## Department of Operations Research and Financial Engineering

#### Chair

Rene A. Carmona

#### Departmental Representative

Alain L. Kornhauser

#### Director of Graduate Studies

William A. Massey

#### Professor

René A. Carmona

Jianqing Fan

Alain L. Kornhauser

William A. Massey

John M. Mulvey

Warren B. Powell

K. Ronnie Sircar

Robert J. Vanderbei

#### Visiting Professor

Frank J. Fabozzi

#### Associate Professor

Patrick Cheridito

Ramon van Handel

#### Assistant Professor

Amir Ali Ahmadi

Samory Kpotufe

Han Liu

Mykhaylo Shkolnikov

Mengdi Wang

#### Associated Faculty

Yacine Aït-Sahalia, Economics

Markus K. Brunnermeier, Economics

Weinan E, Mathematics

Sanjeev R. Kulkarni, Electrical Engineering

H. Vincent Poor, Electrical Engineering

Paul D. Seymour, Mathematics

Christopher A. Sims, Economics

Yakov G. Sinai, Mathematics

John D. Storey, Lewis-Sigler Institute for Integrative Genomics

Wei Xiong, Economics

#### Information and Departmental Plan of Study

Operations research and financial engineering may be considered as the modern form of a liberal education: modern because it is based on science, mathematics, computing and technology, and liberal in the sense that it provides for broad intellectual development and can lead to many different types of careers. By choosing judiciously from courses in engineering, science, mathematics, economics, public policy, and liberal arts, each student may design a program adapted to his or her particular interests.

All students start from a common academic core consisting of statistics, probability and stochastic processes, and optimization. Related courses focus on developing computer skills and exposing students to applications in areas such as finance, operations, transportation, and logistics. Students augment the core program with a coherent sequence of departmental electives. Students may also design specialized programs in areas such as medicine and neuroscience, which must be reviewed and approved by their academic adviser and the departmental representative. Students often draw on courses from economics, computer science, applied mathematics, civil and environmental engineering, mechanical engineering, chemistry, molecular biology, psychology, and the Woodrow Wilson School of Public and International Affairs. Requirements for study in the department follow the general requirements for the School of Engineering and Applied Science and the University.

#### Program of Study

The student's program is planned in consultation with the departmental representative and the student's adviser and requires a year-long thesis or a one-semester senior project. With departmental approval, the exceptional student who wishes to go beyond the science and engineering requirements may select other courses to replace some of the required courses in order to add emphasis in another field of engineering or science or to choose more courses in the area of study. Suggested plans of study and areas of concentration are available from the departmental representative.

In addition to the engineering school requirements, there are three components to the curriculum:

1. The core requirements (six courses). These form the intellectual foundation of the field and cover statistics, probability, stochastic processes, and optimization, along with more advanced courses in mathematical modeling.

2. Departmental electives (eight or nine courses). These are courses that either extend and broaden the core, or expose the student to a significant problem area or application closely related to the core program.

3. Senior independent research. A full-year thesis (or a one semester project) involving an application of the techniques in the program applied to a topic that the student chooses in consultation with a faculty adviser.

*Core requirements *(six courses):

ORF 245 Fundamentals of Engineering Statistics

ORF 307 Optimization

ORF 309 Probability and Stochastic Systems

ORF 335 Introduction to Financial Mathematics

ORF 405 Regression and Applied Time Series

ORF 411 Operations and Information Engineering

*Departmental electives *(eight or nine courses, if a one-semester project is selected but not usually recommended): The departmental electives represent courses that further develop a student's skills in mathematical modeling either by a more in-depth investigation of core disciplines, applying these skills in specific areas of application, or by learning about closely related technologies. Students must choose eight or nine courses, as appropriate, with the following constraints:

1. There must be at least one 300-level math course from the following:

MAE 305 / MAT 391 Mathematics in Engineering I or MAT 427 Ordinary Differential Equations (Both may not be taken because their contents are too similar.)

