Course Descriptions

Elements of topology on the real line. Rigorous treatment of limits, continuity, differentiation, and the Riemann integral. Taylor series. Introduction to metric spaces. Pointwise and uniform convergence for sequences and series of functions. Applications.

Prerequisites:

A good background in linear algebra, and some experience with writing computer programs (in MATLAB, Python or another language). MATLAB will be used as the main language for the course. Alternatively, you can also use Python for the homework assignments. You are encouraged but not required to learn and use a compiled language.

Description:

This course is part of a two-course series meant to introduce graduate students in mathematics to the fundamentals of numerical mathematics (but any Ph.D. student seriously interested in applied mathematics should take it). It will be a demanding course covering a broad range of topics. There will be extensive homework assignments involving a mix of theory and computational experiments, and an in-class final. Topics covered in the class include floating-point arithmetic, solving large linear systems, eigenvalue problems, interpolation and quadrature (approximation theory), nonlinear systems of equations, linear and nonlinear least squares, nonlinear optimization, and Fourier transforms. This course will not cover differential equations, which form the core of the second part of this series, Numerical Methods II.

Recommended Text (Springer books are available online from the NYU network):

• Deuflhard, P. & Hohmann, A. (2003). Numerical Analysis in Modern Scientific Computing. Texts in Applied Mathematiks [Series, Bk. 43]. New York, NY: Springer-Verlag.

Further Reading (available on reserve at the Courant Library):

• Bau III, D., & Trefethen, L.N. (1997). Numerical Linear Algebra. Philadelphia, PA: Society for Industrial & Applied Mathematics.Quarteroni, A., Sacco, R., & Saleri, F. (2006). Numerical Mathematics (2^nd ed.). Texts in Applied Mathematics [Series, Bk. 37]. New York, NY: Springer-Verlag.

If you want to brush up your MATLAB:

• Gander, W., Gander, M.J., & Kwok, F. (2014). Scientific Computing – An Introduction Using Maple and MATLAB. Texts in Computation Science and Engineering [Series, Vol. 11]. New York, NY: Springer-Verlag.
• Moler, C. (2004). Numerical Computing with Matlab. SIAM. Available online.

This seminar will follow up on Numerical Methods II and cover more advanced computational methods (finite difference (FD), volume (FV), and element (FE) schemes including boundary conditions, and boundary integral methods) for solving PDEs that arise in physical sciences. The class will assume familiarity with multi-step and multi-stage (Runge-Kutta) methods for solving systems of ODEs including stability theory, basic finite difference methods for elliptic, parabolic, and hyperbolic PDEs (including von Neumann/Fourier stability analysis), and basic spectral methods (e.g. FFT-based schemes for periodic domains). The class will cover electrostatics (FD and FEM for Poisson including geometric and algebraic multigrid, Laplace in confined domains using boundary integral), elasticity (variational formulation, finite-element methods), linear wave equation (electromagnetism, acoustics, geofluids), and fluid dynamics (FD/FV for 1D conservation laws including dispersion and dissipation, stability, accuracy, and modified equations, FV advection-diffusion including limiters, MAC/FEM for incompressible Navier-Stokes, immersed-boundary method, Stokes flow including boundary-integral methods).

Prerequisites: Procedural programming, some knowledge of Java recommended. 

Description:  This course will introduce students to the software development process, including applications in financial asset trading, research, hedging, portfolio management, and risk management. Students will use the Java programming language to develop object-oriented software, and will focus on the most broadly important elements of programming - superior design, effective problem solving, and the proper use of data structures and algorithms. Students will work with market and historical data to run simulations and test strategies. The course is designed to give students a feel for the practical considerations of software development and deployment. Several key technologies and recent innovations in financial computing will be presented and discussed.

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

Overview: 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 (2^nd 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

Prerequisites: Continuous Time Finance or permission of instructor.

Description:  The classical curriculum of mathematical finance programs generally covers the link between linear parabolic partial differential equations (PDEs) and stochastic differential equations (SDEs), resulting from Feynmam-Kac’s formula. However, the challenges faced by today’s practitioners mostly involve nonlinear PDEs. The aim of this course is to provide the students with the mathematical tools and numerical methods required to tackle these issues, and illustrate the methods with practical case studies like American option pricing, uncertain volatility, uncertain mortality, different rates for borrowing and lending, calibration of models to market smiles, credit valuation adjustment (CVA), transaction costs, illiquid markets, super-replication under delta and gamma constraints, etc.

