Math Finance

  • QST MF 600: Math Refresher
    The Mathematical Finance Program has a very strong quantitative component, one which many incoming students underestimate. Although students admitted to the program have satisfied the prerequisites in Mathematics, the program's prerequisites represent the minimal, not the optimal, background required. Even if you have learned the topics required as prerequisites, reviewing these concepts immediately prior to the start of the program could be enormously helpful and will certainly increase your chance of success in the program. The course will begin with a review of matrix algebra, then proceed to examine the role of calculus in comparative static analysis. Following this, unconstrained and constrained optimization will be covered using multivariate calculus. The second half of the class deals with dynamics, beginning with a review of integration, and continuing with first- and higher-order differential equations.
  • QST MF 601: Launch Week
    Mathematical Finance Launch Week is a combination of orientation activities, academic sessions, and career preparation designed to give students a foundational knowledge of the Mathematical Finance program. The week will include training on various tools students will need for the program, professional development sessions, and social gatherings.
  • QST MF 702: Fundamentals of Finance
    This course covers such topics as: financial markets (bonds, stocks, derivative securities, forward and futures contracts, exchanges, market indexes, and margins); interest rates, present value, yields, term structure of interest rates, duration and immunization of bonds, risk preferences, asset valuation, Arrow-Debreu securities, complete and incomplete markets, pricing by arbitrage, the first and the second fundamental theorems of Finance, option pricing on event trees, risk and return (Sharpe ratios, the risk-premium puzzle), the Capital Asset Pricing Model, and Value-at-Risk.
  • QST MF 703: Programing for Mathematical Finance
    In-depth discussion of object-oriented programming with Python and C++ for finance and data applications. Topics include built-in-types, control structure, classes, constructors, destructors, function overloading, operator functions, friend functions, inheritance, and polymorphism with dynamic binding. Case study looks at the finite differences solutions for the basic models of financial derivatives; as well as the design and development of modular, scalable, and maintainable software for modeling financial derivatives.
  • QST MF 728: Fixed Income Securities
    The course focuses on the valuation, hedging and management of fixed income securities. Fixed income instruments are by far the most important asset class in financial markets. Basic theoretical and empirical term structure concepts are introduced. Short rate models and the Heath-Jarrow-Morton methodology are presented. Market Models and their application for the valuation of forwards, swaps, caps, floors and swaptions, and other interest rate derivatives are discussed in detail.
  • QST MF 730: Portfolio Theory
    A concise introduction to recent results on optimal dynamic consumption-investment problems is provided. Lectures will cover standard mean-variance theory, dynamic asset allocation, asset- liability management, and lifecycle finance. The main focus of this course is to present a financial engineering approach to dynamic asset allocation problems of institutional investors such as pension funds, mutual funds, hedge funds, and sovereign wealth funds. Numerical methods for implementation of asset allocation models will also be presented. The course also focuses on empirical features and practical implementation of dynamic portfolio problems.
  • QST MF 731: Corporate Risk Management
    This course provides an introduction to modern methods of risk management. Lectures cover risk metrics, measurement and estimation of extreme risks, management and control of risk exposures, and monitoring of risk positions. The impact of risk management tools, such as derivative securities, will be examined. Issues pertaining to the efficiency of communication architectures within the firm will be discussed. Regulatory constraints and their impact on risk management will be assessed. The approach to the topic is quantitative. The course is ideal for students with strong quantitative backgrounds who are seeking to understand issues pertaining to risk management and to master modern methods and techniques of risk control.
  • QST MF 770: Advanced Derivatives
    Graduate Prerequisites: GSM MF 795.
    This course provides a comprehensive and in-depth treatment of valuation methods for derivative securities. Extensive use is made of continuous time stochastic processes, stochastic calculus and martingale methods. The main topics to be addressed include (i) European option valuation, (ii) Exotic options, (iii) Multiasset options, (iv) Stochastic interest rate, (v) Stochastic volatility, (vi) American options and (vii) Numerical methods. Additional topics may be covered depending on time constraints.
  • QST MF 772: Credit Risk
    The derivatives market has experienced tremendous growth during the past decade as credit risk has become a major factor fostering rapid financial innovation. This course will provide an in-depth approach to credit risk modelling for the specific purpose of pricing fixed income securities and credit-risk derivatives. The course will explore the nature of factors underlying credit risk and develop models incorporating default risk. Types and structures of credit-derivatives will be presented and discussed. Valuation formulas for popular credit-derivatives will be derived. Numerical methods, for applications involving credit derivative structures and default risks, will be presented.
  • QST MF 793: Statistics for Mathematical Finance
    This course covers the fundamental principles of statistics and econometrics. It is mandatory for all tracks of the MSc. program. The course first reviews the needed concepts in probabilities, properties of random variables, the classic distributions encountered in Finance. Then, we cover the principles of random sampling, properties of estimators, e.g., the standard moment estimators (sample mean, variance, etc..). The next major topic is the regression analysis. We study the OLS and Maximum Likelihood methods, review their properties, in the standard case and when ideal assumptions are not correct. The course ends with a study of time series ARMA models and volatility models such as GARCH and Risk-Metrics. The course makes intensive use of the R package.
