## Courses

**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

**Review of techniques of multivariate calculus, convergence and limits, matrix analysis, vector space basics, and Lagrange multipliers.**

**AMS 510 Analytical Methods for Applied Mathematics and Statistics**Fall, 3 credits, ABCF grading

Prerequisites: A course in linear algebra and in multivariate calculus

**AMS 510 Webpage**

AMS 511, Foundation of Quantitative Finance

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 511, AMS 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*

**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.)**

AMS 562 Introduction to Scientific Programming in C++

AMS 562 Introduction to Scientific Programming in C++

*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