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AMS 507 Introduction to Probability 
The topics include sample spaces, axioms of probability, conditional probability and independence, discrete and continuous random variables, jointly distributed random variables, characteristics of random variables, law of large numbers and central limit theorem, Markov chains. Note: Crosslisted with HPH 696.
Fall, 3 credits, ABCF grading 
AMS 507 Webpage 

AMS 510 Analytical Methods for Applied Mathematics and Statistics 
Review of techniques of multivariate calculus, convergence and limits, matrix analysis, vector space basics, and Lagrange multipliers. 
Fall, 3 credits, ABCF grading
Prerequisites:  A course in linear algebra and in multivariate calculus
AMS 510 Webpage

AMS 511, Foundation of Quantitative Finance 

Introduction to capital markets, securities pricing, and modern portfolio theory, including the organization and operation of securities market, the Efficient Market Hypothesis and its implications, the Capital Asset Pricing Model, the Arbitrage Pricing Theory, and more general factor models. Common stocks and their valuation, statistical analysis, and portfolio selection in a single-period, mean-variance context will be explored along with its solution as a quadratic program. Fixed income securities and their valuation, statistical analysis, and portfolio selection. Discussion of the development and use of financial derivatives. Introduction to risk neutral pricing, stochastic calculus, and the Black-Scholes Formula. Whenever practical, examples will use real market data. Numerical exercises and projects in a high-level programming environment will also be assigned.
Prerequisites:  AMS 510
3 credits, ABCF grading 
AMS 511 webpage 

AMS 512 Capital Markets and Portfolio Theory 
Development of capital markets and portfolio theory in both continuous time and multi-period settings. Utility theory and its application to the determination of optimal consumption and investment policies. Asymptotic growth under conditions of uncertainty. Applications to problems in strategic asset allocation over finite horizons and to problems in public finance. Whenever practical, examples will use real market data. Numerical exercises and projects in a high-level programming environment will also be assigned. 
Prerequisite:  AMS 511
3 credits, ABCF grading 
AMS 512 webpage 

AMS 513 Financial Derivatives and Stochastic Calculus 
Further development of derivative pricing theory including the use of equivalent martingale measures, the Girsanov Theorem, the Radon-Nikodym Derivative, and a deeper, more general understanding of the Arbitrage Theorem. Numerical approaches to solving stochastic PDEÕs will be further developed. Applications involving interest rate sensitive securities and more complex options will be introduced. Whenever practical, examples will use real market data. Numerical exercises and projects in a high-level programming environment will also be assigned. 
Prerequisite:  AMS 511
3 credits, ABCF grading 
AMS 513 webpage 

AMS 514 Computational Finance 
Review of foundations: stochastic calculus, martingales, pricing, and arbitrage. Basic principles of Monte Carlo and the efficiency and effectiveness of simulation estimators. Generation of pseudo- and quasi-random numbers with sampling methods and distributions. Variance reduction techniques such as control variates, antithetic variates, stratified and Latin hypercube sampling, and importance sampling. Discretization methods including first and second order methods, trees, jumps, and barrier crossings. Applications in pricing American options, interest rate sensitive derivatives, mortgage-backed securities and risk management. Whenever practical, examples will use real market data. Extensive numerical exercises and projects in a general programming environment will also be assigned. 
Prerequisite:  AMS 512 and  AMS 513
3 credits, ABCF grading 
AMS 514 webpage 

AMS 515 Case Studies in Computational Finance II 
Actual applications of Quantitative Finance to problems of risk assessment, product design, portfolio management, and securities pricing will be covered. Particular attention will be paid to data collection and analysis, the design and implementation of software, and, most importantly, to differences that occur between Òtheory and practiceÓ in model application, and to the development of practical strategies for handling cases in which Òmodel failureÓ makes the naive use of quantitative techniques dangerous. Extensive use of guest lecturers drawn from the industry will be made. 
Prerequisite:  AMS 512 and  AMS 513
3 credits, ABCF grading 
AMS 515 webpage 

