Course Descriptions

MATHGA.1420001 Introduction To Math Analysis II
3 Points, Tuesdays, Thursdays, 4:556:10PM, Aaditya Rangan
Rigorous treatment of limits and continuity. Riemann integral. Taylor series. Absolute and uniform convergence. Elements of ordinary and partial differential equations. Functions of several variables and their derivatives. The implicit function theorem, optimization, and Lagrange multipliers. Theorems of Gauss, Stokes, and Green. Fourier series and integrals

MATHGA.2020001 Numerical Methods II
3 Points, Mondays, Wednesdays, 11:0012:15PM, Michael O'Neil
Numerical linear algebra. Approximation theory. Quadrature rules and numerical integration. Nonlinear equations and optimization. Ordinary differential equations. Elliptic equations. Iterative methods for large, sparse systems. Parabolic and hyperbolic equations.

MATHGA.2048001 Scientific Computing In Finance
3 Points, Mondays, 7:109:00PM, Richard Lindsey and Mehdi Sonthonnax
This is a version of the course Scientific Computing (MATHGA 2043) designed for applications in quantitative finance. It covers software and algorithmic tools necessary to practical numerical calculation for modern quantitative finance. Specific material includes IEEE arithmetic, sources of error in scientific computing, numerical linear algebra (emphasizing PCA/SVD and conditioning), interpolation and curve building with application to bootstrapping, optimization methods, Monte Carlo methods, and the solution of differential equations.

MATHGA.2120001 Linear Algebra II
3 Points, Mondays, Wednesdays, 2:003:15PM, Fengbo Hang
Linear spaces and mappings. Matrices and linear equations. Eigenvalues and eigenvectors. Jordan form. Special classes of matrices, spectral theory.

MATHGA.2140001 Algebra II
3 Points, Tuesdays, Thursdays, 11:0012:15PM, Yuri Tschinkel
Basic concepts including groups, rings, modules, polynomial rings, field theory, and Galois theory.

MATHGA.2210001 Introduction To Number Theory I
3 Points, Mondays, Wednesdays, 11:0012:15PM, Charles Stine
Introduction to the elementary methods of number theory. Topics: arithmetic functions, congruences, the prime number theorem, primes in arithmetic progression, quadratic reciprocity, the arithmetic of quadratic fields.

MATHGA.2320001 Topology II
3 Points, Mondays, Wednesdays, 6:207:35PM, Sylvain Cappell
Survey of pointset topology. Fundamental groups, homotopy, covering spaces. Singular homology, calculation of homSurvey of pointset topology. Fundamental groups, homotopy, covering spaces. Singular homology, calculation of homology groups, applications. Homology and cohomology of manifolds. Poincar? duality. Vector bundles. De Rham cohomology and differential forms.

MATHGA.2360001 Differential Geometry II
3 Points, Tuesdays, Thursdays, 11:0012:15PM, Robert Ji Wai Young
Theory of curves and surfaces. Riemannian geometry: manifolds, differential forms, and integration. Covariant derivatives and curvature. Differential geometry in the large. Curvature, geodesics, Jacobi fields, comparison theorems, and GaussBonnet theorem.

MATHGA.2440001 Real Variables II
3 Points, Tuesdays, Thursdays, 9:3010:45AM, Hong Wang
Basics of Functional Analysis. Rearrangement Inequalities. Basics of Fourier Analysis. Distributions. Sobolev Spaces. BV Functions. Interpolation. Maximal Function.

MATHGA.2460001 Complex Variables II
3 Points, Mondays, Wednesdays, 9:3010:45AM, Percy Deift
Analytic functions. Cauchy?s theorem and its many consequences. Fractional linear transformations and conformal mappings. Introduction to Riemann surfaces. The Riemann mapping theorems. Entire functions. Special functions.

MATHGA.2470001 Ordinary Differential Equations
3 Points, Tuesdays, Thursdays, 12:301:45PM, Edwin Gerber
Existence, uniqueness, and continuous dependence. Linear ODE. Stability of equilibria. Floquet theory. Poincar?Bendixson theorem. Additional topics may include bifurcation theory, Hamiltonian mechanics, and singular ODE in the complex plane.

MATHGA.2500001 Partial Differential Equations
3 Points, Tuesdays, Thursdays, 2:003:15PM, Guido DePhilippis
Local existence theory: CauchyKowalewsky theorem. Laplaces equation, harmonic functions, maximum principle, single and double layer potential. Fourier transform and distributions. Sobolev spaces. Elliptic boundary value problems. The Cauchy problem for the heat equation, wave equation. Local well posedness for semilinear Cauchy problems.

