phd in math finance

Department of Mathematics

Financial mathematics.

A pioneer in its field, the Financial Mathematics Program offers 15 months of accelerated, integrated coursework that explores the deep-rooted relationship that exists between theoretical and applied mathematics and the ever-evolving world of finance. Their mission is to equip students with a solid foundation in mathematics, and in doing so provide them with practical knowledge that they can successfully apply to complicated financial models. Financial Mathematics students become leaders in their field; program alumni have gone forth to find success at companies like JP Morgan, UBS, and Goldman Sachs. Read more

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DPhil (PhD) studies in Mathematical Finance @ Oxford

The Mathematical and Computational Finance Group (MCFG) at Oxford is one of the largest and most dynamic research environments in mathematical finance in the world.

We combine core mathematical expertise with interdisciplinary approach. We foster lively interactions between researchers coming from different backgrounds and a truly impressive seminar programme, all this within one of the world's top universities, singular through its tradition and unique environment.

If you are passionate about mathematics and research and want to pursue a DPhil in Financial Mathematics, Oxford simply offers one of the best and most exciting places to do it!

 Research Topic and Supervisor Allocation

We welcome students with their own particular ideas of research topic as well as students with a broad interest in the field of Mathematical Finance. You have an opportunity to tell us about your research passions, and indicate potential supervisors, in your application form. This will be followed up during the interview.

In light of this, if you are offered a place, an appropriate supervisor will be proposed prior to your arrival in Oxford. However, there can be some flexibility over this once you arrive.  Keeping with the Oxford tradition, we offer our students independence and respect as early researchers, and always aim to match students with the most appropriate supervisors.

Outstanding students with a strong background in analysis, probability and data science are welcome to apply for our DPhil program. Each year we receive a large number of excellent applications. The selection process is extremely competitive and we can only admit a handful of candidates each year.

In order to apply for DPhil studies in Mathematical & Computational Finance, please indicate your interest in Mathematical and Computational Finance on your application form. Selected applicants will be invited for an interview -- either in person or by video call.

For general information on DPhil please consult our  Doctor of Philosophy (DPhil) admissions pages .

For the CDT Mathematics of Random Systems please consult our  the CDT website .

Or please contact  @email .

Funding for DPhil students is available from a variety of sources. Please note that some funding opportunities have deadlines: it is advised to apply before the deadline in order to maximise your chances of receiving funding.

Funding is also available through the  Centre for Doctoral Training in Mathematics of Random Systems . To apply for this program please How to Apply .

Email:  @email Phone:  +44 (0)1865 615234

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DPhil Graduates

The Mathematical and Computational Finance Program at Stanford University (“MCF”) is one of the oldest and most established programs of its kind in the world. Starting out in the late 1990’s as an interdisciplinary financial mathematics research group, at a time when “quants” started having a greater impact on finance in particular, the program formally admitted masters students starting in 1999. The current MCF program was relaunched under the auspices of the Institute for Computational and Mathematical Engineering in the Stanford School of Engineering in 2014 to better align with changes in industry and to broaden into areas of financial technology in particular. We are excited to remain at the cutting edge of innovation in finance while carrying on our long tradition of excellence.

The MCF Program is designed to have smaller cohorts of exceptional students with diverse interests and viewpoints, and prepare them for impactful roles in finance. We are characterized by our cutting edge curriculum marrying traditional financial mathematics and core fundamentals, with an innovative technical spirit unique to Stanford with preparation in software engineering, data science and machine learning as well as the hands-on practical coursework which is the hallmark skill-set for leaders in present day finance.

Why Study for a Mathematical Finance PhD?

I was emailed by a reader recently asking about mathematical finance PhD programs and the benefits of such a course. If you are considering gaining a PhD in mathematical finance, this article will be of interest to you.

If you are currently near the end of your undergraduate studies or are returning to study after some time in industry, you might consider starting a PhD in mathematical finance. This is an alternative to undertaking a Masters in Financial Engineering (MFE), which is another route into a quantitative role. This article will discuss exactly what you will be studying and what you are likely to get out of a PhD program. Clearly there will be differences between studying in the US, UK or elsewhere. I personally went to grad school in the UK, but I will discuss both UK and US programs.

Mathematical finance PhD programs exist because the techniques within the derivatives pricing industry are becoming more mathematical and rigourous with each passing year. In order to develop new exotic derivatives instruments, as well as price and hedge them, the financial industry has turned to academia. This has lead to the formation of mathematical finance research groups - academics who specialise in derivatives pricing models, risk analysis and quantitative trading.

Graduate school, for those unfamiliar with it, is a very different experience to undergraduate. The idea of grad school is to teach you how to effectively research a concept without any guidance and use that research as a basis for developing your own models. Grad school really consists of a transition from the "spoon fed" undergraduate lecture system to independent study and presentation of material. The taught component of grad school is smaller and the thesis component is far larger. In the US, it is not uncommon to have two years of taught courses before embarking on a thesis (and thus finding a supervisor). In the UK, a PhD program is generally 3-4 years long with either a year of taught courses, or none, and then 3 years of research.

A good mathematical finance PhD program will make extensive use of your undergraduate knowledge and put you through graduate level courses on stochastic analysis, statistical theory and financial engineering. It will also allow you to take courses on general finance, particularly on corporate finance and derivative securities. When you finish the program you will have gained a broad knowledge in most areas of mathematical finance, while specialising in one particular area for your thesis. This "broad and deep" level of knowledge is the hallmark of a good PhD program.

Mathematical Finance research groups study a wide variety of topics. Some of the more common areas include:

• Derivative Securities Pricing/Hedging: The technical term for this is "financial engineering", as "quantitative analysis" now encompasses a wide variety of financial areas. Some of the latest research topics include sophisticated models of options including stochastic volatility models, jump-diffusion models, asymptotic methods as well as investment strategies.
• Stochastic Calculus/Analysis: This is more of a theoretical area, where the basic motivation stems from the need to solve stochastic differential equations. Research groups may look at path-dependent PDEs, functional Ito calculus, measure theory and probability theory.
• Fixed Income Modeling: Research in this area centres on effectively modelling interest rates - such as multi-factor models, multi-curve term structure models as well as interest rate derivatives such as swaptions.
• Numerical Methods: Although not always strictly related to mathematical finance, there is a vast amount of university research carried out to try and develop more effective means of solving equations numerically (i.e. on the computer!). Recent developments include GPU-based Monte Carlo solvers, more efficient matrix solvers as well as Finite Differences on GPUs. These groups will almost certainly possess substantial programming expertise.
• Market Microstructure/High-Frequency Modeling: This type of research is extremely applied and highly valued by funds engaged in this activity. You will find many academics consulting, if not contracting, for specialised hedge funds. Research areas include creating limit order market models, high frequency data statistical modelling, market stability analysis and volatility analysis.
• Credit Risk: Credit risk was a huge concern in the 2007-2008 financial crisis and many research groups are engaged in determining such "counterparty risks". Credit derivatives are still a huge business and so a lot of research goes into collateralisation of securities as well as pricing of exotic credit derivatives.