AMS 516, Statistical Methods in Finance
The course introduces statistical methodologies in quantitative finance. Financial applications and statistical methodologies are intertwined in all lectures. The course will cover regression analysis and applications to the Capital Asset Pricing Model and multifactor pricing models, principal components and multivariate analysis, statistical methods for financial time series; value at risk, smoothing techniques and estimation of yield curves, and estimation and modeling of volatilities.
3 credits, ABCF grading 
AMS 516 webpage 

AMS 517, Risk Management 
Quantitative methods for risk management problems including market risk, credit risk, operational risk and Basel II accord. Multivariate models; extreme value theory; structure and reduced-form models of default; and copula-based models.
Prerequisite:  AMS 511AMS 512, and  AMS 513
3 credits, ABCF grading 
AMS 517 webpage

AMS 518, Advanced Stochastic Models, Risk Assessment, and Portfolio Optimization
The course provides a thorough treatment of advance risk measurement and portfolio optimization, extending the traditional approaches to these topics by combining distributional models with risk or performance measures into one framework. It focuses on, among others, the fundamentals of probability metrics and optimization, new approaches to portfolio optimization and a variety of essential risk measures. Numerical exercises and projects in a high-level programming environment will be assigned. 
Prerequisite:  AMS 512 or instructor consent
Offered Fall semester 
3 credits, ABCF grading 
AMS 518 webpage

AMS 519, Internship in Quantitative Finance
Supervised internship in financial institution. Students will typically work at a trading desk, in an asset management group, or in a risk management group. Students will be supervised by a faculty member and a manager at their internship site. Written and oral reports will be made to both supervisors.
Offered every semester
3-6 credits, S/U Grading
AMS 519 webpage

AMS 520, Machine Learning in Quantitative Finance
This course will merge ML and traditional quantitative finance techniques employed at investment banks, asset management, and securities trading firms. It will provide a systematic introduction to statistical learning and machine learning methods applied in Quantitative Finance. The topics discussed in the course fall broadly into four categories which (as time permits) will be discussed in this order:
(1) Probabilistic Modeling: Bayesian vs. frequentist estimation, bias-variance tradeoff, sequential Bayesian updates, model selection and model averaging; Probabilistic graphical models and mixture models; Multiplicative Weights Update Method Bayesian regression and Gaussian processes.
(2) Feedforward neural networks: Feedforward architecture; Stochastic gradient descent and backpropagation algorithm; Non-Linear Factor Modeling and applications in asset pricing; Convolutional neural networks; Autoencoders.
(3) Sequential Learning: Linear time series models; Probabilistic sequence modeling – Hidden Markov Models and particle filtering; Recurrent Neural Networks; Applications in finance.
(4) Reinforcement Learning: Markov decision process and dynamic programming methods (Bellman equations and Bellman optimality); Reinforcement learning methods (Monte-Carlo methods, policy-based learning, TD-learning, SARSA, and Q-learning); Deep reinforcement learning; Applications of reinforcement learning in finance.
Prerequisite: (AMS 572 and AMS 595) or AMS 561 or Python knowledge with Instructor consent
Offered Fall semester 
3 credits, ABCF grading
AMS 520 webpage 

AMS 522, Bayesian Methods in Finance
The course explores in depth the fundamentals of the Bayesian methodology and the use of the Bayesian theory in portfolio and risk management. It focuses on, among other topics incorporating the prior views of analysts and investors into the asset allocation process, estimating and predicting volatility, improving risk forecasts, and combining the conclusions of different models. Numerical exercises and projects in a high-level programming environment will be assigned.
Prerequisite:  AMS 512 or instructor consent
Offered Spring semester 
3 credits, ABCF grading
AMS 522 webpage 

AMS 523, Mathematics of High Frequency Finance
The course explores Elements of real and complex linear spaces. Fourier series and transforms, the Laplace transform and z-transform. Elements of complex analysis including Cauchy theory, residue calculus, conformal mapping and Möbius transformations. Introduction to convex sets and analysis in finite dimensions, the Legendre transform and duality. Examples are given in terms of applications to high frequency finance. 
Offered Fall semester
3 credits, ABCF grading
AMS 523 webpage