We will strive to make this course reasonably comprehensive, and to find the right balance between ideas, mathematical theory, and numerical implementations. We will spend some time on the theory: optimal stopping, stochastic control, backward stochastic differential equations (BSDEs), McKean SDEs, branching diffusions. But the main focus will deliberately be on ideas and numerical examples, which we believe help a lot in understanding the tools and building intuition.

PDE methods suffer from the curse of dimensionality. Since most quantitative finance problems are highdimensional,

we will mostly focus on simulation-based methods (a.k.a. Monte Carlo algorithms). This course exposes the students with a wide variety of Machine Learning techniques, old and new, including parametric regression, nonparametric regression, neural networks, kernel trick, etc. These techniques allow us to compute some quantities that are key ingredients of the nonlinear Monte Carlo algorithms.

The Python programming language will be used to provide simple numerical simulations illustrating the methods presented in the course. Homeworks will allow the students to check their understanding of the course by solving exercises inspired by our experience as quantitative analysts, and will involve some coding in Python.

Recommended text: 

• Guyon, J. and Henry-Labordère, P.: Nonlinear Option Pricing, Chapman & Hall/CRC Financial Mathematics Series, 2014.

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. 

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. 

Course 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. 

For the syllabus for the course, click HERE.

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. 

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.

Prerequisites: Undergraduate linear algebra.

Description: Linear algebra is two things in one: a general methodology for solving linear systems, and a beautiful abstract structure underlying much of mathematics and the sciences. This course will try to strike a balance between both. We will follow the book of our own Peter Lax, which does a superb job in describing the mathematical structure of linear algebra, and complement it with applications and computing. The most advanced topics include spectral theory, convexity, duality, and various matrix decompositions.

Text: Lax, P.D. (2007). Pure and Applied Mathematics: A Wiley Series of Texts, Monographs and Tracts [Series, Bk. 78]. Linear Algebra and Its Applications (2^nd ed.). Hoboken, NJ: John Wiley & Sons/ Wiley-Interscience.

Recommended Text: Strang, G. (2005). Linear Algebra and Its Applications (4^th ed.). Stamford, CT: Cengage Learning.

Prerequisites: Elements of linear algebra and the theory of rings and fields.

Description: Basic concepts of groups, rings and fields. Symmetry groups, linear groups, Sylow theorems; quotient rings, polynomial rings, ideals, unique factorization, Nullstellensatz; field extensions, finite fields.

Recommended Texts:

• Artin, M. (2010). Featured Titles for Abstract Alagebra [Series]. Algebra (2^nd ed.). Upper Saddle River, NJ: Pearson
• Chambert-Loir, A. (2004). Undergraduate Texts in Mathematics [Series]. A Field Guide to Algebra (2005 ed.). New York, NY: Springer-Verlag
• Serre, J-P. (1996). Graduate Texts in Mathematics [Series, Vol. 7]. A Course in Arithmetic (Corr. 3^rd printing 1996 ed.). New York, NY: Springer-Verlag

Prerequisites: Any knowledge of groups, rings, vector spaces and multivariable calculus is helpful. Undergraduate students planning to take this course must have V63.0343 Algebra I or permission of the Department.

Description: After introducing metric and general topological spaces, the emphasis will be on the algebraic topology of manifolds and cell complexes. Elements of algebraic topology to be covered include fundamental groups and covering spaces, homotopy and the degree of maps and its applications. Some differential topology will be introduced including transversality and intersection theory. Some examples will be taken from knot theory.

Recommended Texts:

• Hatcher, A. (2002). Algebraic Topology. New York, NY: Cambridge University Press
• Munkres, J. (2000). Topology (2^nd ed.). Upper Saddle River, NJ: Prentice-Hall/ Pearson Education
• Guillemin, V., Pollack, A. (1974). Differential Topology. Englewood Cliffs, NJ: Prentice-Hall
• Milnor, J.W. (1997). Princeton Landmarks in Mathematics [Series]. Topology from a Differentiable Viewpoint (Rev. ed.). Princeton, NJ: Princeton University Press

Prerequisites: Multivariable calculus and linear algebra.