These are only a fraction of the total areas that are studied within mathematical finance. The best place to find out more about research topics is to visit the websites of all the universities which have a mathematical finance research group, which is typically found within the mathematics, statistics or economics faculty.

The benefits of undertaking a PhD program are numerous:

• Employment Prospects: A PhD program sets you apart from candidates who only possess an undergraduate or Masters level ability. By successfully defending a thesis, you have shown independence in your research ability, a skill highly valued by numerate employers. Many funds (and to a lesser extent, banks) will only hire PhD level candidates for their mathematical finance positions, so in a pragmatic sense it is often a necessary "rubber stamp". In investment banks, this is not the case so much anymore, as programming ability is generally prized more. However, in funds, it is still often a requirement. Upon being hired you will likely be at "associate" level rather than "analyst" level, which is common of undergraduates. Your starting salary will reflect this too.
• Knowledge: You will spend a large amount of time becoming familiar with many aspects of mathematical finance and derivatives theory. This will give you a holistic view into the industry and a more transferable skill set than an undergraduate degree as you progress up the career ladder. In addition, you will have a great deal of time to learn how to program models effectively (without the day-to-day pressure to get something implemented any way possible!), so by the time you're employed, you will be "ahead of the game" and will know best practices. This aspect is down to you, however!
• Intellectual Prospects: You are far more likely to gain a position at a fund after completing a PhD than without one. Funds are often better environments to work in. There is usually less stress and a more relaxed "collegiate" environment. Compare this to working on a noisy trading floor, where research might be harder to carry out and be perceived as less important.

I would highly recommend a mathematical finance PhD, so long as you are extremely sure that a career in quantitative finance is for you. If you are still unsure of your potential career options, then a more general mathematics, physics or engineering PhD might be a better choice.

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The PhD in Mathematical Finance is for students seeking careers in research and academia. Doctoral candidates will have a strong affinity for quantitative reasoning and the ability to connect advanced mathematical theories with real-world phenomena. They will have an interest in the creation of complex models and financial instruments as well as a passion for in-depth analysis.

Learning Outcomes

The PhD curriculum has the following learning goals. Students will:

• Demonstrate advanced knowledge of literature, theory, and methods in their field.
• Be prepared to teach at the undergraduate, master’s, and/or doctoral level in a business school or mathematics department.
• Produce original research of quality appropriate for publication in scholarly journals.

After matriculation into the PhD program, a candidate for the degree must register for and satisfactorily complete a minimum of 16 graduate-level courses at Boston University. More courses may be needed, depending on departmental requirements.

PhD in Mathematical Finance Curriculum

The curriculum for the PhD in Mathematical Finance is tailored to each incoming student, based on their academic background. Students will begin the program with a full course load to build a solid foundation in not only math and finance but also the interplay between them in the financial world. As technology plays an increasingly larger role in financial models, computer programming is also a part of the core coursework.

Once a foundation has been established, students work toward a dissertation. Working closely with a faculty advisor in a mutual area of interest, students will embark on in-depth research. It is also expected that doctoral students will perform teaching assistant duties, which may include lectures to master’s-level classes.

Course Requirements

The minimum course requirement is 16 courses (between 48 and 64 credits, depending on whether the courses are 3 or 4 credits each). Students’ course choices must be approved by the Mathematical Finance Director prior to registration each semester. The following is a typical program of courses.

• GRS EC 701 Microeconomic Theory
• GRS MA 711 Real Analysis
• GRS MA 779 Probability Theory I
• QST FE 918 Doctoral Seminar in Finance
• GRS EC 703 Advanced Microeconomic Theory
• GRS MA 776 Partial Differential Equations
• GRS MA 781 Probability Theory 2
• QST FE 920 Advanced Capital Market Theory
• GRS EC 702 Macroeconomic Theory
• GRS MA 783 Advanced Stochastic Processes
• QST MF 850 Advanced Computational Methods
• QST MF 922 Advanced Mathematical Finance
• GRS EC 704 Advanced Microeconomic Theory
• GRS MA 751 Statistical Machine Learning
• QST MF 810 FinTech Programming
• QST MF 921 Topics in Dynamic Asset Pricing

Additional Requirements

Qualifying examination.

Students must appear for a qualifying examination after completion of all coursework to demonstrate that they have:

• acquired advanced knowledge of literature and theory in their area of specialization;
• acquired advanced knowledge of research techniques; and
• developed adequate ability to craft a research proposal.

Guidelines for the examination are available from the departments. Students who do not pass either the written and/or oral comprehensive examination upon first try will be given a second opportunity to pass the exam. Should the student fail a second time, the student’s case will be reviewed by the Mathematical Finance Program Development Committee (MF PDC), which will determine if the student will be withdrawn from the PhD program. In addition, the PhD fellowship (if applicable) of any student who does not pass either the written and/or oral comprehensive examination after two attempts will be suspended the semester after the exam was attempted.

Dissertation

Following successful completion of the qualifying examination, the student will develop a research proposal for the dissertation. The final phase of the doctoral program is the completion of an approved dissertation. The dissertation must be based on an original investigation that makes a substantive contribution to knowledge and demonstrates capacity for independent, scholarly research.

Doctoral candidates must register as continuing students for DS 999 Dissertation, a 2-credit course, for each subsequent regular semester until all requirements for the degree have been completed. PhD students graduating in September are required to register for Dissertation in Summer Session II preceding graduation.

Academic Standards

Time limit for degree completion.

After matriculation into the PhD program, a candidate for the degree must meet certain milestones within specified time periods (as noted in the table below) and complete all degree requirements within six years of the date of first registration. Those who fail to meet the milestones within the specified time, or who do not complete all requirements within six years, will be reviewed by the PhD PDC and may be dismissed from the program. A Leave of Absence does not extend the six-year time limit for degree completion.

Performance Review

The Mathematical Finance Program Development Committee will review the progress of each doctoral candidate. Students must maintain a 3.30 cumulative grade point average in all courses to remain in good academic standing. Students who are not in good academic standing will be allowed one semester to correct their status. Prior to the start of the semester, the student must submit a letter to the Faculty Director (who will forward it to the PDC) explaining why the student has fallen short of the CGPA requirement and how the student plans to correct the situation. Failure to increase the CGPA to acceptable levels may result in probation or withdrawal from the program, at the discretion of the PhD Program Development Committee (PDC).

Graduation Application

Students must submit a graduation application at least seven months before the date they expect to complete degree requirements. It is the student’s responsibility to initiate the process for graduation. The application is available online and should be submitted through the Specialty Master’s & PhD Center website for graduation in January, May, or August.

If graduation must be postponed beyond the semester for which the application is submitted, students should contact the Specialty Master’s & PhD Center to defer the date. If students wish to postpone their graduation date past the six-year time limit for completion, they must formally petition the PhD Program Development Committee (PDC) for an extension. The petition, which must include the reason(s) for the extension as well as a detailed timetable for completion, is subject to departmental and PDC approval.

