Course Descriptions

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).

Description: 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 (2nd 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.

Prerequisites: a graduate-level PDE course, Numerical Methods II (or equivalent, with approval of syllabus by instructor(s)), and programming experience.

Description: This course follows on Numerical Methods II and covers theoretical and practical aspects of advanced computational methods for the numerical solution of partial differential equations. The first part will focus on finite element methods (FEMs), and the second part on finite volume methods (FVMs) including discontinuous Galerkin (FE+FV) methods. In addition to setting up the numerical and functional analysis theory behind these methods, the course will also illustrate how these methods can be implemented and used in practice for solving partial differential equations in two and three dimensions. Example PDEs will include the Poisson equation, linear elasticity, advection-diffusion(-reaction) equations, the shallow-water equations, the incompressible Navier-Stokes equation, and others if time permits. Students will complete a final project that includes using, developing, and/or implementing state-of-the-art solvers.

In the Fall of 2023, Georg Stadler will teach the first half of this course and cover FEMs, and Aleks Donev will teach in the second half of the course and cover FVMs. 

Literature:

1. Elman, Silvester, and Wathen: Finite Elements and Fast Iterative Solvers, Oxford University Press, 2014.
2. Farrell: Finite Element Methods for PDEs, lecture notes, 2021.
3. Hundsdorfer & Verwer: Numerical Solution of Time-Dependent Advection-Diffusion-Reaction Equations, Springer-Verlag, 2003.
4. Leveque: Finite Volume Methods for Hyperbolic Problems, Cambridge Press, 2002.

Prerequisites: Completion of a Python summer assignment and relevant self-study.

Description:  The purpose of this course is threefold: It will teach students the popular Python programming language. Students will learn the five most important concepts of modern, object-oriented software development: testing, data structures, design, working with data, and distributed computing. 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 realtime database and using Python to work with SQL. Students will make extensive use of the Anki study system to gauge their own progress and prepare for tests. Please note that students who sign up for this course should prepare by completing a summer assignment.

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

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. 

Prerequisties:  Student needs to have taken math courses in multivariate calculus, linear algebra, and calculus-based probability. In addition, they need to have taken a computer science / programming course.

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 or permission of the instructor.

Description:

Linear spaces, subspaces. Linear dependence, linear independence; span, basis, dimension, isomorphism. Quotient spaces. Linear functionals, Dual spaces. Linear mappings, null space, range, fundamental theorem of linear algebra. Underdetermined systems of linear equations. Composition, inverse, transpose of linear maps, algebra of linear maps. Similarity transformations. Matrices, matrix Multiplication, Matrix Inverse, Matrix Representation of Linear Maps, Determinant, Laplace expansion, Cramer's rule. Eigenvalue problem, eigenvalues and eigenvectors, characteristic polynomial, Cayley-Hamilton theorem. Diagonalization.

Prerequisites: Undergraduate linear algebra.

Description: Linear algebra is two things in one: a general methodology for solving linear systems, and an abstract structure underlying much of mathematics and the sciences. The course will assume that students have already had a course on linear algebra, and will be more advanced, focusing on analytical issues such as the behavior of eigenvalues

Recommended Text: Strang, G. (2005). Linear Algebra and Its Applications (4^th ed.). Stamford, CT: Cengage Learning. Lax, P.D. (2007). Pure and Applied Mathematics: A Wiley Series of Texts, Monographs and Tracts [Series, Bk. 78]. 

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.

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.

Background on analysis on manifolds, convex integration, including Nash C^1 -isometric immersion theorem, De Lellis-Székelyhidi construction of weak solutions of the Euler equation and the generalized Nash implicit function theorem with applications.

 Several methods for the proofs of the classical isoperimetric inequality and its generalizations including Federer-Fleming inequality, Almgren’s Sharp Inequality, Gaussian isoperimetric inequality, the Brunn-Minkowski and Alexandrov-Fenchel inequalities, Poincare -Sobolev inequalities, Shannon, Loomis-Whitney and Rodgers-Brascamp-Lieb inequalities, general concentration of measure theorems and probabilistic construction of expanders and related spaces.

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.

Prerequisites: Math: familiarity with applied differential equations or permission of instructor; Neurobiology: background will be provided.

Description:  This half-term course will involve the formulation and analysis of differential equation models for neuronal ensembles and neuronal computations. Spiking and firing rate mechanistic treatments of network dynamics as well as probabilistic behavioral descriptions will be covered.  We will consider mechanisms of coupling, synaptic dynamics, rhythmogenesis, synchronization, bistability, adaptation,…  Applications will likely include: competition and perceptual bistability, decision-making, working memory, brain oscillations (fast: gamma-like; slow: up-down states), time estimation and learning a rhythm. There will be a few homework exercises.