AMS 526 Numerical Analysis I 
Direct and indirect methods for solving simultaneous linear equations and matrix inversion, conditioning, and round-off errors. Computation of eigenvalues and eigenvectors. 
Co-requisite:   AMS 510  and   AMS 595
Fall, 3 credits, ABCF grading
AMS 526 Webpage

AMS 527 Numerical Analysis II 
Numerical methods based upon functional approximation: polynomial interpolation and approximation; and numerical differentiation and integration. Solution methods for ordinary differential equations. AMS 527 may be taken whether or not the student has completed AMS 526. 
Spring, 3 credits, ABCF grading 
AMS 527 Webpage

AMS 528 Numerical Analysis III 
An introduction to scientific computation, this course considers the basic numerical techniques designed to solve problems of physical and engineering interest. Finite difference methods are covered for the three major classes of partial differential equations: parabolic, elliptic, and hyperbolic. Practical implementation will be discussed. The student is also introduced to the important packages of scientific software algorithms. AMS 528 may be taken whether or not the student has completed AMS 526 or AMS 527. 
Spring, 3 credits, ABCF grading 
AMS 528 Webpage

AMS 530 Principles in Parallel Computing 
This course is designed for both academic and industrial scientists interested in parallel computing and its applications to large-scale scientific and engineering problems. It focuses on the three main issues in parallel computing: analysis of parallel hardware and software systems, design and implementation of parallel algorithms, and applications of parallel computing to selected problems in physical science and engineering. The course emphasizes hands-on practice and understanding of algorithmic concepts of parallel computing. 
Prerequisite: A course in basic computer science such as operating systems or architectures or some programming experience 
Spring, 3 credits, ABCF grading 
AMS 530 Webpage 

AMS 540 Linear Programming 
Formulation of linear programming problems and solutions by simplex method. Duality, sensitivity analysis, dual simplex algorithm, decomposition. Applications to the transportation problem, two-person games, assignment problem, and introduction to integer and nonlinear programming. This course is offered as both MBA 540 and AMS 540.
Prerequisite: A course in linear algebra 
3 credits, ABCF grading 
AMS 540 Webpage

AMS 542 Analysis of Algorithms 
Techniques for designing efficient algorithms, including choice of data structures, recursion, branch and bound, divide and conquer, and dynamic programming. Complexity analysis of searching, sorting, matrix multiplication, and graph algorithms. Standard NP-complete problems and polynomial transformation techniques. This course is offered as both AMS 542 and CSE 548.
Spring, 3 credits, ABCF grading 
AMS 542 Webpage

AMS 550 Operations Research: Stochastic Models 
Includes Poisson processes, renewal theory, discrete-time and continuous-time Markov processes, Brownian motion, applications to queues, statistics, and other problems of engineering and social sciences. 
Prerequisite:   AMS 507  
Spring, 3 credits, ABCF grading 
AMS 550 Webpage

AMS 553 Simulation and Modeling 
A comprehensive course in formulation, implementation, and application of simulation models. Topics include data structures, simulation languages, statistical analysis, pseudorandom number generation, and design of simulation experiments. Students apply simulation modeling methods to problems of their own design. This course is offered as CSE 529, AMS 553, and MBA 553.
Prerequisite: CSE 214 or equivalent; AMS 310 or AMS 507 or equivalent; or permission of instructor 
Spring, 3 credits, ABCF grading 
AMS 553 Webpage 

AMS 560 Big Data Systems, Algorithms and Networks

Recent progress on big data systems, algorithms and networks. Topics include the web graph, search engines, targeted advertisements, online algorithms and competitive analysis, and analytics, storage, resource allocation, and security in big data systems. Offered in the Spring Semester
3 credits, Letter graded (A, A-, B+, etc.)
AMS 560 Webpage