MAE 306 / MAT 392 Mathematics in Engineering II

MAT 320 Introduction to Real Analysis

MAT 322 Introduction to Differential Equations

MAT 306 Introduction to Graph Theory

MAT 377 Combinatorial Mathematics

MAT 378 Theory of Games

MAT 385 Probability Theory

MAT 427 Ordinary Differential Equations

MAT 486 Random Process

2. There must be at least two courses from the Department of Operations Research and Financial Engineering (ORF).

3. There can be no more than three courses from any one department (excluding ORF).

A list of all other departmental electives may be found in the departmental undergraduate academic guide; see the department website.

Students in the department often participate in the following certificate programs and laboratories:

**Certificate in Finance. **The department cooperates with the Bendheim Center for Finance, which offers a certificate program in finance.

**Certificate Program in Engineering and Management Systems. **The department sponsors a certificate program for students majoring in other departments who complete a significant part of the core of the undergraduate program.

**Certificate in Applied and Computational Mathematics. **Students seeking a strong mathematical foundation can combine courses from the department with supporting courses which develop more fundamental mathematical skills.

The department maintains several research laboratories which may be used as part of undergraduate research projects.

**Princeton Autonomous Vehicle Engineering (PAVE).** This extracurricular undergraduate activity focuses on the implementation of advanced sensing and control technologies for optimal autonomous decision making in vehicles. The current objective is the development of an autonomous vehicle that can pass the New Jersey State Driving Test.

**Computational and Stochastic Transportation and Logistics Engineering Laboratory. **The CASTLE Laboratory works on problems in dynamic resource management with ongoing projects in chemical distribution, railroads, trucking, and the airlift mobility command. Through this lab, students gain access to data and specialized tools to aid them in their research into transportation and logistics.

**Princeton Laboratory for Energy Systems Analysis**. PENSA is the home of the SAP Initiative for Energy Systems Research at Princeton University. Our goal is to bring advanced analytical thinking to the development of new energy technologies, the rigorous study of energy policy, and the efficient management of energy resources.

**Financial Engineering Laboratory. **This facility provides students with access to specialized software packages and to financial data and news services. Research in the laboratory is concerned with the analysis of the various forms of financial risk and the development of new financial instruments intended to control the risk exposure of insurance and reinsurance companies.

### Courses

ORF 105 The Science and Technology of Decision Making (also EGR 106) Not offered this year QR

An individual makes decisions every day. In addition, other people are making decisions that have an impact on the individual. In this course we will consider both how these decisions are made and how they should be made. In particular, we will focus on the use of advanced computing and information technology in the decision-making process.
* Staff*

ORF 245 Fundamentals of Statistics (also EGR 245) Fall, Spring QR

A first introduction to probability and statistics. This course will provide the foundations of rigorous statistical analysis including estimation, confidence intervals, hypothesis testing and regression and classification. Applicability and limitations of these methods will be illustrated using a variety of real-world data sets. Prerequisite MAT 201 Equivalent or concurrent. Three lectures, one preceptorial.
*
R. Vanderbei**,
S. Kpotufe*

ORF 307 Optimization (also EGR 307) Spring

Many real-world problems involve maximizing a linear function subject to linear inequality constraints. Such problems are called Linear Programming (LP) problems. Examples include min-cost network flows, portfolio optimization, options pricing, assignment problems and two-person zero-sum games to name but a few. The theory of linear programming will be developed with a special emphasis on duality theory, which is used to derive algorithms for solving LP problems. These algorithms will be illustrated on real-world examples such as those mentioned. Two 90 minute lectures, one preceptorial. Prerequisite MAT 202.
*
R. Vanderbei*

ORF 309 Probability and Stochastic Systems (also EGR 309/MAT 380) Fall

An introduction to probability and its applications. Random variables, expectation, and independence. Poisson processes, Markov chains, Markov processes, and Brownian motion. Stochastic models of queues, communication systems, random signals, and reliability. Prerequisite: MAT 201, 203, 217, or instructor's permission.
*
R. van Handel*