Description:  This is an advanced graduate class on discrete probabilistic methods. Familiarity with undergraduate-level of probability and combinatorics will be assumed. For the first couple of weeks, we will cover basic techniques based on the following references:

• Alon, Spencer, The Probabilistic Method
• Janson, Łuczak, Rucinski, Random Graphs

Prerequisites:  Some Riemannian geometry, algebraic topology, and group theory.

Description:  ​Geometric group theory is based on the idea that we can study discrete groups by looking at geometric objects on which they act. A single group can act on many different spaces, but all of these spaces share certain large-scale geometric properties and asymptotic behavior. In this course, we will introduce some of the techniques of geometric group theory and study their applications, especially hyperbolic groups and groups with nonpositive curvature.

References:

• B. Bowditch, A course on geometric group theory
•  C. Druţu, M. Kapovich, Geometric group theory
• Clara Löh, Geometric Group Theory: an Introduction

Description: The course answers the most frequent question: how much can we say and do about something we don’t know? It teaches how to measure uncertainty by different entropic measures, from Boltzmann and Gibbs entropies to mutual information and entanglement entropy. Different ways of entropy growth (information forgetting) will be discussed: from dynamical chaos to renormalization group. Bayesian framework for hypotheses testing will be explained. Examples will be taken from physics, linguistics, biology and mathematics. Brief introduction into quantum information theory will be given as well. No prior knowledge of information theory or statistics is required.

Lecture notes to the course page: https://www.weizmann.ac.il/complex/falkovich/sites/complex.falkovich/files/uploads/PNI23.pdf

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: Real analysis at the graduate level.

Description: 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.
 

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

Description:  This is a special topics course in Analysis/PDEs. We shall discuss mainly the analytic aspect of the theory of harmonic maps, in particular, the energy minimizing, energy stationary and weakly harmonic maps and the related regularity (partial regularity) theory. We shall also study of the singular sets of such maps, and also discuss the associated gradient and other type flows. Applications and related theory of liquid crystals and Ginzburg-Landau approximations will be also addressed. This is a Full semester course.
 

Main References: F.H.Lin and C.Y.Wang: Analysis of Harmonic Maps and its Gradient Flows. World Scientific Publ. Company.  Other references and research papers (monographs) will be added later on.

Prerequisites: Undergraduate Linear Algebra and ODE. Also,
PDE strongly recommended.
 

There is no assigned textbook for the course, but this book contains a
fair cross-section of topics: MH Holmes, Introduction to Perturbation Methods, Springer, 2nd edition 2013. Free download
from NYU via SpringerLink

Syllabus:
Regular and singular perturbations of algebraic equations, asymptotic expansions,
integral asymptotics. Dimensional analysis, scaling. Method of multiple scales for ODEs, averaging, WKB
solution, Kapitza’s pendulum. Similarity solutions for PDEs. Matched asymptotic expansions, boundary layers,
matching rules.
Green’s function asymptotics, near-field, far-field, and multipole expansions.
Fourier methods for dispersive PDEs, group velocity, stationary phase asymptotics.
Geometric wave theory, eikonal and transport equations, ray tracing for inhomogeneous
media, caustics. Possible additional topics: homogenization theory, Gaussian random functions, stochastic processes.

Asymptotic and exact solution to a dispersive PDE
Prerequisites: elementary linear algebra and differential
equations.

This is a first-year graduate course for all incoming PhD and
Master students interested in pursuing research in Applied
Mathematics.

This course 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, Fourier transform methods, geometric wave theory, and calculus
of variations.

Grading: this course will be graded as a regular course with a grad

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.

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: 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: Linear Algebra, Multi-dimensional calculus (e.g. gradients, Hessians), Probability Theory. Recommended but not required: experience training DNNs, high dimensional probability, convex optimization.

Description:  Deep Neural Networks (DNNs) are often said to be black boxes, whose inner workings are mysterious. This notion has been challenged in recent years by a number of theoretical results describing the behavior of DNNs mathematically. Though the current state of research is still far from a complete 'Theory of Deep Learning', many important ideas and concepts have been identified. This course is targeted towards students who are interested in mathematical techniques to prove and understand DNNs, but also students who are interested in designing and fine-tuning DNNs according to theoretical principles.

This course will give an overview of the many distinct training dynamics of DNNs and their impact on the function learned by the network. The focus will mainly be on mathematical analysis of deep networks with a large number of neurons. More generally, the goal is to give some mathematical intuition for the many different dynamics that can be observed in DNNs and the mechanism behind them.

Subjects covered: Neural Tangent Kernel (NTK), spectral Bias of DNNs, generalization and curse of dimensionality, transition from NTK to active regimes, implicit bias towards sparsity in linear and nonlinear networks, feature learning/symmetry learning, Saddle-to-Saddle dynamics.

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.