Course Descriptions

This course and its sequel MA-GY 6223 rigorously treat the basic concepts and results in real analysis. Course topics include limits of sequences, topological concepts of sets for real numbers, properties of continuous functions and differentiable functions. Important concepts and theorems include supremum and infimum, Bolzano-Weierstrass theorem, Cauchy sequences, open sets, closed sets, compact sets, topological characterization of continuity, intermediate value theorem, uniform continuity, mean value theorems and inverse function theorem. | Offered in the fall.

Prerequisite(s): MA-UY 2122 or permission of adviser.

Prerequisites

Undergraduate-level linear algebra, multivariate calculus, probability and statistics. Basic program-
ming knowledge (Python).

Description
The goal of this class is to provide students with the fundamentals underlying modern computational
statistics and to understand how these methods can be effectively implemented in practical problems
of inference and estimation. Much of the material covered in the class is also found at the basis of
machine learning and will thus be useful to students following classes in statistical learning theory,
convex optimization and kernel methods. The course will include reminders on probability and random
variables, before presenting numerical methods for root finding, function minimization, function
approximation and numerical linear algebra. We will then move to numerical integration, random
variable generation and sampling methods, notably Markov chain Monte Carlo (MCMC) methods.
The final block will be elements of classical statistics : parametric regression and classification, Gaussian processes, CDF estimation, the bootstrap and the jacknife, density fitting and non- parametric regression. If time permits, we will briefly review variational inference as a complementary approach to sampling.

This course covers basic ideas of linear algebra: Groups, rings, fields, vector spaces, basis, dependence, independence, dimension. Relation to solving systems of linear equations and matrices. Homomorphisms, duality, inner products, adjoints and similarity. | Offered in the fall.

Prerequisite(s): MA-UY 2034  and MA-UY 2114 or Graduate Standing.

• 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.

Description: This class concerns the design of Monte Carlo sampling techniques for estimation of averages with respect to high dimensional probability distributions. Standard simulation tools such as importance sampling, Gibbs and Metropolis-Hastings sampling, Langevin dynamics, and hybrid Monte Carlo will be motivated and introduced. We will discuss the qualitative advantages and disadvantages of various schemes. Particular attention will be paid to the major complicating issues like conditioning and rare events along with methods to address them (e.g. tempering, interacting particle methods, and free-energy methods). This class does not cover in-depth mathematical convergence analysis of sampling algorithms.

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: 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. 

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.

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:   Undergraduate analysis and algebra at the level of MATH-UA 325 Analysis and MATH-UA 343 Algebra are strongly recommended.  Undergraduate students planning to take this course must have MATH-UA 343 Algebra and MATH-UA 325 Analysis (or the respective Honors versions) or permission of the Department.

Course Description:  After introducing metric spaces and 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, and homotopy.  Additional material may be covered at the discretion of the instructor, such as degree theory, transversality and intersection theory, and examples from knot theory.

Prerequisite: Basic knowledge of homology theory.

Description:  This one semester course will discuss invariants of manifold and of vector bundles, including theories of characteristic classes (e.g., Stiefel-Whitney classes, Chern classes and Pontryagin classes) and their applications to a variety of geometrical problems. These subjects are foundational for aspects of differential geometry, algebraic geometry, geometrical topology, symplectic geometry, combinatorial geometry and global analysis. 

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.

Description: The aim of this course is to introduce some key ideas and techniques in the theory of minimal surfaces. Topics will include the Douglas-Rado solutions to the Plateau problem, first and second variation of the area, curvature estimates and the Bernstein problem. If time permits, we will also discuss applications of minimal surfaces in other geometric problems.

Description: 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.

Description:  Recent progress on Ricci curvature including an overview of: quantitative behavior of singular sets and their rectifiability, for noncollapsed Gromov Hausdorff limit space with Ricci curvature bounded below, the proof of the codimension 4 conjecture on the singular set of noncollapsed limits of Einstein manifolds.

Description:  In these seven courses, we will first introduce the classical Gaussian Hermitian ensembles and calculate their local spectral statistics. The main goal for this class will then be to present a self-contained proof of universality of these statistics for Wigner matrices, through:

1. estimates on the resolvent up to small scales; 
2. distribution of the eigenvalues;
3. distribution of the eigenvectors.