Description: Differentiable manifolds, tangent bundle, embedding theorems, vector fields and differential forms. Introduction to Riemannian metrics, connections and geodesics.

Text: Lee, J.M. (2009). Graduate Studies in Mathematics [Series, Vol. 107]. Manifolds and Differential Geometry. Providence, RI: American Mathematical Society.

As part of our new NSF research training group (RTG) in Modeling & Simulation, we will be organizing a lunchtime group meeting for students, postdocs, and faculty working in applied mathematics who do modeling & simulation. The aim is to create a space to discuss applied mathematics research in an informal setting: to (a) give students and postdocs a chance to present their research (or a topic of common interest) and get feedback from the group, (b) learn about other ongoing and future research activities in applied math at the Institute, and (c) discuss important open problems and research challenges.

Description: The Atmosphere Ocean Science Student Seminar focuses on research and presentation skills. The course is spread across two semesters, and participants are expected to participate in both to earn the full 3 credits. Participants will prepare and present a full length (45-50 minute) talk on their research each semester, for a total of two over the duration of the course. In addition, short “elevator talks” are developed and given in the second semester, the goal being to encapsulate the key points of your research in under 5 minutes. A main goal of the course is learning to present your research to different audiences. We consider overview talks, appropriate for a department wide colloquium, specialty talks, as would be given in a focused seminar, and a broad pitch you would give when meeting people and entering the job market. When not presenting, students are expected to engage with the speaker, asking questions and providing feedback at the end of the talk.

In this course I  will discuss basic properties of the geometry of projective curves and the fields of rational functions on the curves.

There is a lot of literature on the subject: 

I will use the book of Arbarello, Cornalba, Griffiths, Harris, Geomtry of Algebraic curves and  the book Bogomolov Petrov `Algebraic curves and One-dimensional fields' Courant lecture Notes

Note: Master's students need permission of course instructor before registering for this course.

Prerequisites: A familiarity with rigorous mathematics, proof writing, and the epsilon-delta approach to analysis, preferably at the level of MATH-GA 1410, 1420 Introduction to Mathematical Analysis I, II.

Description: Measure theory and integration. Lebesgue measure on the line and abstract measure spaces. Absolute continuity, Lebesgue differentiation, and the Radon-Nikodym theorem. Product measures, the Fubini theorem, etc. L^p spaces, Hilbert spaces, and the Riesz representation theorem. Fourier series.

Main Text: Folland's Real Analysis: Modern Techniques and Their Applications

Secondary Text: Bass' Real Analysis for Graduate Students

Prerequisites: Advanced calculus (or equivalent).

Description: Complex numbers; analytic functions; Cauchy-Riemann equations; Cauchy's theorem; Laurent expansion; analytic continuation; calculus of residues; conformal mappings.

Text: Marsden and Hoffman, Basic Complex Analysis, 3d edition

Note: Master's students need permission of course instructor before registering for this course.

Prerequisites: Complex Variables I (or equivalent) and MATH-GA 1410 Introduction to Math Analysis I.

Description: Complex numbers, the complex plane. Power series, differentiability of convergent power series. Cauchy-Riemann equations, harmonic functions. conformal mapping, linear fractional transformation. Integration, Cauchy integral theorem, Cauchy integral formula. Morera's theorem. Taylor series, residue calculus. Maximum modulus theorem. Poisson formula. Liouville theorem. Rouche's theorem. Weierstrass and Mittag-Leffler representation theorems. Singularities of analytic functions, poles, branch points, essential singularities, branch points. Analytic continuation, monodromy theorem, Schwarz reflection principle. Compactness of families of uniformly bounded analytic functions. Integral representations of special functions. Distribution of function values of entire functions.

Text: Ahlfors, L. (1979). International Series in Pure and Applied Mathematics [Series, Bk. 7]. Complex Analysis (4thin ed.). New York, NY: McGraw-Hill.

Note: Master's students should consult course instructor before registering for PDE II in the spring.