MATHGA.2660001 Advanced Topics In Analysis: TBA
3 Points, Mondays, Wednesdays, 12:301:45PM, LaiSang Young
Description TBA 
MATHGA.2751001 Risk & Portfolio Management
3 Points, Wednesdays, 7:109:00PM, Gordon Ritter
A mathematically sophisticated introduction to the analysis of investments. Core topics include expected utility, risk and return, meanvariance analysis, equilibrium asset pricing models, and arbitrage pricing theory.

MATHGA.2752001 Active Portfolio Management
3 Points, Wednesdays, 7:109:00PM, Jerome Benveniste
Theoretical aspects of portfolio construction and optimization, focusing on advanced techniques in portfolio construction, addressing the extensions to traditional meanvariance optimization including robust optimization, dynamical programming and Bayesian choice. Econometric issues associated with portfolio optimization, including estimation of returns, covariance structure, predictability, and the necessary econometric techniques to succeed in portfolio management will be covered.

MATHGA.2753001 Advanced Risk Management
3 Points, Wednesdays, 5:107:00PM, Ken Abbott and Irena Khrebtova
Measuring and managing the risk of trading and investment positions: interest rate positions, vanilla options positions, and exotic options positions. The portfolio risk management technique of ValueatRisk, stress testing, and credit risk modeling.

MATHGA.2791001 Financial Securities And Markets
3 Points, Mondays, 7:109:00PM, Mehdi Sonthonnax
A first course in derivatives valuation. Arbitrage, risk neutral pricing, binomial trees. BlackScholes theory, early exercise, barriers, interest rate models, floors, caps, swaptions. Introduction to creditbased instruments.

MATHGA.2793001 Dynamic Asset Pricing (2nd Half Of Semester)
3 Points, Tuesdays, 5:107:00PM, Bruno Dupire and Montacer Essid
This is an advanced course on asset pricing and trading of derivative securities. Using tools and techniques from stochastic calculus, we cover (1) BlackScholesMerton 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 termstructure models such as Vasicek and HullWhite. It is important that students taking this course have good working knowledge of calculusbased 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.

MATHGA.2798001 Interest Rate & Fx Models
3 Points, Thursdays, 5:107:00PM, Fabio Mercurio and Alexey Kuptsov
The course is divided into two parts. The first addresses the fixedincome models most frequently used in the finance industry, and their applications to the pricing and hedging of interestbased derivatives. The second part covers the foreign exchange derivatives markets, with a focus on vanilla options and firstgeneration (flow) exotics. Throughout both parts, the emphasis is on practical aspects of modeling, and the significance of the models for the valuation and risk management of widelyused derivative instruments

MATHGA.2799001 Modeling And Risk Management Of Bonds And Securitized Products (2nd Half Of Semester)
3 Points, Mondays, 5:107:00PM, Rodney SunadaWong
This halfsemester course is designed for students interested in Fixed Income roles in frontoffice trading, market risk management, model development (“Quants”, “Strats”), or model validation. We begin by modeling the cash flows of a generic bond, emphasizing how the bond reacts to changes in markets, how traders may position themselves given their views on the markets, and how risk managers think about the risks of a bond. We then focus on Mortgages, covering the fundamentals of Residential Mortgages, and MortgageBacked Securities. Students will build pricing models for mortgages, passthroughs, sequentials and CMO’s that generate cash flows and that take into account interest rates, prepayments and credit spreads (OAS). The goals are for students to develop: (1) an understanding of how to build these models and how assumptions create “model risk”, and (2) a trader’s and risk manager’s intuition for how these instruments behave as markets change, and (3) a knowledge how to hedge these products. We will graph cash flows and changes in market values to enhance our intuition (e.g. in Excel, Python or by using another graphing tool). In the course we also review the structures of CLO’s, Commercial Mortgage Backed Securities (CMBS), Auto Asset Backed Securities (ABS), Credit Card ABS, subprime mortgages and CDO’s and credit derivatives such as CDX, CMBX and ABX. We discuss the modeling risks of these products and the drivers of the Financial Crisis of 2008. As time permits, we touch briefly on Peertopeer / MarketPlace Lending.