PhD degree requirements are complete only when copies of the dissertation have been certified as meeting the standards of Questrom School of Business and have been accepted by Mugar Memorial Library.

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PhD Program

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Wharton’s PhD program in Finance provides students with a solid foundation in the theoretical and empirical tools of modern finance, drawing heavily on the discipline of economics.

The department prepares students for careers in research and teaching at the world’s leading academic institutions, focusing on Asset Pricing and Portfolio Management, Corporate Finance, International Finance, Financial Institutions and Macroeconomics.

Wharton’s Finance faculty, widely recognized as the finest in the world, has been at the forefront of several areas of research. For example, members of the faculty have led modern innovations in theories of portfolio choice and savings behavior, which have significantly impacted the asset pricing techniques used by researchers, practitioners, and policymakers. Another example is the contribution by faculty members to the analysis of financial institutions and markets, which is fundamental to our understanding of the trade-offs between economic systems and their implications for financial fragility and crises.

Faculty research, both empirical and theoretical, includes such areas as:

• Structure of financial markets
• Formation and behavior of financial asset prices
• Banking and monetary systems
• Corporate control and capital structure
• Saving and capital formation
• International financial markets

Candidates with undergraduate training in economics, mathematics, engineering, statistics, and other quantitative disciplines have an ideal background for doctoral studies in this field.

Effective 2023, The Wharton Finance PhD Program is now STEM certified.

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The field of finance covers the economics of claims on resources. Financial economists study the valuation of these claims, the markets in which they are traded, and their use by individuals, corporations, and the society at large.

At Stanford GSB, finance faculty and doctoral students study a wide spectrum of financial topics, including the pricing and valuation of assets, the behavior of financial markets, and the structure and financial decision-making of firms and financial intermediaries.

Investigation of issues arising in these areas is pursued both through the development of theoretical models and through the empirical testing of those models. The PhD Program is designed to give students a good understanding of the methods used in theoretical modeling and empirical testing.

Preparation and Qualifications

All students are required to have, or to obtain during their first year, mathematical skills at the level of one year of calculus and one course each in linear algebra and matrix theory, theory of probability, and statistical inference.

Students are expected to have familiarity with programming and data analysis using tools and software such as MATLAB, Stata, R, Python, or Julia, or to correct any deficiencies before enrolling at Stanford.

The PhD program in finance involves a great deal of very hard work, and there is keen competition for admission. For both these reasons, the faculty is selective in offering admission. Prospective applicants must have an aptitude for quantitative work and be at ease in handling formal models. A strong background in economics and college-level mathematics is desirable.

It is particularly important to realize that a PhD in finance is not a higher-level MBA, but an advanced, academically oriented degree in financial economics, with a reflective and analytical, rather than operational, viewpoint.

Faculty in Finance

Anat r. admati, juliane begenau, jonathan b. berk, greg buchak, antonio coppola, peter m. demarzo, darrell duffie, steven grenadier, benjamin hébert, arvind krishnamurthy, hanno lustig, matteo maggiori, paul pfleiderer, joshua d. rauh, claudia robles-garcia, ilya a. strebulaev, vikrant vig, jeffrey zwiebel, emeriti faculty, robert l. joss, george g.c. parker, myron s. scholes, william f. sharpe, kenneth j. singleton, james c. van horne, recent publications in finance, behavioral responses to state income taxation of high earners: evidence from california, beyond the balance sheet model of banking: implications for bank regulation and monetary policy, fee variation in private equity, recent insights by stanford business, nine stories to get you through tax season, “geoeconomics” explains how countries flex their financial muscles, car loans are a hidden driver of the ride-sharing economy.

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Mathematical Finance

A rapidly growing area of mathematical finance is quantitative behavioral finance. The high-tech boom and bust of the late 1990s followed by the housing and financial upheavals of 2008 have made a convincing case for the necessity of adopting broader assumptions in finance. These include considering motivations beyond valuation considerations, and an asset base that is not infinite. The asset flow system of ordinary differential equations developed by Prof. Caginalp and collaborators in the 1990s has been an active part of this research at Pitt. These equations are being used to understand the dynamics and stability. A related component involves large scale studies (e.g. over 100,000 data points) of market data that can be used to deduce underlying motivational effects. By extracting the valuation, recent studies have shown that momentum trading (buying on uptrend) plays a strong role, as do money supply, changes in volume and several other variable. Furthermore, with suitable modeling, one can deduce nonlinear effects. In particular, a recent uptrend that is too steep has a negative influence on prices. These topics have been the focus of the PhD thesis of Mark DeSantis at the University of Pittsburgh.

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PhD Program in Finance

2023-24 curriculum outline.

The MIT Sloan Finance Group offers a doctoral program specialization in Finance for students interested in research careers in academic finance. The requirements of the program may be loosely divided into five categories: coursework, the Finance Seminar, the general examination, the research paper, and the dissertation. Attendance at the weekly Finance Seminar is mandatory in the second year and beyond and is encouraged in the first year.  During the first two years, students are engaged primarily in coursework, taking both required and elective courses in preparation for their general examination at the end of the second year.  Students are required to complete a research paper by the end of their fifth semester, present it in front of the faculty committee and receive a passing grade.  After that, students are required to find a formal thesis advisor and form a thesis committee by the end of their eighth semester. The Thesis Committee should consist of at least one tenured faculty from the MIT Sloan Finance Group.

Required Courses

The following set of required courses is designed to furnish each student with a sound and well-rounded understanding of the theoretical and empirical foundations of finance, as well as the tools necessary to make original contributions in each of these areas. Finance PhD courses (15.470, 15.471, 15.472, 15.473, 15.474) in which the student does not receive a grade of B or higher must be retaken.

First Year - Summer

Math Camp begins on the second Monday in August. 

First Year - Fall Semester

14.121/14.122 Micro Theory I/II

14.451/14.452 Macro Theory I/II ( strongly recommended)

14.380/14.381 — Statistics/Applied Econometrics

15.470 — Asset Pricing

First Year - Spring Semester

14.123/14.124 Micro Theory III/IV

14.453/14.454 Macro Theory III/IV (strongly recommended)

14.382 – Econometrics

15.471 – Corporate Finance

Second Year - Fall Semester

15.472 — Advanced Asset Pricing

  14.384 — Time-Series Analysis or  14.385 — Nonlinear Econometric Analysis  (Enrolled students receive a one-semester waiver from attending the Finance Seminar due to a scheduling conflict)

15.475 — Current Research in Financial Economics

Second Year - Spring Semester

15.473 — Advanced Corporate Finance

 15.474 — Current Topics in Finance (strongly encouraged to take multiple times)

15.475 — Current Research in Financial Economics

Recommended Elective Courses

Beyond these required courses, students are expected to enroll in elective courses determined by their primary area of interest. There are two informal “tracks” in Financial Economics: Corporate Finance and Asset Pricing. Recommended electives are designed to deepen the student's grasp of material that will be central to the writing of his/her dissertation. Students also have the opportunity to take courses at Harvard University. There is no formal requirement to select one track or another, and students are free to take any of the electives.