Prerequisites: Knowledge of undergraduate level linear algebra and ODE; also some exposure to complex variables (can be taken concurrently).

Description: A basic introduction to PDEs, designed for a broad range of students whose goals may range from theory to applications. This course emphasizes examples, representation formulas, and properties that can be understood using relatively elementary tools. We will take a broad viewpoint, including how the equations we consider emerge from applications, and how they can be solved numerically. Topics will include: the heat equation; the wave equation; Laplace's equation; conservation laws; and Hamilton-Jacobi equations. Methods introduced through these topics will include: fundamental solutions and Green's functions; energy principles; maximum principles; separation of variables; Duhamel's principle; the method of characteristics; numerical schemes involving finite differences or Galerkin approximation; and many more.

See the syllabus for more information (including a tentative semester plan).

Recommended Texts:

• Guenther, R.B., & Lee, J.W. (1996). Partial Differential Equations of Mathematical Physics and Integral Equations. Mineola, NY: Dover Publications.
• Evans, L.C. (2010). Graduate Studies in Mathematics [Series, Bk. 19]. Partial Differential Equations (2^nd ed.). Providence, RI: American Mathematical Society.

Prerequisites: MATH-GA 2500 (Partial Differential Equations), or a comparable introduction to PDE using Sobolev spaces and functional analysis.

Selected PDE topics of broad importance and applicability, including:  boundary integral methods for elliptic PDE; regularity via Schauder estimates; steepest-descent and dynamical systems perspectives on some nonlinear parabolic equations; weak and strong solutions of the Navier-Stokes equations; and topics from the calculus of variations, including homogenization and Gamma convergence.

Prerequisites:

Real analysis; basic knowledge of complex variables and functional analysis.

Description:

Fourier series and integrals, Hardy-Littlewood maximal function, interpolation theory, Hilbert transform, singular integrals and Calderon-Zygmund theory, oscillatory integrals, Littlewood-Paley theory, pseudo-differential operators and Sobolev spaces. If time allows: paradifferential calculus, T1 theorem. 

The course will follow the book

Fourier Analysis by Javier Duoandikoetxea, Graduate Studies in Mathematics, AMS, 2001, as well as the lecture notes by Terry Tao

https://www.math.ucla.edu/~tao/247a.1.06f/
https://www.math.ucla.edu/~tao/247b.1.07w/

Enhanced dissipation (or diffusion) refers to the interaction of a first-order advection term with a second-order diffusive term in a parabolic PDE, resulting in an increase in "effective" diffusion or energy dissipation on certain time and length scales. This phenomenon is particularly important in fluids but arises in other contexts as well, such as in probability (mixing, ergodicity). In this course, we will present some basic models exhibiting enhanced dissipation and review various techniques for analyzing it, such as homogenization and hypo-coercivity methods. The course will be examples-based (both classical and latest research papers) and focused on building a portfolio of useful techniques. 

This course is an introduction to ergodic theory, a probabilistic approach  to dynamical systems. No prior knowledge of the subject is assumed.  Topics include ergodicity, the Ergodic Theorems, mixing properties,  entropy; ergodic theory of continuous and differentiable maps including Lyapunov exponents. Class notes will be made available in advance, and students are expected to go over the material beforehand so more class time can be devoted to discussion.

Prerequisites: Real analysis at the graduate level.

Recommended Text: Walters, P. (2000). Graduate Texts in Mathematics [Series, Bk. 79]. An Introduction to Ergodic Theory. New York, NY: Springer-Verlag.

Prerequisites: Elementary linear algebra and differential equations.

Description: This is a first-year course for all incoming PhD and Masters students interested in pursuing research in applied mathematics. It provides a concise and self-contained introduction to advanced mathematical methods, especially in the asymptotic analysis of differential equations. Topics include scaling, perturbation methods, multi-scale asymptotics, transform methods, geometric wave theory, and calculus of variations

Recommended Texts:

