simulation Methods for Finance. Prerequisites: 16:958:563, and 16:198:443 or equivalent C++  course or permission of instructor. Modern simulation methods and advanced statistical computing techniques for financial applications. Introduction to Monte-Carlo simulation methods, variance reduction technique, the bootstrap methods, Markov chain Monte Carlo methods, Sequential Monte Carlo method, hidden Markov models, Bayesian methods, etc. Expect to use C++ and R for programming and data analysis.  Emphasis on examples and applications from finance and risk management.

FSRM 16:958:694 Asset Allocation and Portfolio ManagementThe course will develop a general quantitative approach to modern portfolio theory, optimization, and trading. Topics to include: factor models and Arbitrage Pricing Theory (APT); modeling risk including VaR, expected shortfall, variance decompositions, contributions to risk, dynamic volatilities and correlations, etc. We then discuss valuation including fundamental analysis and dividend discount models; forecasting; event studies and cross-sectional studies; the information ratio and information horizons; Fundamental Law of Active Management, Information Coefficient, Transfer Coefficient and related issues. We go on to give a mathematical background in optimization including Lagrange multipliers and the dual, primal-dual and interior point methods, the Barrier method, and second-order cone methods. We then apply this to portfolio optimization in particular, including multi-period optimization with transaction costs and constraints. As special cases, we will discuss mean-variance, Black-Litterman and bayesian generalizations, and optimal dynamic hedging of derivatives via the offsetting replicating portfolio. We then move on to study market microstructure theory and optimal execution, including standard broker execution algos and models of the limit order book dynamics. Time permitting we may discuss various advanced topics such as the Ross Recovery Theorem and inferring information about the underlying instrument from the derivatives markets. In each case, the focus will be on using advanced statistics to achieve a deeper understanding of the model and the data. Where appropriate, we will apply the relevant statistical models to real financial data, and in the part of the course dealing with intraday data, we will discuss efficient implementations of statistical estimation procedures on large data sets.

FSRM 16:958:536 Financial Risk Evaluation and Management.  Prerequisites: 16:958:590, 16:958:534. This course deals with the practical application of risk management in financial institutions. Leading practitioners from industry teach case studies on the application of market, credit and operational risk techniques and the institutional processes for managing their implementation, as well as regulatory requirements for managing model risk in model development and validation . Makes extensive use of redacted client data in the form of case studies and projects.

### Selected non-FSRM Elective Course Descriptions

Stat 16:960:542 Life Data Analysis. Prerequisites: One year of calculus, level V statistics or permission of instructor. Statistical methodology for survival and reliability data. Topics include life table techniques; competing risk analysis; parametric and nonparametric inference for lifetime distributions; regression with censored data; Poisson and renewal processes; multi-state survival models and goodness of fit tests. Statistical software will be used.

Stat 16:960:554 Applied Stochastic Processes. Prerequisites: Advanced calculus, 16:960:582 or equivalent. Markov chains, recurrence, random walk, gambler's ruin, ergodic theory and stationary distributions, continuous time Markov chains, queuing problems, renewal processes, martingales, Markov processes, Brownian motion, concepts in stochastic calculus, Ito's formula

Stat 16:960:567 Applied Multivariate Analysis. Prerequisites: Level V statistics or permission of instructor. Methods for reduction of dimensionality, including principal components analysis, factor analysis, and multidimensional scaling; correlation techniques, including partial, multiple and canonical correlation; classification and clustering methods. Emphasis on data analytic issues, concepts and methods (e.g., graphical techniques) and on applications drawn from several areas, including behavioral, management, physical and engineering sciences

CS 16:198:513 Design and Analysis of Data Structures and Algorithms I. Prerequisites: Familiarity with Prim and Kruskal minimum spanning tree algorithms and Dijkstra shortest path algorithm. Discussion of representative algorithms and data structures encountered in applications. Worst case, average case, and amortized analysis. Data structures: search trees, hash tables, heaps, Fibonacci heaps, union-find. Algorithms: string matching, sorting and ordering statistics, graph algorithms. NP-completeness.

CS 16:198:514 Design and Analysis of Data Structures and Algorithms II. Prerequisites: 16:198:513. Advanced data structures such as splay trees, link-cut dynamic trees, and finger search trees. Models of parallel computation; selected parallel algorithms. Approximation algorithms and their performance guarantees. Probabilistic algorithms and their analysis. Primality testing. The algorithm for unification. Algorithms for computing the convex hull of a set of points in the plane. Best-first search and variations (hot-node, branch-and-bound), min-max and alpha-beta, and search on game trees. Cocke-Kasami-Younger and Earley's parsing algorithms.