Mathematics (PHD) – Financial Mathematics Track

Program at a glance.

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Learn more about the cost to attend UCF.

The Financial Mathematics track in the Mathematics PhD program is designed to prepare students for research and leadership positions in industry, government, non-governmental organizations, and academia requiring employment of financial mathematics.

The Mathematics PhD program consists of at least 75 credit hours of course work beyond the bachelor's degree, of which a minimum of 48 hours of formal course work, exclusive of independent study, are required. The program requires 36 credit hours of core courses and 15 credit hours of dissertation research (7980).

Total Credit Hours Required: 75 Credit Hours Minimum beyond the Bachelor's Degree

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Track Prerequisites

Bachelor's degree in related field.

Students entering the graduate program with regular status are assumed to have a working knowledge of undergraduate calculus, differential equations, linear algebra (or matrix theory), boundary value problems, statistics, computer programming, and maturity in the language of advanced calculus (at the level of MAA 4226).

Degree Requirements

Required courses.

• All students are required to complete the following courses with grade of "B" or better.
• MAA5237 - Mathematical Analysis (3)
• MAS5145 - Advanced Linear Algebra and Matrix Theory (3)
• MAP5641 - Financial Mathematics I (3)
• MAP6642 - Financial Mathematics II (3)
• MAP5612 - Computational Methods for Financial Mathematics I (3)
• MAP6616 - Computational Methods for Financial Mathematics II (3)
• MAP6646 - Risk Management for Financial Mathematics (3)
• MAP5606 - Differential Equations for Financial Mathematics (3)
• MAP6195 - Mathematical Foundations for Massive Data Modeling and Analysis (3)
• MAP6207 - Optimization Theory (3)
• STA6857 - Applied Time Series Analysis (3)
• MAP5931 - Proseminar for Financial Mathematics (1)
• MAP5933 - Seminar in Financial Mathematics (2)
• The remaining credit hours consist of additional dissertation research (7980 or 7919), at least 15 credit hours of regular classroom elective courses, and at most 12 credit hours of independent study or independent directed research. Students who pass the qualifying examination may substitute some of the core courses with the approval of the adviser and the graduate program director.

Elective Courses

• Earn at least 24 credits from the following types of courses: Elective courses require the approval of the adviser and the graduate program director; up to 12 credit hours of elective courses may be taken outside the department. At least one-half of the program courses must be taken at the 6000 level. At least 12 hours of elective course work must be formal course work, exclusive of independent study. Electives are chosen in consultation with the student's advisory committee and may be chosen from the suggested options: Discrete Mathematics, General Applied Mathematics, Mathematical Computer Tomography, Image Processing and Computer Graphics, Mathematical Finance, Mathematical Physics, Pure Mathematics, Data Science, and Mathematical Statistics. A list of elective course options can be obtained from the graduate program director. Courses that are taken outside the Mathematics department must be approved by both the adviser and graduate program director. These courses are selected in consultation with the student's advisory committee.

Dissertation

• Earn at least 15 credits from the following types of courses: MAP 7980 - Dissertation Research 15 Credit Hours (minimum) After passing the candidacy examination and meeting the other requirements that are required for admission to candidacy, the student can register for Doctoral Dissertation (MAP 7980). A minimum of 15 Doctoral Dissertation credit hours are required for the degree.

Examinations

Qualifying examination.

• The qualifying/comprehensive examination is based on the core course work. To continue in the PhD program, students must pass the examination at the PhD level. Two attempts are permitted. The examination will be administered twice a year: one in the Fall semester and the other in the Spring semester. To take the examination, students must have earned a "B" or better in each core course, must have a minimum grade point average of 3.0 (out of 4.0) in the program, or must obtain permission from the graduate program director. Students will normally take the examination after the first year and are expected to have passed it by the end of the second year of study, unless a written request for a postponement has been approved by the Graduate Committee at least two months before the examination date. The student must pass the Qualifying Examination in at most two attempts. It is strongly recommended that the student select a dissertation adviser by the completion of 18 credit hours of course work, and it is strongly recommended that the student works with the dissertation adviser to form a dissertation committee within two semesters of passing the Qualifying Examination.

Candidacy Examination

• The Candidacy Examination consists of a written examination based on the materials from two selected two-semester sequence courses taken by the students. A committee formed or selected by the Graduate Committee or the graduate program director is responsible for preparing and grading the written examinations. Each sequence that is selected for the candidacy examination must be approved by the dissertation adviser, the dissertation committee, and the graduate program director. Students in the Financial Mathematics Track will ordinarily select one of the sequences for their candidacy examination to be MAP 5641/MAP6642 Financial Mathematics I and II, and MAP5612/MAP6616 Computational Methods for Financial Mathematics I and II. The Candidacy Examination can be attempted after passing the qualifying examination. The Candidacy Examination must be completed within three years after passing the qualifying examination. A student must successfully pass the Candidacy Examination within at most two attempts.

Admission to Candidacy

• The following are required to be admitted to candidacy and enroll in dissertation hours: Completion of all course work, except for dissertation hours. Successful completion of the candidacy examination. The dissertation advisory committee is formed, consisting of approved graduate faculty and graduate faculty scholars. Submittal of an approved program of study.

Dissertation Proposal Examination

• After passing the candidacy examination, the student will prepare a dissertation proposal and orally present it to the dissertation advisory committee for approval. The proposal will include a description of the research performed to date and an agenda for the research planned to be completed for the dissertation. In addition to standards of correctness, indicating a suitable level of mastery of the material of the area of the dissertation, and suitability of the proposed dissertation topic, the presentation must meet current standards for professional presentations within the discipline of mathematics. For the successful completion of the Dissertation Proposal Examination, the presentation must be judged as passing the requirements for the examination by the majority of the dissertation committee. This exam must be passed within 18 months of passing the candidacy examination and not later than the end of the sixth year of graduate study. A candidate must pass this examination within at most two attempts.

Dissertation Defense

• Upon completion of a student's research, the student's committee schedules an oral defense of the dissertation. Most students complete the program within five years after obtaining their bachelor's degree. Students are expected to complete the dissertation in no more than seven years from the date of admission to the program.

Independent Learning

• The required 15 credit hours of dissertation will provide ample opportunities for students to gain the independent learning experience through studying published research papers and deriving, on their own, new and meaningful research results.

Grand Total Credits: 75

Application requirements, financial information.

Graduate students may receive financial assistance through fellowships, assistantships, tuition support, or loans. For more information, see the College of Graduate Studies Funding website, which describes the types of financial assistance available at UCF and provides general guidance in planning your graduate finances. The Financial Information section of the Graduate Catalog is another key resource.

Fellowship Information

Fellowships are awarded based on academic merit to highly qualified students. They are paid to students through the Office of Student Financial Assistance, based on instructions provided by the College of Graduate Studies. Fellowships are given to support a student's graduate study and do not have a work obligation. For more information, see UCF Graduate Fellowships, which includes descriptions of university fellowships and what you should do to be considered for a fellowship.