• Barenblatt, G.I. (1996). Cambridge Texts in Applied Mathematics [Series, Bk. 14]. Scaling, Self-similarity, and Intermediate Asymptotics: Dimensional Analysis and Intermediate Asymptotics. New York, NY: Cambridge University Press
• Hinch, E.J. (1991). Cambridge Texts in Applied Mathematics [Series, Bk. 6]. Perturbation Methods. New York, NY: Cambridge University Press
• Bender, C.M., & Orszag, S.A. (1999). Advanced Mathematical Methods for Scientists and Engineers [Series, Vol. 1]. Asymptotic Methods and Perturbation Theory. New York, NY: Springer-Verlag
• Whitham, G.B. (1999). Pure and Applied Mathematics: A Wiley Series of Texts, Monographs and Tracts [Series Bk. 42]. Linear and Nonlinear Waves (Reprint ed.). New York, NY: John Wiley & Sons/ Wiley-Interscience
• Gelfand, I.M., & Fomin, S.V. (2000). Calculus of Variations. Mineola, NY: Dover Publications

Prerequisites: Introductory complex variable and partial differential equations.

Description: The course will expose students to basic fluid dynamics from a mathematical and physical perspectives, covering both compressible and incompressible flows. Topics: conservation of mass, momentum, and Energy. Eulerian and Lagrangian formulations. Basic theory of inviscid incompressible and compressible fluids, including the formation of shock waves. Kinematics and dynamics of vorticity and circulation. Special solutions to the Euler equations: potential flows, rotational flows, irrotational flows and conformal mapping methods. The Navier-Stokes equations, boundary conditions, boundary layer theory. The Stokes Equations.

Text: Childress, S. Courant Lecture Notes in Mathematics [Series, Bk. 19]. An Introduction to Theoretical Fluid Mechanics. Providence, RI: American Mathematical Society/ Courant Institute of Mathematical Sciences.

Recommended Text: Acheson, D.J. (1990). Oxford Applied Mathematics & Computing Science Series [Series]. Elementary Fluid Dynamics. New York, NY: Oxford University Press.

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.

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

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.

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.  

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

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.

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

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?

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.

Prerequisites:

Calculus through partial derivatives and multiple integrals; no previous knowledge of probability is required.

Description:

The course introduces the basic concepts and methods of probability.

Topics include: probability spaces, random variables, distributions, law of large numbers, central limit theorem, random walk, Markov chains and martingales in discrete time, and if time allows diffusion processes including Brownian motion.

Required text:

Probability and Random Processes, 3rd edition by G.Grimmett and D. Stirzaker, Oxford Press 2001 (Note: this is NOT the newer 4th edition).

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

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.

Prerequisites:

A first course in probability, familiarity with Lebesgue integral, or MATH-GA 2430 Real Variables as mandatory co-requisite.

Description:

First semester in an annual sequence of Probability Theory, aimed primarily for Ph.D. students. Topics include laws of large numbers, weak convergence, central limit theorems, conditional expectation, martingales and Markov chains.

Recommended Text:

S.R.S. Varadhan, Probability Theory (2001).

Description:

This course serves as an introduction to the fundamentals of geophysical fluid dynamics. No prior knowledge of fluid dynamics will be assumed, but the course will move quickly into the subtopic of rapidly rotating, stratified flows. Topics to be covered include (but are not limited to): the advective derivative, momentum conservation and continuity, the rotating Navier-Stokes equations and non-dimensional parameters, equations of state and thermodynamics of Newtonian fluids, atmospheric and oceanic basic states, the fundamental balances (thermal wind, geostrophic and hydrostatic), the rotating shallow water model, vorticity and potential vorticity, inertia-gravity waves, geostrophic adjustment, the quasi-geostrophic approximation and other small-Rossby number limits, Rossby waves, baroclinic and barotropic instabilities, Rayleigh and Charney-Stern theorems, geostrophic turbulence. Students will be assigned bi-weekly homework assignments and some computer exercises, and will be expected to complete a final project. This course will be supplemented with out-of-class instruction.

Recommended Texts:

• Vallis, G.K. (2006). Atmospheric and Oceanic Fluid Dynamics: Fundamentals and Large-scale Circulation. New York, NY: Cambridge University Press.
• Salmon, R. (1998). Lectures on Geophysical Fluid Dynamics. New York, NY: Oxford University Press.
• Pedlosky, J. (1992). Geophysical Fluid Dynamics (2nd ed.). New York, NY: Springer-Verlag.