AMS 561 Introduction to Computational and Data Science
This course provides a foundation of knowledge and basic skills for the successful application in graduate research of modern techniques in computational and data science relevant to engineering, the humanities, and the physical, life and social sciences. It is consciously crafted to provide a rich, project-oriented, multidisciplinary experience that establishes a common vocabulary and skill set. Centered around the popular programming language Python, the course will serve as an introduction to programming including data structures, algorithms, numerical methods, basic concepts in computer architecture, and elements of object-oriented design.  Also introduced will be important concepts and tools associated with the analysis and management of data, both big and small, including basic statistical modeling in R, aspects of machine learning and data mining, data management, and visualization. No previous computing experience is assumed. Students are assumed to have taken some introductory courses in two of these three math subjects: linear algebra, calculus, and probability.       3 credits, ABCF grading
Antirequisite:   AMS 595
Pre-requisite: Instructor Consent Required
Offered in the Spring Semester
AMS 561 Webpage

AMS 562 Introduction to Scientific Programming in C++
This course provides students with foundational skills and knowledge in practical scientific programming relevant for scientists and engineers. The primary language is C++ since it is a widely-used object-oriented language, includes C as a subset, and is a powerful tool for writing robust, complex, high-performance software. Elements of Python, Bash, and other languages will be introduced to complement the capabilities of C++, and essential tools for software development and engineering will be employed throughout the course (e.g., makefiles, version control, online code repositories, debugging, etc.)    This course is controlled and owned by the Institute for Advanced Computational Science (IACS).
3 credits, ABCF grading
Offered in the Fall Semester
AMS 562 Webpage

AMS 569 Probability Theory I 
Probability spaces and sigma-algebras. Random variables as measurable mappings. Borel-Cantelli lemmas. Expectation using simple functions. Monotone and dominated convergence theorems. Inequalities. Stochastic convergence. Characteristic functions. Laws of large numbers and the central limit theorem. 
Prerequisite:   AMS 510  
AMS 569 Webpage
3 credits, ABCF grading

AMS 570 Introduction to Mathematical Statistics 
Probability and distributions; multivariate distributions; distributions of functions of random variables; sampling distributions; limiting distributions; point estimation; confidence intervals; sufficient statistics; Bayesian estimation; maximum likelihood estimation; statistical tests.
Prerequisite:   AMS 507
Spring, 3 credits, ABCF grading 
AMS 570 Webpage 

AMS 572  Data Analysis
Introduction to basic statistical procedures. Survey of elementary statistical procedures such as the t-test and chi-square test. Procedures to verify that assumptions are satisfied. Extensions of simple procedures to more complex situations and introduction to one-way analysis of variance. Basic exploratory data analysis procedures (stem and leaf plots, straightening regression lines, and techniques to establish equal variance). 
3 credits, ABCF grading 

AMS 578 Regression Theory 
Classical least-squares theory for regression including the Gauss-Markov theorem and classical normal statistical theory. An introduction to stepwise regression, procedures, and exploratory data analysis techniques. Analysis of variance problems as a subject of regression. Brief discussions of robustness of estimation and robustness of design. 
Prerequisite:   AMS 572
Spring, 3 credits, ABCF grading 
AMS 578 Webpage 

AMS 580 Statistical Learning
This course teaches the following fundamental topics: (1) General and Generalized Linear Models; (2) Basics of Multivariate Statistical Analysis including dimension reduction methods, and multivariate regression analysis; (3) Supervised and unsupervised statistical learning.
Spring, 3 credits, ABCF grading
AMS 580 Webpage

AMS 588 Failure and Survival Data Analysis
Statistical techniques for planning and analyzing medical studies. Planning and conducting clinical trials and retrospective and prospective epidemiological studies. Analysis of survival times including singly censored and doubly censored data. Quantitative and quantal bioassays, two-stage assays, routine bioassays. Quality control for medical studies. 
3 credits, ABCF grading
AMS 588 Webpage

AMS 595 Fundamentals of Computing 
Introduction to programming in MATLAB, Python, and C/C++, including scripting, basic data structures, algorithms, scientific computing, software engineering and programming tools.  No previous programming experience is required.
Anti-requisite:   AMS 561
Fall, 1-9 credits, ABCF grading 
AMS 595 Webpage 

AMS 603 Risk Measures for Finance & Data Analysis
Students will work on projects in quantitative finance.
1-3 credits;  ABCF grading
AMS 603 Webpage