The department offers over 20 Graduate Teaching Assistantships every year on a competitive basis. A few Graduate Research Assistantships are also available for qualified students.

The PhD concentration in finance emphasizes theoretical economics and provides a rigorous, analytically-grounded education. The Finance Department has a long and prominent intellectual history. Ideas that we now take as commonplace, such as moral hazard problems caused by deposit insurance and the Hansen-Jagannathan bounds in asset-pricing, have their origin at the Carlson School.

About the Program

Faculty members are also dedicated to producing top-flight scholars by offering both doctoral courses that focus on cutting edge research as well as collaborative research opportunities. Students who have strong interests and abilities in quantitative methods, mathematics, and economics will find this program both challenging and stimulating.

Our faculty members are more than just educators; they are accomplished leaders in the finance industry and dedicated researchers shaping the future of finance.

The PhD finance concentration requires a strong mastery of economic theory.

Get to know current students in the PhD Finance program.

Learn more about their educational background, expertise, and research interests. 

Finance Seminars & Conferences

Learn more about the events and conferences presented by the Carlson School of Management's Finance Department.

Alumni Perspectives

Hongda Zhong

"I enjoyed each and every day of my five-year PhD study at the Carlson School of Management. Here, we have world-leading scholars who care and nurture PhD students. Faculty and students constantly discuss research ideas and collaborate on joint projects in a collegial atmosphere. The rigorous academic standard and the patient guidance from my advisory team prepared me for my future career in academia. PhD life in Minnesota is fun as well, camping, hiking, road trip, just to name a few. Even in the winter, life is never boring. I miss my time with my classmates to go ice fishing and skiing. The helpful staff in the school also made all administrative processes very straightforward. In short, Carlson offers everything I wished for as a PhD student and I wish you the opportunity to share my joy."

Assistant Professor of Finance Naveen Jindal School of Management at the University of Texas at Dallas

More about Hongda Zhong

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• The Department of Mathematics hosts and runs the program with support from faculty members in Statistics, Economics, Finance, and Computer Science.
• Fourteen Mathematics faculty members support the program, along with faculty from Economics, Statistics, Finance, Scientific Computing, Computer Science, and Risk Management.
• Industry professionals and alumni visit each Spring for our annual two-day Financial Math Quant Symposium .
• Graduates of the program find employment as quants in industry and government, as actuaries, and in academia (see the alumni page ).
• The MS degree may be earned as a terminal degree or en route to the PhD , and carries the Professional Science Master's designation, as well as the STEM designation.
• There are four track options within the two-year MS program for Financial Mathematics: 1. Quantitative Finance track (PhD preparation option); 2. Quantitative Finance track (terminal MS option); 3. Actuarial Science track; 4. Data Science track.

Wan-Yu Tsai completed her PhD in Financial Mathematics from FSU in 2017. After graduation she joined Bank of America as a Quantitative Finance Analyst in Charlotte, NC.

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Mathematical Sciences

Mellon college of science, ph.d. programs, doctor of philosophy in mathematical sciences.

Students seeking a Ph.D. in Mathematical Sciences are expected to show a broad grasp of mathematics and demonstrate a genuine ability to do mathematical research. The Doctor of Philosophy in Mathematical Sciences is a traditional research degree, and its requirements are representative of all doctoral programs.

After being admitted to graduate status by the Department, a student seeking a Ph.D. must be admitted to candidacy for this degree by fulfilling the appropriate program requirements.

The most important requirement for the Ph.D. degree is timely completion and public defense of an original Ph.D. thesis. The Ph.D. thesis is expected to display depth and originality and be publishable by a refereed journal.

Doctor of Arts in Mathematical Sciences

The Doctor of Arts degree shares all requirements and standards with the Ph.D., except with regard to the thesis. The D.A. thesis is not expected to display the sort of original research required for a Ph.D. thesis, but rather to demonstrate an ability to organize, understand, and present mathematical ideas in a scholarly way, usually with sufficient innovation and worth to produce a publishable work. Whenever practical, the department provides D.A. candidates with the opportunity to use materials developed to teach a course. While a typical Ph.D. recipient will seek a position that has a substantial research component, the D.A. recipient will usually seek a position where research is not central.

Doctor of Philosophy in Algorithms, Combinatorics, and Optimization (ACO)

This program is administered jointly by the Department of Mathematical Sciences, the Department of Computer Science, and the Tepper School of Business. It focuses on discrete mathematics and algorithmic issues arising in computer science and operations research, particularly the mathematical analysis of these issues. The participating units evaluate applicants separately. The requirements for this degree and information on participating faculty are available at the ACO page .

Doctor of Philosophy in Pure and Applied Logic (PAL)

This is an interdisciplinary program with faculty from the Department of Mathematical Sciences, the Department of Philosophy, and the School of Computer Science. The participating units evaluate applicants separately and set their own program requirements. Students who have been admitted to the PAL program, and who complete the requirements for the Ph.D. in Mathematical Sciences with a thesis in the area of logic, can choose to receive either a Ph.D. in Pure and Applied Logic or a Ph.D. in Mathematical Sciences. The choice of which degree to receive is usually based on the intended career path.

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The CMU Rales Fellow Program is dedicated to developing a diverse community of STEM leaders from underrepresented and underresourced backgrounds by eliminating cost as a barrier to education. Learn more about this program for master's and Ph.D. students. Learn more

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ARE YOU READY TO BEGIN YOUR CAREER AS A QUANT? LET’S GET TO WORK.

Mathematics in finance at nyu courant is for those committed to launching careers in the financial industry and putting in the work to make it happen..

LEARN. CONNECT. SUCCEED.

Join a community committed to rigorous academic exploration and robust professional networking—and watch your future take shape., be the most capable quant in the room, immerse yourself in the foundations—and the future—of mathematical finance and financial data science—and prepare to lead the financial industry into a better tomorrow..

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Study in the world’s foremost financial hub alongside some of the financial industry’s leading minds—and discover what real-life experience really looks like. , find our student resume books here., learn more about mathematics in finance at nyu courant., a curriculum that is one of a kind.

Courses here are developed and designed for one reason: to help you start your career in the financial industry.

MATH-GA.2041-001

Computing In Finance

Prerequisites: Procedural programming, some knowledge of Java recommended.

Description: The purpose of this course is threefold: 1) It will teach students the popular Python programming language. 2) Students will learn the five most important concepts of modern, object-oriented software development, which are testing, data structures, design, working with data, and distributed computing. 3) All of the examples used in class will have a financial context. Projects we will work on include developing a toy exchange, building a framework for managing live price data, and tools for preparing high frequency data for simulations and backtests. Additional topics include Google’s Firebase real-time database and using Python to work with SQL. Extensive use of the Anki study system will be made to allow students to gauge their own progress and prepare for tests.