CS 16:198:515 Programming Languages and Compilers I. Prerequisites: Familiarity with an imperative programming language (e.g., C), an undergraduate or graduate compilers course, and an undergraduate or graduate data structures/algorithms course. This course covers topics in programming languages and compilers such as: LL(1) and LR parsing techniques with error handling; attribute grammars and their use in syntax-directed translation; type systems and polymorphism; models of programming language semantics (i.e., operational semantics through closure interpreters ); data abstraction; functional, logical, and object-oriented paradigms; intermediate representations of programs; examples of novel programming models for cyber-physical systems; parallel programming models; automatic parallelization.

CS 16:198:516 Programming Languages and Compilers II. Prerequisites: 16:198:515. Focus on advanced, optimizing compiler design and typically includes a programming project to write an optimizing compiler.

CS 16:198:527 Computer Methods for Partial Differential Equations. Prerequisites: 16:198:510. Classes of computer methods: methods of points, methods of lines, finite elements, Ritz-Galerkin-type methods. Examples of simple computer programs. Stability, Consistency and Convergence. Hyperbolic equations. Basic properties, methods of solution, including the method of characteristics. Computer implementation, the relations between boundary conditions and extraneous solutions. Parabolic equations. Difference methods. Numerical stability ADIP method. Galerkin and finite element methods. Elliptic equations. Introduction to finite difference methods - detailed analysis of the finite element method.

Math 16:642:623 Computational Finance. Prerequisites: 16:642:621, 16:642:573, and 16:332:503, or equivalent courses. Students learn how to implement financial option-pricing and risk-management models using C++, building on previous and concurrent courses on object-oriented programming with C++, numerical analysis, and mathematical finance. MATLAB, Python, and Excel-VBA may also be used, though primarily as tools for benchmarking and C++ code interfacing. Numerical methods discussed include Monte Carlo simulation, finite difference, finite element, and spectral element solution of partial differential equations, binomial and trinomial trees, the fast Fourier transform (FFT). Asset classes discussed include equities, fixed income and interest rates, foreign exchange, and commodities, though the majority ofapplications will be for equity derivatives for simplicity and access to market data.

Math 16:642:624 Credit Risk Modeling. Prerequisites: 16:642:622 and 16:642:573 or 16:642:574. In addition to equity, interest rates, FX, and commodity derivatives, credit derivatives play an increasingly important role in financial markets. The course will include a review of jump processes; the basic theory of single name credit derivative modeling; structural, reduced form or intensity models; credit default swaps; default correlation, multiname credit derivative modeling; top down versus bottom up models; basket credit derivatives; collaterized debt obligations; and tranche options. The goal of the course is to cover most of the material in "Credit Risk Modeling" by David Lando (Princeton University Press, 2004) or "Credit Derivatives Pricing Models" by Philipp Schonbucher (Wiley, 2004).

Math 16:642:625 Portfolio Theory and Applications. 16:642:622 and 16:960:563, or an equivalent graduate course on regression analysis. The course will introduce discuss quantitative portfolio theory and related topics. It will begin with classical Markowitz theory and related analytics in a variety of real-world contexts - constrained optimization, benchmark/active optimization, risk-managed, etc. Following this, topics will include Bayesian mathematics, Black-Litterman, parameter estimation, and alternative risk measures. A heavy emphasis will be placed on programming and analytics; students will construct and manage their own portfolios under a variety of assumptions. Applications discussed during the course are implemented in MATLAB (see software section below).

ECE 16:332:503 Programming Methodology for Finance. This is a design oriented course that meets in a computer lab/classroom for maximum emphasis on hands-on programming. Lectures will be reinforced with small programming examples during the lecture, followed by homeworks and lab exercises that will focus on numerical computing and computational financial applications. The course is broken up into three major parts: the first part covers the basics of C++ syntax, data types and program structure. The second introduces object oriented programming concepts. The third part of the course covers data structures and advanced program design. All concepts and topics covered will be demonstrated using financial or numerical computing applications. The course will culminate in a final project. The textbook for the course is "How to Program in C++"by Deitel & Deitel. It is a general guide for C++ and has a supplemental lab manual.

Econ 16:220:501 Microeconomics I. Prerequisites: 16:220:500 or permission of instructor. General equilibrium theory; the Arrow-Debreu model, decision making under uncertainty; the Von Neumann-Morgenstern theory, risk aversion, applications to insurance problems and portfolio choice, applications to competitive equilibrium with uncertainty.

Econ 16:220:502 Microeconomics II. Prerequisites: 16:220:501. Introduction to the theory of games and related economic models with informational asymmetries. Topics include non-cooperative games and models of moral hazard and adverse selection.

Econ 16:220:504 Macroeconomics I. Prerequisites: 16:220:503 or permission of instructor. Introduction to economic dynamics, economic growth, business cycles, and the role of macroeconomic policy.

Econ 16:220:505 Macroeconomics II. Prerequisites: 16:220:504. General equilibrium modeling of the macroeconomy. Topics will include the stochastic growth model and multiple equilibrium. Empirical validation will also be stressed.