# Masters Program in Financial Mathematics

The Department of Mathematics offers a separate Master of Science in Financial Mathematics degree. The Financial Mathematics Program is designed to produce graduates with a good understanding of the theoretical background of pricing models for financial derivatives, but more importantly a real understanding of the underlying assumptions and an ability to critically ascertain the applicability and limitations of the various models. A significant part of the program will be taught by professionals from the financial industry and will be devoted to examining how models behave in practice under a variety of market conditions, to examine how realistic the underlying assumptions are and to understand what happens when these assumption are violated. Students will learn to use the models to set up hedges and to evaluate the effectiveness of these hedges by simulating various market conditions.

The program consists of four components: Mathematics, Probability Theory and Economics, and Financial Applications and Simulations.

The Mathematics component runs over three quarters, Probability Theory runs over two quarters and Economics over one quarter. The Financial Applications and Simulations is a three quarter component. Courses in each component meet for three hours per week for a total of nine hours of instruction per week. The Mathematics and Probability Theory will be taught by faculty members from the Departments of Mathematics and Statistics , respectively. The Economics course will be taught by a faculty member from the Department of Economics . The Financial Applications courses will be taught by professionals from financial institutions and will also include a computer lab.

The contents and curriculum for the program has been worked out jointly by faculty members at the University and by practitioners in the field to insure the relevance of the material. The teaching of the program relies heavily on the use of computer simulations to illustrate the material. This both makes it possible to cover more material and teaches students to implement the theory at every stage.

Various software packages are licensed to the program and will be provided free of charge for the course work. Course material and assignments will be available and submitted on line.

The program has a three quarter-course requirement for obtaining the Master of Science degree. The program is structured to allow part-time enrollment to complete the program over two or three years. The courses will be taught evenings at the main campus of the University located in Hyde Park.

The requirements for acceptance to the program are a solid undergraduate background in mathematics, ideally a major in mathematics or science/engineering, with some background also in probability theory. Some experience in C/C++ programming will also be useful. Persons with practical experience in the financial industry but with less of a mathematical background will be considered but may be required to acquire additional skills in mathematics.

The following are the required courses to graduate:

FINM 33000 | Mathematical Foundations of Option Pricing | 100 |

FINM 33400 | Statistical Risk Management | 100 |

FINM 36700 | Portfolio Theory and Risk Management 1 | 050 |

FINM 33603 | Fixed Income Derivatives 1 | 050 |

FINM 34500 | Stochastic Calculus | 100 |

FINM 33150 | Regression Analysis & Quantitative Trading Strategies | 100 |

FINM 33604 | Fixed Income Derivatives 2 | 050 |

FINM 37300 | Foreign Exchange/Fixed Income Derivatives | 050 |

FINM 32000 | Numerical Methods | 100 |

FINM 35000 | Topics in Economics | 100 |

FINM 37400 | Advanced Option Pricing | 050 |

FINM 36702 | Portfolio Theory and Risk Management 2 | 050 |

These courses may change and/or be revised each year. In addition, those who do not pass the computer programming placement exam are required to take and pass the Computing for Finance sequence.

### Mathematics - Financial Mathematics Courses

**FINM 32000. Numerical Methods. 100 Units.**

Implementing the theory introduced in FINM 33000, this course takes a numerical/computational approach to the pricing and hedging of financial derivatives. Topics include: Trees as diffusion approximations; Finite difference methods for PDE solution; Monte Carlo methods for simulation; Fourier transform methods for pricing.

Instructor(s): Roger Leee Terms Offered: Spring

**FINM 32200. Computing for Finance 1. 050 Units.**

As the first course in a three-part series, no previous programming knowledge is assumed. In Computing for Finance I, we will introduce the syntax and semantics of C++ and basics of OO programming. As part of the course work, students will develop an OO option pricer using the Monte Carlo technique. Classes are taught using a combination of lectures and in class hands-on lab sessions.

Instructor(s): Chanaka Liyanaarachchi Terms Offered: Fall

**FINM 32300. Computing for Finance 2. 050 Units.**

We will discuss new programming techniques, including more OO features and Templates in C++. We will also examine the use of the Standard Library in C++. Students will extend the option pricer to use Tree methods. Classes are taught using a combination of lectures and in class hands-on lab sessions.