MATH-GA.2070-001

Data Science And Data-Driven Modeling

Prerequisites: N/A

Description: This is a half-semester course covering practical aspects of econometrics/statistics and data science/machine learning in an integrated and unified way as they are applied in the financial industry. We examine statistical inference for linear models, supervised learning (Lasso, ridge and elastic-net), and unsupervised learning (PCA- and SVD-based) machine learning techniques, applying these to solve common problems in finance. In addition, we cover model selection via cross-validation; manipulating, merging and cleaning large datasets in Python; and web-scraping of publicly available data.

MATH-GA.2791-001

Financial Securities And Markets

Prerequisites: Multivariate calculus, linear algebra, and calculus-based probability.

Description: This course provides a quantitative introduction to financial securities for students who are aspiring to careers in the financial industry. We study how securities traded, priced and hedged in the financial markets. Topics include: arbitrage; risk-neutral valuation; the log-normal hypothesis; binomial trees; the Black-Scholes formula and applications; the Black-Scholes partial differential equation; American options; one-factor interest rate models; swaps, caps, floors, swaptions, and other interest-based derivatives; credit risk and credit derivatives; clearing; valuation adjustment and capital requirements. It is important that students taking this course have good working knowledge of multivariate calculus, linear algebra and calculus-based probability.

MATH-GA.2751-001

Risk & Portfolio Management

Description: Risk management is arguably one of the most important tools for managing investment portfolios and trading books and quantifying the effects of leverage and diversification (or lack thereof). This course is an introduction to portfolio and risk management techniques for portfolios of (i) equities, delta-1 securities, and futures and (ii) basic fixed income securities. A systematic approach to the subject is adopted, based on selection of risk factors, econometric analysis, extreme-value theory for tail estimation, correlation analysis, and copulas to estimate joint factor distributions. We will cover the construction of risk measures (e.g. VaR and Expected Shortfall) and portfolios (e.g. portfolio optimization and risk). As part of the course, we review current risk models and practices used by large financial institutions. It is important that students taking this course have good working knowledge of multivariate calculus, linear algebra and calculus-based probability.

MATH-GA.2903-001

Stochastic Calculus

Description: The goal of this half-semester course is for students to develop an understanding of the techniques of stochastic processes and stochastic calculus as it is applied in financial applications. We begin by constructing the Brownian motion (BM) and the Ito integral, studying their properties. Then we turn to Ito’s lemma and Girsanov’s theorem, covering several practical applications. Towards the end of the course, we study the linkage between SDEs and PDEs through the Feynman-Kac equation. It is important that students taking this course have good working knowledge of calculus-based probability.

MATH-GA.2071-001

Machine Learning & Computational Statistics

Prerequisites: Multivariate calculus, linear algebra, and calculus-based probability. Students should also have working knowledge of basic statistics and machine learning (such as what is covered in Data Science & Data-Driven Modeling).

Description: Description TBA

MATH-GA.2793-001

Dynamic Asset Pricing

Prerequisites: Calculus-based probability, Stochastic Calculus, and a one semester course on derivative pricing (such as what is covered in Financial Securities and Markets).

Description: This is an advanced course on asset pricing and trading of derivative securities. Using tools and techniques from stochastic calculus, we cover (1) Black-Scholes-Merton option pricing; (2) the martingale approach to arbitrage pricing; (3) incomplete markets; and (4) the general option pricing formula using the change of numeraire technique. As an important example of incomplete markets, we discuss bond markets, interest rates and basic term-structure models such as Vasicek and Hull-White. It is important that students taking this course have good working knowledge of calculus-based probability and stochastic calculus. Students should also have taken the course “Financial Securities and Markets” previously. In addition, we recommend an intermediate course on mathematical statistics or engineering statistics as an optional prerequisite for this class.

MATH-GA.2755-001

Project & Presentation

Description: Students in the Mathematics in Finance program conduct research projects individually or in small groups under the supervision of finance professionals. The course culminates in oral and written presentations of the research results.

MATH-GA.2043-001

Scientific Computing

Prerequisites: Undergraduate multivariate calculus and linear algebra. Programming experience strongly recommended but not required.

Description: This course is intended to provide a practical introduction to computational problem solving. Topics covered include: the notion of well-conditioned and poorly conditioned problems, with examples drawn from linear algebra; the concepts of forward and backward stability of an algorithm, with examples drawn from floating point arithmetic and linear-algebra; basic techniques for the numerical solution of linear and nonlinear equations, and for numerical optimization, with examples taken from linear algebra and linear programming; principles of numerical interpolation, differentiation and integration, with examples such as splines and quadrature schemes; an introduction to numerical methods for solving ordinary differential equations, with examples such as multistep, Runge Kutta and collocation methods, along with a basic introduction of concepts such as convergence and linear stability; An introduction to basic matrix factorizations, such as the SVD; techniques for computing matrix factorizations, with examples such as the QR method for finding eigenvectors; Basic principles of the discrete/fast Fourier transform, with applications to signal processing, data compression and the solution of differential equations. This is not a programming course but programming in homework projects with MATLAB/Octave and/or C is an important part of the course work. As many of the class handouts are in the form of MATLAB/Octave scripts, students are strongly encouraged to obtain access to and familiarize themselves with these programming environments. Recommended Texts: Bau III, D., & Trefethen, L.N. (1997). Numerical Linear Algebra. Philadelphia, PA: Society for Industrial & Applied Mathematics Quarteroni, A.M., & Saleri, F. (2006). Texts in Computational Science & Engineering [Series, Bk. 2]. Scientific Computing with MATLAB and Octave (2nd ed.). New York, NY: Springer-Verlag Otto, S.R., & Denier, J.P. (2005). An Introduction to Programming and Numerical Methods in MATLAB. London: Springer-Verlag London

MATH-GA.2046-001

Advanced Statistical Inference And Machine Learning

Prerequisites: The following four courses, or equivalent: (1) Data Science and Data-Driven Modeling, (2) Financial Securities and Markets, (3) Machine Learning & Computational Statistics, and (4) Risk and Portfolio Management. It is important you have experience with the Python stack.

Description: A rigorous background in Bayesian statistics geared towards applications in finance. The early part of the course will cover the Bayesian approach to modeling, inference, point estimation, and forecasting, sufficient statistics, exponential families and conjugate priors, and the posterior predictive density. We will then undertake a detailed treatment of multivariate regression including Bayesian regression, variable selection techniques, multilevel/hierarchical regression models, and generalized linear models (GLMs). We will continue to discuss Bayesian networks and belief propagation with applications to machine learning and prediction tasks. Solution techniques include Markov Chain Monte Carlo methods, Gibbs Sampling, the EM algorithm, and variational mean field theory. We shall then introduce reinforcement learning with applications to transaction cost minimization and realistic optimal hedging of derivatives. Real world examples will be given throughout the course, including portfolio optimization with transaction costs, and a selection of the most important prediction tasks arising in buy-side quant trading.