  • QST MF 794: Stochastic Methods of Asset Pricing II
    The course covers: Feynman-Kac formula and Fokker-Plank equation, Stochastic calculus with jumps, Levy processes and jump diffusion models in finance, Bellman's Principle of Dynamic Programming and Hamilton-Jacobi-Bellman equation, classical problems for optimal control in finance (Merton's problem, etc.), investment-consumption decisions with transaction costs, the connection between asset pricing and free-boundary problems for PDEs, optimal stopping problems and the exercise of American-style derivatives, capital structure and valuation of real options and corporate debt, exchange options, stochastic volatility models, and Dupire's formula.
  • QST MF 795: Stochastic Methods of Asset Pricing I
    This course develops the basic tools from measure-theoretic probability theory and stochastic calculus that are needed for an in-depth study of continuous time finance. Some related tools from asset pricing (e.g., risk-preferences and state-price densities) are introduced as well, and the basic ingredients of continuous time financial modeling are developed. The following topics are covered: probability and measure, the coin-toss space and the random walk, random variables and convergence, Gaussian distribution, martingales, Brownian motion, stochastic integration for semimartingales and Ito formula, Girsanov's theorem, stochastic differential equations, continuous time market models and pricing by arbitrage, resume of Malliavin calculus, replication and pricing of contingent claims, market completeness and the fundamental theorems of asset pricing.
  • QST MF 796: Computational Methods of Mathematical Finance
    This course introduces common algorithmic and numerical schemes that are used in practice for pricing and hedging financial derivative products. Among others, the course covers Monte-Carlo simulation methods (generation of random variables, exact simulation, discretization schemes), finite difference schemes to solve partial differential equations, numerical integration, and Fourier transforms. Special attention is given to the computational requirements of these different methods, and the trade-off between computational effort and accuracy.
  • QST MF 805: Ptflo Construct
  • QST MF 821: Algorithmic and High-Frequency Trading
    With the availability of high frequency financial data, new areas of research in stochastic modeling and stochastic control have opened up. This course will introduce some of the concepts, questions and methods that arise in this domain. We will begin with a description of the basic elements of electronic markets, some of the features of the data, its empirical implications and simple microeconomic models. Next, we will investigate algorithmic trading problems from the stochastic optimal control perspective, including the optimal execution problem and show how to modify the classical approaches to include order-flow information and the effect that dark pools have on trading. Electronic markets would not exist without market markets, so we next investigate how market markers with various trading goals and constraints post orders optimally. Trading pairs of assets that mean-revert is another important algorithmic strategy, and we will see how stochastic control methods can be utilized to inform agents how to optimally trade.
  • QST MF 840: Data Analysis and Empirical Methods
    This is the second course of the econometrics sequence in the Mathematical Finance program. The course quickly reviews OLS, GLS, the Maximum Likelihood principle (MLE). Then, the core of the course concentrates on Bayesian Inference, now an unavoidable mainstay of Financial Econometrics. After learning the principles of Bayesian Inference, we study their implementation for key models in finance, especially related to portfolio design and volatility forecasting. Over the last twenty years, radical developments in simulation methods, such as Markov Chain Monte Carlo (MCMC) have extended the capabilities of Bayesian methods. Therefore, after studying direct Monte Carlo simulation methods, the course covers non-trivial methods of simulation such as Markov Chain Monte Carlo (MCMC), applying them to implement models such as stochastic volatility.
  • QST MF 850: Advanced Computational Methods
    Graduate Prerequisites: GSM MF 796.
    This course explores algorithmic and numerical schemes used in practice for the pricing and hedging of financial derivative products in stochastic models of multiple dimensions with jumps. The focus of this course lies on Monte-Carlo simulation methods, and variance reduction techniques to control for the accuracy of such estimators. Other topics covered include multivariate numerical integration, optimization routines, and root-finding methods.
  • QST MF 921: Topics in Dynamic Asset Pricing
    This course provides a comprehensive and in-depth treatment of modern asset pricing theories. Extensive use is made of continuous time stochastic processes, stochastic calculus and optimal control. Particular emphasis will be placed on (i) consumption-portfolio choice problems and (ii) equilibrium asset pricing models. Advances involving non-separable preferences, incomplete information and heterogeneous agents will be discussed.
  • QST MF 930: Advanced Corporate Finance
    This doctoral level class on corporate finance covers both theoretical and empirical work. Rather than explaining the underpinnings of basic corporate research (e.g., model/applications dealing with asymmetric information, agency problems, and capital market frictions), we go deeper in understanding "how to operationalize" research on concrete topics that are central to contemporary corporate finance, such as bankruptcy, capital structure, mergers and acquisitions, the firm boundaries, investment, and much more. The class also looks at the interface between corporate finance and other research areas, such as asset pricing and banking. The course is a blend of new approaches to modeling in corporate research (e.g., dynamic, structural models of financial policy that generate typically quantitative predictions) and new approaches to testing design (e.g., regression discontinuities and natural experiments). The goal is to expose the students to the "state-of-the-art" of research in corporate finance and prepare them to do research in corporate finance using new methods and tools.
  • QST MF 998: Directed Study: Mathematical Finance
    Graduate Prerequisites: Consent of instructor and the program director
    PhD-level directed study in Mathematical Finance. 1, 2, or 3 cr. Application available on the Graduate Program Office website.