The Dyson Brownian motion will play a key role.

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.

1. Lecture one: topology of Riemann surfaces, genus, Complex Riemann surfaces as quotients Fields of meromorphic functions on complex compact Riemann surfaces and first properties
2. Lecture two: One-dimensional fields, valuations of the field and points on a projective curve- i.e. compact complex Riemann surface if the ground field is the field of complex numbers
3. Lecture 3: Line bundles and coherent sheaves on curves,  divisors Classes of divisors. Elliptic and Hyperelliptic curves.
4. Lecture 4: Cohomology of coherent sheaves on  curves Hollmorpic differentials.
5. Lecture 5: Jacobian variety of the projective curve
6. Lecture 6: Duality, Riemann- Roch theorem Cohomology calculation,
7. Lecture 7: Curves over finite fields.

Description:

1. Two ways of turbulence onset in fluid flows: temporal and spatial. Temporal chaotization in unstable flows and dynamical chaos. Spatial chaotization in stable flows and percolation transition.
2. Developed turbulence and phenomenology of a cascade by Richardson and Kolmogorov. Turbulent diffusion.
3. Explicit realization of the cascade scenario in turbulence of weakly interacting waves. Kinetic wave equation and its general properties: conservation laws and entropy growth.
4. Exact stationary solutions of the kinetic wave equations in isotropic scale-invariant media. Locality of interaction and cascades.
5. Stationary turbulence spectra in anisotropic media and applications to geophysics and astrophysics.  Matching turbulence spectra with sources and sinks. Bottleneck effect.
6. Strong wave turbulence. Solitons, vortices and shocks.
7. Intermittency of fluid vortex turbulence. Kolmogorov multipliers.

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 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).

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.

Overview of the course: The plan is to cover the transport equation, the method of char-
acteristics, and the fundamental second order PDEs: the wave, Laplace and heat equations. Time permitting we will discuss: transform methods, Sobolev spaces, weak solutions, and some nonlinear PDEs.

Textbook: Evans, L.C. Partial Differential Equations (2nd ed), 2010. Graduate Studies in
Mathematics. Providence, RI: American Mathematical Society.

 

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

Description: 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/

Prerequisite: For topics (1) and (2), analysis of several variables is a must, basic knowledge of manifolds helpful; measure theory is assumed for topic (3).

Description:  This course introduces the student to the first fundamental ideas of differentiable dynamical systems, focusing on hyperbolic dynamics.  Hyperbolicity in dynamical systems refers to the fast separation of nearby orbits. No prior knowledge of the subject is assumed.  I will begin with otivating examples. The three main topics I plan to cover are (1) local theory (stable/unstable/center manifolds) at fixed points, (2) geometric theory of chaotic systems (horseshoes, homoclinic orbits, attractors); and (3) ergodic theory (ergodicity, mixing, Lyapunov exponents).

Recommended text: I will not be following any text. Class notes will be made available. The closest text is Introduction to Dynamical systems by Brin and Stuck.

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.

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.  

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.

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

 Course 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.

 

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: The first few weeks will be devoted to a crash course on large deviations, with emphasis on the role of entropy. The second half of the course will be focused on models of interfaces (Gaussian and non-Gaussian) and their inter-relations (GFF, Villain, Coulomb gases), associated entropic repulsion, CLT's, and extremes.

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.

Description:

The Earth’s atmosphere and oceans have been warming for the last 50 years, and are expected to do so until at least the end of this Century. Human activities, primarily the emission of carbon dioxide due to the burning of fossil fuel, are the main driving forces behind these changes. We will review the physical basis for anthropogenic climate change, study the evidence for climate changes over the last few decades, and analyze some of its potential future impacts.

As an advanced topics course, the class will also invite students to be active participants. About one third of the course will be dedicated to more formal lectures, one third to class discussion of recent scientific papers, and one third to student-led research projects. Among potential research topics are assessing climate sensitivity from recent warming, investigating recent trends in weather extremes, and analyzing the response of the atmospheric circulation.