# Mathematical Finance

• QST MF 600: Mathematics and Statistics Review
Mathematical Finance as a discipline borrows concepts from probability theory, statistics, linear algebra, calculus and optimization, ordinary and partial differential equations, computer science and financial economics. This course reviews the essential prerequisites in mathematics, probability and statistics to prepare students for the MS in Mathematical Finance program. The course begins with a review of probability and the fundamentals of stochastic processes. Following this, statistical inference and linear algebra are reviewed. A review of comparative statics, optimization theory and dynamics, beginning with a review of integration, and continuing with first- and higher- order differential equations concludes the course.
• QST MF 601: Launch
Mathematical Finance Launch is an orientation program for students entering the MS in Mathematical Finance program.
• QST MF 610: Mathematical Finance Career Management
This course prepares students in the MS Mathematical Finance program for the global employment market in quantitative finance. The course has the following objectives: to familiarize students with the foundational mathematics and statistics required for the MSMF program, to develop sound networking and job search strategies, to prepare students for 'quant' interviews, to develop good career management habits, and to familiarize students with important developments in financial markets and issues of the day that affect the global financial services industry.
• QST MF 702: Fundamentals of Finance
This course covers such topics as: financial markets (bonds, stocks, derivative securities, forward and futures contracts, exchanges, market indices, 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, risk premia), and the Capital Asset Pricing Model. (Mathematical Finance courses are reserved for students enrolled in the Mathematical Finance program.)
• 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. (Mathematical Finance courses are reserved for students enrolled in the Mathematical Finance program.)
• QST MF 728: Fixed Income Securities
The course focuses on the valuation, hedging and management of fixed income securities. 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. (Mathematical Finance courses are reserved for students enrolled in the Mathematical Finance program.)
• 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 covers empirical features and practical implementation of dynamic portfolio problems. (Mathematical Finance courses are reserved for students enrolled in the Mathematical Finance program.)
• QST MF 731: Corporate Risk Management
This course is an introduction to modern methods of risk management. The first half of the course focuses on market risk. Here, lectures cover risk measures (such as Value at Risk and Expected Shortfall), with a focus on computation of such measures in a dynamic, multi-asset environment using real-world data. In particular, students will learn to compute, back-test, and account for risk measures when both monitoring and constructing portfolios. Additionally, lectures cover scenario analysis, stress testing, and the measurement of severe tail risk via extreme value theory. In the second half of the course, lectures cover alternate types of risk. These include operational, liquidity, model, and counter-party credit risk. In particular, students will derive formulas for the valuation adjustments due to counter-party default. 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. (Mathematical Finance courses are reserved for students enrolled in the Mathematical Finance program.)
• QST MF 740: Economics of FinTech
The course covers the following topics: introduction to Blockchains and cryptocurrencies; contract theory for initial coin offerings; robo-advising; crowd wisdom; and privacy issues. Although the course introduces some Blockchain programming languages, e.g. Solidity, the emphasis of the course is on the economics of FinTech rather than on programming. Students are expected to be familiar with basic financial economics, econometrics, and stochastic processes.
• QST MF 770: Advanced Derivatives
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. (Mathematical Finance courses are reserved for students enrolled in the Mathematical Finance program.)
• 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. (Mathematical Finance courses are reserved for students enrolled in the Mathematical Finance program.)
• 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 GLS principles, 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. (Mathematical Finance courses are reserved for students enrolled in the Mathematical Finance program.)
• QST MF 794: Stochastic Methods in Asset Pricing II
The course covers: the Feynman-Kac formula and the Fokker-Plank equation, stochastic calculus with jumps, Levy processes and jump diffusion models in finance, Bellman's principle of dynamic programming and the 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. (Mathematical Finance courses are reserved for students enrolled in the Mathematical Finance program.)
• QST MF 795: Stochastic Methods in 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 semi-martingales 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. (Mathematical Finance courses are reserved for students enrolled in the Mathematical Finance program.)
• 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. (Mathematical Finance courses are reserved for students enrolled in the Mathematical Finance program.)
• QST MF 810: Advanced Programming: Data Structures and Algorithms
The course introduces students to a number of efficient algorithms and data structures for computational problems across a variety of areas within FinTech. In the first half of the course, a special programming language for blockchains, such as Solidity, is taught, and TensorFlow, a special Python library for deep learning models, is used to solve stochastic control problems in finance. In the second half of the course, advanced techniques for improving computational performance, including the use of parallel computation and GPU acceleration are surveyed; frameworks for big data analysis such as Apache Hadoop and Apache Spark are studied. Students will have the opportunity to employ these techniques and gain hands-on experience developing advanced applications. (This course is reserved for students enrolled in the Graduate Certificate in Financial Technology.)
• QST MF 815: Advanced Machine Learning Applications for Finance
This course surveys applications of machine learning techniques to various types of financial datasets. This course starts with financial data structure and features, then introduces deep learning and advanced supervised learning techniques. We will examine several machine learning applications in pricing, hedging, and portfolio management. Advanced methods for clustering and classification such as support vector machine and unsupervised learning will be introduced. Reinforcement learning and its connection with optimal control will be discussed. Text data will be introduced and analyzed using text mining techniques. Machine learning techniques will be applied to asset allocation. Strategy back-testing and strategy risk will also be discussed. (This course is reserved for students enrolled in the Graduate Certificate in Financial Technology.)
• QST MF 821: Algorithmic and High-Frequency Trading
This course will introduce concepts of electronic markets, and statistical and optimal control techniques to model and trade in these markets. 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 study statistical tools to estimate and predict price and volatility of the high-frequency price. Then 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. 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. (Mathematical Finance courses are reserved for students enrolled in the Mathematical Finance program.)
• QST MF 825: Advanced Topics in Investments
This course is designed for students seeking to work as quants in a quantitative finance investments group. It covers utility theory, portfolio optimization, asset pricing, and some aspects of factor models, incorporating the impact of parameter uncertainty. The course does not cover risk management or fixed income instruments, nor does it describe how the financial services industry works. Rather, it teaches how a quant should optimize a portfolio. The course makes extensive use of R (Excel or VBA are not substitutes), optimization theory, statistics, regression theory (OLS, GLS, testing theory), and matrix algebra. Students should be very comfortable with these concepts before taking the course; further, students should already have taken a finance course covering expected returns models (CAPM), options and futures. The course emphasizes the ability to prove theoretical results and their validity, an essential trait for investments quants. Students who completed QST FE825 may not take this course for credit. (Mathematical Finance courses are reserved for students enrolled in the Mathematical Finance program.)
• QST MF 840: Data Analysis and Financial Econometrics