MATHGA.2800001 Trading Energy Derivatives (1st Half Of Semester)
3 Points, Tuesdays, 5:107:00PM, Ilia Bouchouev
The course provides a comprehensive overview of most commonly traded quantitative strategies in energy markets. The class bridges quantitative finance and energy economics covering theories of storage, net hedging pressure, optimal risk transfer, and derivatives pricing models. Throughout the course, the emphasis is placed on understanding the behavior of various market participants and trading strategies designed to monetize inefficiencies resulting from their activities and hedging needs. We discuss in detail recent structural changes related to financialization of energy commodities, crossmarket spillovers, and linkages to other financial asset classes. Trading strategies include traditional risk premia, volatility, correlation, and higherorder options Greeks. Examples and case studies are based on actual market episodes using real market data.

MATHGA.2801001 Advanced Topics In Equity Derivatives (2nd Half Of Semester)
3 Points, Wednesdays, 7:109:00PM, Alireza Javaheri
Description TBA 
MATHGA.2802001 Market Microstructure (1st Half Of Semester)
3 Points, Wednesdays, 7:109:00PM, Merrell Hora
This is a halfsemester course covering topics of interest to both buyside traders and sellside execution quants. The course will provide a detailed look at how the trading process actually occurs and how to optimally interact with a continuous limitorder book market. We begin with a review of early models, which assume competitive suppliers of liquidity whose revenues, corresponding to the spread, reflect the costs they incur. We discuss the structure of modern electronic limit order book markets and exchanges, including queue priority mechanisms, order types and hidden liquidity. We examine technological solutions that facilitate trading such as matching engines, ECNs, dark pools, multiple venue problems and smart order routers. The second part of the course is dedicated pretrade market impact estimation, posttrade slippage analysis, optimal execution strategies and dynamic noarbitrage models. We cover AlmgrenChriss model for optimal execution, Gatheral’s nodynamicarbitrage principle and the fundamental relationship between the average response of the market price to traded quantity, and properties of the decay of market impact. Homework assignments will supplement the topics discussed in lecture. Some coding in Java will be required and students will learn to write their own simple limitorderbook simulator and analyze real NYSE TAQ data.

MATHGA.2840001 Advanced Topics In Applied Math: TBA
3 Points, Tuesdays, Thursdays, 11:0012:45PM, Eric Vanden Eijnden
Description TBA 
MATHGA.2901001 Essentials Of Probability
3 Points, Mondays, Wednesdays, 3:304:45PM, Yuri Bakhtin
Probability as a tool in computer science, finance, statistics, and the natural and social sciences. Independence. Random variables and their distributions. Conditional probability. Laws of large numbers. Central limit theorem. Random walk, Markov chains, and Brownian motion. Selected applications.

MATHGA.2903001 Stochastic Calculus (2nd Half Of Semester)
3 Points, Mondays, 5:107:00PM, David Li
Review of basic probability and useful tools. Bernoulli trials and random walk. Law of large numbers and central limit theorem. Conditional expectation and martingales. Brownian motion and its simplest properties. Diffusion in general: forward and backward Kolmogorov equations, stochastic differential equations and the Ito calculus. FeynmanKac and CameronMartin Formulas

MATHGA.2912001 Probability Theory II
3 Points, Mondays, Wednesdays, 12:301:45PM, Paul Bourgade
The classical limit theorems: laws of large numbers, central limit theorem, iterated logarithm, arcsine law. Further topics: large deviation theory, martingales, Birkhoff?s ergodic theorem, Markov chains, Shannon?s theory of information, infinitely divisible and stable laws, Poisson processes, and Brownian motion. Applications.

MATHGA.2962001 Mathematical Statistics
3 Points, Wednesdays, 6:008:30PM, Yisong Yang
Principles and methods of statistical inference. Topics: large sample theory, minimum variance unbiased estimates, method of maximum likelihood, sufficient statistics, NeymanPearson theory of hypothesis testing, confidence intervals, regression, nonparametric methods.

MATHGA.3003001 Ocean Dynamics
3 Points, Tuesdays, Thursdays, 3:304:45PM, Shafer Smith
The goal of this course is to introduce students to modern dynamical oceanography, with a focus on mathematical models for observed phenomena. The lectures cover the observed structure of the ocean, the thermodynamics of seawater, the equations of motion for rotatingstratified flow, and the most useful approximations thereof: the primitive, planetary geostrophic, and quasigeostrophic equations. The lectures demonstrate how these approximations can be used to understand boundary layers, winddriven circulation, buoyancydriven circulation, oceanic waves (Rossby, Kelvin, and inertiagravity), potential vorticity dynamics, theories for the observed upperocean stratification (the thermocline), and for the abyssal circulation. Students should have some knowledge in geophysical fluid dynamics before taking this course. Throughout the lectures, the interplay between observational, theoretical, and modeling approaches to problems in oceanography are highlighted.