ORF 311 Optimization Under Uncertainty Not offered this year

A survey of quantitative approaches for making optimal decisions involving uncertainty and complexity including decision trees, Monte Carlo simulation, and stochastic programming. Forecasting and planning systems are integrated with a focus on financial applications. Prerequisites: ORF 307 or MAT 305, and 309. Two 90-minute classes, one preceptorial.
*
J. Mulvey*

ORF 322 Human-Machine Interaction (see PSY 322)

ORF 335 Introduction to Financial Mathematics (also ECO 364) Spring QR

This course introduces the basics of quantitative finance, particularly the use of stochastic models to value and hedge risks from options, futures and other derivative securities. The models studied include binomial trees in discrete time, and the Black-Scholes theory is introduced in continuous-time models. Computational methods are introduced in Matlab. The second half of the class looks at modern topics such as credit risk, stochastic volatility, portfolio optimization, as well as lessons from the financial crisis. Prerequisites: ORF 309, ECO 100, and MAT 104. R. Sircar
*
R. Sircar*

ORF 374 Special Topics in Operations Research and Financial Engineering Not offered this year

A course covering special topics in operations research or financial engineering. Subjects may vary from year to year.
*
J. Mulvey*

ORF 375 Independent Research Project Fall

Independent research or investigation resulting in a substantial formal report in the student's area of interest under the supervision of a faculty member. Open to sophomores and juniors.
*
A. Kornhauser*

ORF 376 Independent Research Project Spring

Independent research or investigation resulting in a substantial formal report in the student's area of interest under the supervision of a faculty member. Open to sophomores and juniors.
*
A. Kornhauser*

ORF 401 Electronic Commerce Spring

Electronic commerce, traditionally the buying and selling of goods using electronic technologies, extends to essentially all facets of human interaction when extended to services, particularly information. The course focuses on both the software and the hardware aspects of traditional aspects as well as the broader aspects of the creation, dissemination and human consumption electronic services. Covered will be the physical, financial and social aspects of these technologies. Two 90-minute lectures, one 50-minute preceptorial.
*
A. Kornhauser*

ORF 405 Regression and Applied Time Series Fall

Statistical Analysis of financial data: Density estimation, heavy tail distributions and dependence. Regression: linear, nonlinear, nonparametric. Time series analysis: classical models (AR, MA, ARMA), state space systems and filtering, and stochastic volatility models (ARCH, GARCH). Prerequsites: ORF 245 and MAT 202.
*
R. Carmona*

ORF 406 Statistical Design of Experiments Not offered this year

Major methods of statistics as applied to the engineering and physical sciences. The central theme is the construction of empirical models, the design of experiments for elucidating models, and the applications of models for forecasting and decision making under uncertainty. Three lectures. Prerequisite: 245 or equivalent.
* Staff*

ORF 407 Fundamentals of Queueing Theory Spring QR

An introduction to the fundamental results of queuing theory. Topics covered include: the classical traffic, offered load, loss, and delay models for communication systems. The theory of Markov chains, Poisson processes, and renewal theory are discussed through concrete examples and motivations. Fundamental queuing results such as the Erlang blocking and delay formulae, Jackson networks, Little's law and Lindley's equation are presented. Applications are drawn from classical problems in voice and data network performance, to modern issues in healthcare operations. Prerequisite: ORF 309 or equivalent. Two 90-minute lectures.
*
W. Massey*

ORF 409 Introduction to Monte Carlo Simulation Fall

Introduction to the uses of simulation and computation in the analysis of stochastic models and interpreting real phenomena. Topics include generating discrete and continuous random variables, the statistical analysis of simulated data, variance reduction techniques, statistical validation techniques, stochastic ordering, nonstationary Markov chains, and Markov chain Monte Carlo methods. Applications are drawn from problems in finance, insurance, manufacturing, and communication networks. Students will be encouraged to program in Python. Precept offered to help students with the language. Prerequisite: ORF 309. Two 90-minute lectures.
*
W. Massey*