Instructor(s): Chanaka Liyanaarachchi Terms Offered: Winter

**FINM 32400. Computing for Finance 3. 050 Units.**

We will discuss topics relevant to implementing a basic electronic trading system using programming techniques covered in Part 1 and Part 2 of this course series. Topics discussed include the implementation of a trading algorithm, handling the connectivity to an exchange/brokerage house and issues related to performance. Different design choices and tradeoffs between those different choices; concurrent and parallel programming will be discussed within the context of this project. Classes are taught using a combination of lectures and in class hands-on lab sessions.

Instructor(s): Chanaka Liyanaarachchi Terms Offered: Spring

**FINM 33000. Mathematical Foundations of Option Pricing. 100 Units.**

Introduction to the theory of arbitrage-free pricing and hedging of financial derivatives. Topics include: Arbitrage; Fundamental theorems of asset pricing; Binomial and other discrete models; Black-Scholes and other continuous-time Gaussian models in one-dimensional and multidimensional settings; PDE and martingale methods; Change of numeraire.

Instructor(s): Roger Lee Terms Offered: Autumn

**FINM 33150. Regression Analysis & Quantitative Trading Strategies. 100 Units.**

The course covers Linear and Non-linear Regression methods for estimating parameters of models. We will cover topics like Method of Moments, Generalized Linear Regression, Gauss-Newton Regression, Instruments, Generalized method of Moments. These methods will be used to develop factor models for securities returns.

Instructor(s): Brian Boonstra Terms Offered: Winter

**FINM 33170. Statistics of High-Frequency Financial Data. 100 Units.**

This course is an introduction to the econometric analysis of high-frequency financial data. This is where the stochastic models of quantitative finance meet the reality of how the process really evolves. The course is focused on the statistical theory of how to connect the two, but there will also be some data analysis. With some additional statistical background (which can be acquired after the course), the participants will be able to read articles in the area. The statistical theory is longitudinal, and it thus complements cross-sectional calibration methods (implied volatility, etc.). The course also discusses volatility clustering and market microstructure.

Terms Offered: Spring

Prerequisite(s): STAT 39000/FINM 34500, also some statistics/econometrics background as in STAT 24400–24500, or FINM 33150 and FINM 33400, or equivalent, or consent of instructor.

Equivalent Course(s): STAT 33970

**FINM 33400. Statistical Risk Management. 100 Units.**

The course starts at a rather introductory level, but the progress is swift. It covers a brief survey of basic probability theory, and provides an introduction to some useful statistical distributions, both univariate and multivariate. A discussion of copulas and various correlation measures. Risk measures and ideas behind a reasonable risk measure. A few elements from Monte Carlo simulation. Statistical estimation, the maximum likelihood method and nonparametric methods. Asymptotic properties of estimators. Goodness of fit tests and model selection. Extreme value theory.

Instructor(s): Jostein Paulsen Terms Offered: Autumn

**FINM 33602. Advanced Fixed Income Derivatives. 100 Units.**

The course will focus on additional chapters of fixed income derivatives that could not be included in the basic Fixed Income Derivatives, Part I and II courses. The topics include term curve bootstrapping and smoothing; in-depth derivation of the HJM framework; Black's model and forward measure; the statistical model and HJM; market models calibration; volatility skew adjustments for interest rate models; CVA counterparty risk; risk management with the statistical model; numerical methods for Hull-White model: trinomial trees, Monte Carlo and finite difference methods.

Instructor(s): Yuri Balasanov Terms Offered: Winter

Prerequisite(s): students will be required to have a solid understanding of the material covered in Fixed Income Derivatives, Part I (33603) and Mathematical Foundations of Option Pricing (33000). Students who wish to take the course must also complete and pass a placement exam.

**FINM 33603. Fixed Income Derivatives 1. 050 Units.**

This is part one of a two-part course on Fixed Income Derivatives. The topics will include an introduction to fixed income markets, a detailed review of fixed income derivative instruments, and a general approach to bootstrapping the LIBOR term curve from available market quotes. We also discuss the application of the Black-Scholes-Merton model to pricing European swaptions and caps/floors. Students will study a statistical approach to building a foundation for the Heath-Jarrow-Morton framework of interest rate models, covered in the second part of the course.

This is a 5-week course at the second-half of the quarter.