MATH-GA.2049-001

Alternative Data In Quantitative Finance

Prerequisites: Risk and Portfolio Management; and Computing in Finance. In addition, students should have a working knowledge of statistics, finance, and basic machine learning. Students should have working experience with the Python stack (numpy/pandas/scikit-learn).

Description: This half-semester elective course examines techniques dealing with the challenges of the alternative data ecosystem in quantitative and fundamental investment processes. We will address the quantitative tools and technique for alternative data including identifier mapping, stable panel creation, dataset evaluation and sensitive information extraction. We will go through the quantitative process of transferring raw data into investment data and tradable signals using text mining, time series analysis and machine learning. It is important that students taking this course have working experience with Python Stack. We will analyze real-world datasets and model them in Python using techniques from statistics, quantitative finance and machine learning.

MATH-GA.2047-001

Trends In Financial Data Science

Description: This is a full semester course covering recent and relevant topics in alternative data, machine learning and data science relevant to financial modeling and quantitative finance. This is an advanced course that is suitable for students who have taken the more basic graduate machine learning and finance courses Data Science and Data-Driven Modeling, and Machine Learning & Computational Statistics, Financial Securities and Markets, and Risk and Portfolio Management.

MATH-GA.2803-001

Fixed Income Derivatives: Models & Strategies In Practice

Prerequisites: Familiarity with the foundational mathematical tools of finance; basic understanding of the motivation for and the machinery of pricing models in the interest-rate domain; programming skills; basic proficiency in Excel. Some product knowledge of interest-rate products is helpful but not required.

Description: Armed with a foundation in bond math and the theory and implementation of interest-rate models, many fixed-income quants are challenged to understand how these concepts and tools are deployed in the sales/trading environment. Often the economic content of a simple trade idea gets obscured by market jargon, especially in a competitive transactional environment. The class will focus on the practical workings of the fixed-income and rates-derivatives markets. The content is motivated by a representative set of real-world trading, investment, and hedging objectives. Each situation will be examined from the ground level; risk and reward attributes will be identified. This strategy will reinforce the link from underlying market views to the applicable product set and to the tools for managing the position. Common threads among products – structural or model-based – will be emphasized. We plan on covering bonds, swaps, flow options, semi-exotics, and some structured products. This problem-oriented holistic view is a productive way to understand the line from product creation to modeling, marketing, trading, and hedging. We hope to convey intuition about both the power and limitations of models. How do sell-side practitioners manage the various constraints and imperfections in the context of changing market backdrops and customer demands?

MATH-GA.2707-001

Time Series Analysis & Statistical Arbitrage

Prerequisites: Financial Securities and Markets; Scientific Computing in Finance (or Scientific Computing); and familiarity with basic probability.

Description: The term "statistical arbitrage" covers any trading strategy that uses statistical tools and time series analysis to identify approximate arbitrage opportunities while evaluating the risks inherent in the trades (considering the transaction costs and other practical aspects). This course starts with a review of Time Series models and addresses econometric aspects of financial markets such as volatility and correlation models. We will review several stochastic volatility models and their estimation and calibration techniques as well as their applications in volatility based trading strategies. We will then focus on statistical arbitrage trading strategies based on cointegration, and review pairs trading strategies. We will present several key concepts of market microstructure, including models of market impact, which will be discussed in the context of developing strategies for optimal execution. We will also present practical constraints in trading strategies and further practical issues in simulation techniques. Finally, we will review several algorithmic trading strategies frequently used by practitioners.

MATH-GA.2805-001

Trends In Sell-Side Modeling: Xva, Capital And Credit Derivatives

Prerequisites: Advanced Risk Management; Financial Securities and Markets, or equivalent familiarity with market and credit risk models; and Computing in Finance, or equivalent programming experience.

Description: This class explores technical and regulatory aspects of counterparty credit risk, with an emphasis on model building and computational methods. The first part of the class will provide technical foundation, including the mathematical tools needed to define and compute valuation adjustments such as CVA and DVA. The second part of the class will move from pricing to regulation, with an emphasis on the computational aspects of regulatory credit risk capital under Basel 3. A variety of highly topical subjects will be discussed during the course, including: funding costs, XVA metrics, initial margin, credit risk mitigation, central clearing, and balance sheet management. Students will get to build a realistic computer system for counterparty risk management of collateralized fixed income portfolios, and will be exposed to modern frameworks for interest rate simulation and capital management.

MATH-GA.2752-001

Active Portfolio Management

Prerequisites: Risk & Portfolio Management and Computing in Finance.

Description: The first part of the course will cover the theoretical aspects of portfolio construction and optimization. The focus will be on advanced techniques in portfolio construction, addressing the extensions to traditional mean-variance optimization including robust optimization, dynamical programming and Bayesian choice. The second part of the course will focus on the econometric issues associated with portfolio optimization. Issues such as estimation of returns, covariance structure, predictability, and the necessary econometric techniques to succeed in portfolio management will be covered. Readings will be drawn from the literature and extensive class notes.

MATH-GA.2753-001

Advanced Risk Management

Prerequisites: Financial Securities and Markets, and Computing in Finance or equivalent programming experience.

Description: The importance of financial risk management has been increasingly recognized over the last several years. This course gives a broad overview of the field, from the perspective of both a risk management department and of a trading desk manager, with an emphasis on the role of financial mathematics and modeling in quantifying risk. The course will discuss how key players such as regulators, risk managers, and senior managers interact with trading. Specific techniques for measuring and managing the risk of trading and investment positions will be discussed for positions in equities, credit, interest rates, foreign exchange, commodities, vanilla options, and exotic options. Students will be trained in developing risk sensitivity reports and using them to explain income, design static and dynamic hedges, and measure value-at-risk and stress tests. Students will create Monte Carlo simulations to determine hedge effectiveness. Extensive use will be made of examples drawn from real trading experience, with a particular emphasis on lessons to be learned from trading disasters. Text: Allen, S.L. (2003). Wiley Finance [Series, Bk. 119]. Financial Risk Management: A Practitioner’s Guide to Managing Market and Credit Risk. Hoboken, NJ: John Wiley & Sons.

MATH-GA.2801-001

Advanced Topics In Equity Derivatives

Prerequisites: Financial Securities and Markets, Stochastic Calculus, and Computing in Finance or equivalent programming experience.

Description: This half-semester course will give a practitioner’s perspective on a variety of advanced topics with a particular focus on equity derivatives instruments, including volatility and correlation modeling and trading, and exotic options and structured products. Some meta-mathematical topics such as the practical and regulatory aspects of setting up a hedge fund will also be covered.

MATH-GA.2708-001

Algorithmic Trading & Quantitative Strategies

Prerequisites: Computing in Finance, and Risk and Portfolio Management, or equivalent.

Description: In this course we develop a quantitative investment and trading framework. In the first part of the course, we study the mechanics of trading in the financial markets, some typical trading strategies, and how to work with and model high frequency data. Then we turn to transaction costs and market impact models, portfolio construction and robust optimization, and optimal betting and execution strategies. In the last part of the course, we focus on simulation techniques, back-testing strategies, and performance measurement. We use advanced econometric tools and model risk mitigation techniques throughout the course. Handouts and/or references will be provided on each topic.