ORF 411 Operations and Information Engineering Fall

The management of complex systems through the control of physical, financial and informational resources. The course focuses on developing mathematical models for resource allocation, with an emphasis on capturing the role of information in decisions. The course seeks to integrate skills in statistics, stochastics and optimization using applications drawn from problems in dynamic resource management which tests modeling skills and teamwork. Prerequisites: ORF 245, ORF 307 and ORF 309, or equivalents. Two 90 minute lectures, preceptorial.
*
W. Powell*

ORF 417 Dynamic Programming Not offered this year

An introduction to stochastic dynamic programming and stochastic control. The course deals with discrete and continuous-state dynamic programs, finite and infinite horizons, stationary and nonstationary data. Applications drawn from inventory management, sequential games, stochastic shortest path, dynamic resource allocation problems. Solution algorithms include classical policy and value iteration for smaller problems and stochastic approximation methods for large-scale applications. Prerequisites: 307 and 309.
* Staff*

ORF 418 Optimal Learning Spring QR

Addresses the problem of collecting information used to estimate statistics or fit a model which is then used to make decisions. Of particular interest are sequential problems where decisions adapt to information as it is learned. The course introduces students to a wide range of applications, demonstrates how to express the problem formally, and describes a variety of practical solution strategies. Prerequisite: ORF 245, ORF 309. Two 90-minute lectures, one preceptorial.
*
W. Powell*

ORF 435 Financial Risk Management Fall

This course covers the basic concepts of modeling, measuring and managing different types of financial risks. Topics include portfolio optimization (mean-variance approach and expected utility), interest rate risk, pricing and hedging in complete and incomplete markets, indifference pricing, risk measures, systemic risk. Prerequisites: ORF 245, ORF 335 or ECO 465 (concurrent enrollment is acceptable) or instructor's permission. Two 90-minute lectures, one preceptorial.
*
P. Cheridito*

ORF 455 Energy and Commodities Markets

This course is an introduction to commodities markets (energy, metals, agricultural products) and issues related to renewable energy sources such as solar and wind power, and carbon emissions. Energy and other commodities represent an increasingly important asset class, in addition to significantly impacting the economy and policy decisions. Emphasis will be on the application of Financial Mathematics to a variety of different products and markets. Topics include: energy prices (including oil and electricity); cap and trade markets; storable vs non-storable commodities; financialization of commodities markets; applications of game theory.
*
R. Sircar*

ORF 467 Transportation Systems Analysis Fall

Studied is the transportation sector of the economy from a technology and policy planning perspective. The focus is on the methodologies and analytical tools that underpin policy formulation, capital and operations planning, and real-time operational decision making within the transportation industry. Case studies of innovative concepts such as dynamic "value pricing", real-time fleet management and control, GPS-based route guidance systems, automated transit networks and the emergence of Smart Driving / Autonomous Cars. Two 90-minute lectures, one preceptorial.
*
A. Kornhauser*

ORF 473 Special Topics in Operations Research and Financial Engineering Not offered this year

A course covering one or more advanced topics in operations research and financial engineering. Subjects may vary from year to year.
* Staff*

ORF 474 Special Topics in Operations Research and Financial Engineering Not offered this year

A course covering one or more advanced topics in operations research and financial engineering. Subjects may vary from year to year.
* Staff*

ORF 478 Senior Thesis Spring

A formal report on research involving analysis, synthesis, and design, directed toward improved understanding and resolution of a significant problem. The research is conducted under the supervision of a faculty member, and the thesis is defended by the student at a public examination before a faculty committee. The senior thesis is equivalent to a year-long study and is recorded as a double course in the Spring.
*
A. Kornhauser*

ORF 479 Senior Project Spring

A one-semester project that fulfills the departmental independent work requirement for concentrators. Topics are chosen by students in consultation with members of the faculty. A written report is required at the end of the term.
*
A. Kornhauser*