Instructor(s): Yuri Balasanov, Lida Doloc, Jeffrey Greco Terms Offered: Autumn

**FINM 33604. Fixed Income Derivatives 2. 050 Units.**

This is part two of a two-part course on Fixed Income Derivatives. The topics covered will include a derivation of the Heath-Jarrow-Morton family of models using methods of arbitrage pricing theory and an in-depth case study of the Hull-White interest rate model (an HJM model). Additionally, students will learn about the role of forward measure in pricing fixed income derivatives and LIBOR market models.

This is a 5-week course at the first-half of the quarter.

Instructor(s): Yuri Balasanov, Lida Doloc, Jeffrey Greco Terms Offered: Winter

Prerequisite(s): Fixed Income Derivatives, Part I. Students should be prepared for the extensive use of Stochastic Calculus.

**FINM 34500. Stochastic Calculus. 100 Units.**

The course starts with a quick introduction to martingales in discrete time, and then Brownian motion and the Ito integral are defined carefully. The main tools of stochastic calculus (Ito's formula, Feynman-Kac formula, Girsanov theorem, etc.) are developed. The treatment includes discussions of simulation and the relationship with partial differential equations. Some applications are given to option pricing, but much more on this is done in other courses. The course ends with an introduction to jump process (Levy processes) and the corresponding integration theory.

Instructor(s): Greg Lawler Terms Offered: Winter

Equivalent Course(s): STAT 39000

**FINM 35000. Topics in Economics. 100 Units.**

This course explores the economics of asset pricing. Going beyond no-arbitrage valuation, students learn how asset prices can be linked to economic fundamentals. As the recent recession and financial crisis show, there are important links between financial markets and the real economy. This course gives students a systematic way for understanding these links. Several important areas and puzzles of financial economics are presented. Topics in equity pricing include return-predictability, excess volatility, and factor-models. In fixed income, the course covers the empirical evidence of the term structure and how it compares to the Expectations Hypothesis, as well as how these facts fit with classes of common term-structures models. In international finance, the course covers the carry trade, the home-equity bias, and the currency trilemma.

Instructor(s): Mark Hendricks Terms Offered: Spring

**FINM 36700. Portfolio Theory and Risk Management 1. 050 Units.**

The course introduces investment analysis, allocation, risk control. The course begins with classic topics such as mean-variance analysis, priced and un-priced risk, hedging, and the efficient frontier of investment opportunities. Factor models are used to understand the relation between risk and expected return. Examples covered in the course include the CAPM, Black-Litterman, and principal component factors. Finally, the course discusses modern risk control, including risks from interest-rates, liquidity, and credit. Value-at-risk, and expected shortfall are discussed.

This is a 5-week course at the first-half of the quarter.

Instructor(s): Mark Hendricks Terms Offered: Autumn

**FINM 36702. Portfolio Theory and Risk Management 2. 050 Units.**

This course combines a technical topic with an analysis of situations that produce outsized losses. Students gain familiarity with the credit portfolio loss models that are used to limit trading, allocate costs, and determine required bank capital. They also review the interplay between the technical and human factors that has led to prominent risk control failures. Unique in the Financial Math program, students make in-class presentations that detail the optimal responses of various market participants to unexpected circumstances.

This is a 5-week course at the second-half of the quarter.

Instructor(s): Jon Frye Terms Offered: Spring

**FINM 37300. Foreign Exchange/Fixed Income Derivatives. 050 Units.**

This course will examine international currency markets, financial products, applications of quantitative models and FX risk management with an emphasis on the derivative products and quantitative methods in common use today. Topics will include a) the behavior of FX rates: exchange rate regimes, international monetary systems, FX modeling and forecasting, b) FX markets and products: spot, forward, futures, deposits, cross-currency swaps, non-deliverable contracts, FX options, exotic options, hybrid products and structured notes, and c) Risk management: from the trading book, trading institution, global asset manager and multinational corporation perspectives.

This is a 5-week course at the second-half of the quarter.

Instructor(s): Tony Capozzoli Terms Offered: Winter

**FINM 37400. Advanced Option Pricing. 050 Units.**

This course covers several areas oriented towards pricing and application of various non-standard derivative securities, structured notes and credit derivatives. In addition, fixed income applications such as option adjusted analysis and hedging applications are covered. The course includes live Reuters Eikon and Bloomberg screens.

This is a 5-week course at the first-half of the quarter.

Instructor(s): Jack Mosevich, Izzy Nelken Terms Offered: Spring