MATH-GA.2798-001

Interest Rate & Fx Models

Prerequisites: Financial Securities and Markets, Stochastic Calculus, and Computing in Finance (or equivalent familiarity with financial models, stochastic methods, and computing skills).

Description: The course is divided into two parts. The first addresses the fixed-income models most frequently used in the finance industry, and their applications to the pricing and hedging of interest-based derivatives. The second part covers the foreign exchange derivatives markets, with a focus on vanilla options and first-generation (flow) exotics. Throughout both parts, the emphasis is on practical aspects of modeling, and the significance of the models for the valuation and risk management of widely-used derivative instruments.

MATH-GA.2802-001

Market Microstructure

Prerequisites: Financial Securities and Markets, Risk and Portfolio Management, and Computing in Finance or equivalent programming experience.

Description: This is a half-semester course covering topics of interest to both buy-side traders and sell-side execution quants. The course will provide a detailed look at how the trading process actually occurs and how to optimally interact with a continuous limit-order book market. We begin with a review of early models, which assume competitive suppliers of liquidity whose revenues, corresponding to the spread, reflect the costs they incur. We discuss the structure of modern electronic limit order book markets and exchanges, including queue priority mechanisms, order types and hidden liquidity. We examine technological solutions that facilitate trading such as matching engines, ECNs, dark pools, multiple venue problems and smart order routers. The second part of the course is dedicated pre-trade market impact estimation, post-trade slippage analysis, optimal execution strategies and dynamic no-arbitrage models. We cover Almgren-Chriss model for optimal execution, Gatheral’s no-dynamic-arbitrage principle and the fundamental relationship between the average response of the market price to traded quantity, and properties of the decay of market impact. Homework assignments will supplement the topics discussed in lecture. Some coding in Java will be required and students will learn to write their own simple limit-order-book simulator and analyze real NYSE TAQ data.

MATH-GA.2799-001

Modeling And Risk Management Of Bonds And Securitized Products

Prerequisites: Stochastic Calculus, and Financial Securities and Markets or equivalent knowledge of basic bond mathematics and bond risk measures (duration and convexity).

Description: This half-semester course is designed for students interested in Fixed Income roles in front-office trading, market risk management, model development (“Quants”, “Strats”), or model validation. We begin by modeling the cash flows of a generic bond, emphasizing how the bond reacts to changes in markets, how traders may position themselves given their views on the markets, and how risk managers think about the risks of a bond. We then focus on Mortgages, covering the fundamentals of Residential Mortgages, and Mortgage-Backed Securities. Students will build pricing models for mortgages, pass-throughs, sequentials and CMO’s that generate cash flows and that take into account interest rates, prepayments and credit spreads (OAS). The goals are for students to develop: (1) an understanding of how to build these models and how assumptions create “model risk”, and (2) a trader’s and risk manager’s intuition for how these instruments behave as markets change, and (3) a knowledge how to hedge these products. We will graph cash flows and changes in market values to enhance our intuition (e.g. in Excel, Python or by using another graphing tool). In the course we also review the structures of CLO’s, Commercial Mortgage Backed Securities (CMBS), Auto Asset Backed Securities (ABS), Credit Card ABS, subprime mortgages and CDO’s and credit derivatives such as CDX, CMBX and ABX. We discuss the modeling risks of these products and the drivers of the Financial Crisis of 2008. As time permits, we touch briefly on Peer-to-peer / MarketPlace Lending.

MATH-GA.1234-001

Cryptocurrency And Blockchains: Mathematics And Technologies

Prerequisites: Multivariate calculus and calculus-based probability. Students should have completed Computing in Finance (MATH-GA-2401) or equivalent, have strong coding skills in Python, and working experience with the Python stack (numpy/pandas/scikit-learn).

Description: This course is only offered in the Fall Semester. This half-semester course examines the building technologies and concepts in distributed ledger technologies and the workings of crypto financial markets. We begin by an overview of the traditional central banking system and the mechanics of central bank money and commercial bank lending as the two dominant mechanisms of money creation. We explore the current network of banking in traditional finance (TradFi) and its hierarchy of commercial banks, central banks, correspondent banks, settlement and clearing mechanism, and the instruments used to create and transmit money. We cover the principles of private and public key cryptography and its usage in encryption, digital signature, and message authentication. Hash functions serve as one-way functions that play a prominent role in creating message digests and solving the cryptographic puzzle in proof-of-work-based blockchains. We cover the main challenges of secure communication and typical attacks such as replay, man-in-the-middle, Sybil attacks and the cryptographic techniques used to tackle them. Next, we take a deep-dive in the original Bitcoin whitepaper and show how the integration of cryptographic digital signatures, recursive blockchains, hash-based proof-of-work consensus mechanism to solve the 51% attack, and double-spend problem gave rise to the pioneering Bitcoin blockchain. The Ethereum blockchain and its smart contracts have given rise to a variety of distributed apps (dApps), prominent among them decentralized exchanges (DEX) using constant function demand curves for creating automatic market-making. We cover the mechanics of these markets and concepts of swapping, liquidity pairs, yield farming and the general landscape of decentralized finance (DeFi). Blockchain data is public by design and there is a wealth of real-time and historical data. We discuss some of the data analysis and machine learning methods utilized to analyze this type of data. Given that blockchain is a software protocol, it is important that students taking this course have strong coding skills in Python and working experience with the Python stack (numpy/pandas/scikit-learn).

MATH-GA.2800-001

Trading Energy Derivatives

Prerequisites: Financial Securities and Markets, and Stochastic Calculus.

Description: The course provides a comprehensive overview of most commonly traded quantitative strategies in energy markets. The class bridges quantitative finance and energy economics covering theories of storage, net hedging pressure, optimal risk transfer, and derivatives pricing models. Throughout the course, the emphasis is placed on understanding the behavior of various market participants and trading strategies designed to monetize inefficiencies resulting from their activities and hedging needs. We discuss in detail recent structural changes related to financialization of energy commodities, crossmarket spillovers, and linkages to other financial asset classes. Trading strategies include traditional risk premia, volatility, correlation, and higher-order options Greeks. Examples and case studies are based on actual market episodes using real market data.

MATH-GA.2048-001

Scientific Computing In Finance

Prerequisites: Risk and Portfolio Management, Financial Securities and Markets, and Computing in Finance.

Description: This is a version of the course Scientific Computing (MATH-GA 2043.001) designed for applications in quantitative finance. It covers software and algorithmic tools necessary to practical numerical calculation for modern quantitative finance. Specific material includes IEEE arithmetic, sources of error in scientific computing, numerical linear algebra (emphasizing PCA/SVD and conditioning), interpolation and curve building with application to bootstrapping, optimization methods, Monte Carlo methods, and the solution of differential equations. Please Note: Students may not receive credit for both MATH-GA 2043.001 and MATH-GA 2048.001

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