Course Descriptions

Below are representative sample course descriptions. The actual content of courses varies from year to year.

Introduction to Financial Mathematics, MAP 5601 (3, Fall)
Graduate Bulletin Description:

Introduction to probability, measure and integration, binomial pricing model and arbitrage pricing theory, introduction to stochastic calculus and the Black-Scholes equation.

Course Rationale:

Students must understand mathematical concepts early in the program that may not be available in the typical undergraduate curriculum and some of which are spread over many graduate mathematics courses. The mathematical topics are prerequisite for the study of concrete financial problems; finance terminology will be at times employed to assure early familiarity.

Intended Audience:

First year students in Financial Mathematics. With permission may be elected by students from a variety of disciplines interested in financial mathematics. This course or the equivalent is required in master’s degree option (e).

Prerequisites:

Calculus through multivariate, ordinary differential equations, probability, linear algebra.

Co- or pre-requisite:

FIN 5515

References: Grading:

Two One-Hour Exams and a Comprehensive Final.

Course Outline
I. Introduction
a. Elementary definitions from probability
b. Discrete processes and martingales
c. Independence
d. The binomial model of asset pricing and derivative evaluation
II. Introduction to Measure Theory
a. Measures, s-fields and integration
b. Stopping times and American options in the binomial model
c. Conditional expectation and the Radon-Nikodym theorem
d. Introduction to Brownian motion
e. The log-normal random process and an introduction to stochastic calculus
f. Derivation of the Black-Scholes equation
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Introduction to Computational Finance, MAP 5611 (3, Spring)
Graduate Bulletin Description:

Computational methods for solving mathematical problems in finance. Basic numerical methods. Numerical solution of parabolic partial differential equations, including convergence and stability. Solution of the Black-Scholes equation. Boundary conditions for American options. Binomial and random walk methods.

Course Objective:

Sophisticated mathematical models, whose solution often requires computers, are important in finance. This course will give students the basic numerical tools and practice to solve financial problems on computers. Students will be expected to implement basic algorithms in a high level programming language.

Intended Audience:

Students from a variety of disciplines interested in financial mathematics, with priority to Financial Math students.

Prerequisites:

Introduction to Financial Mathematics (MAP 5601) or equivalent; competence in a high level programming language such as C, C++ or Fortran.

Texts:
  • (Recommended) The Mathematics of Financial Derivatives, by Wilmott, Howison, and Dewynne, Cambridge Univ. Press, 1995; and
  • Mathematics of Scientific Computing, Second Edition, by David Kincaid and Ward Cheney, Brooks/Cole, 2002.
Grading:

The grade will be based on periodic assignments, mostly computational and requiring computer programming, a midterm and a final exam.

Course Outline:
I. Basic Numerical Methods
a. Errors and conditioning
b. Solution of nonlinear algebraic equations
c. Interpolation, differentiation and quadrature
d. Fast Fourier Transforms
e. Solution of ordinary differential equations
f. Monte Carlo Methods
II. Numerical Solution of Parabolic Partial Differential Equations
a. Finite difference methods
b. Stability, convergence, error
c. The Black-Scholes equation
d. European options
e. Methods for American options
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Financial Engineering, MAP 6621 (3, Fall)
Graduate Bulletin Description:

Topics may include portfolio theory, methods and issues of active portfolio management, the martingale approach to derivative pricing, interest rate models and derivative pricing for stochastic interest rates.

Course Description:

This is a required course normally taken in the second year Financial Mathematics master’s degree program and is required for doctoral-intenders. The mathematics of financial instruments for students who have completed Introduction to Financial Mathematics and (MBA) Investment Management and Analysis. The course extends the students’ mathematical understanding of financial problems, and will place greater emphasis on practical market issues.

Prerequisites:

MAP 5601, Introduction to Financial Mathematics; and FIN 5515, Investments.

Representative Texts:
  • Baxter and Rennie, Financial Calculus, Cambridge, 1996.
  • R.C. Grinold and R.N. Kahn, Active Portfolio Management, 2nd ed, McGraw-Hill, 2000.
Grading:

Mid-term, final, plus homework.

Representative Topics
I. Portfolio Theory
a. portfolio construction
b. transaction costs
c. performance analysis and statistical significance
d. characteristic portfolio theory
II. Discrete models of asset pricing
a. review of binary tree model
b. general discrete models: arbitrage, martingale measures, forwards & futures
III. Continuous time asset pricing
a. Ito calculus
b. Girsanov and Martingale Representation theorems
c. interest rate models: short rate, HJM, multifactor
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Monte Carlo Methods in Financial Mathematics, MAP 5615 (3, Spring)
Prerequisites:

MAP 5601; and competence in a programming language for scientific computing.

Graduate Bulletin Description:

This course examines how the theory of Monte Carlo Methods is developed in the context of topics selected from computational finance, such as pricing exotic derivatives, American option pricing, and estimating sensitivities. The theory includes pseudorandom numbers, generation of random variables, variance reduction techniques, low-discrepancy sequences, and randomized quasi-Monte Carlo methods.

Representative Text:

Monte Carlo Methods in Financial Engineering by Paul Glasserman, Springer, 2004

Grading:

Letter graded.

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Financial Mathematics Projects, MAP 6437 (3, Spring)
Prerequisites:

This course is only open to students in the Financial Mathematics graduate program who have passed MAP 6621 and expect to complete the master’s degree requirements in the same calendar year.

The purpose of this capstone course is for students to bring together knowledge from previous courses to read current research, formulate specific project ideas, develop computational experiments to support their own conclusions, hone written and oral presentation skills, and practice teamwork to produce a polished final product under time-limited conditions.

The main objective is the completion of an individual research project on a topic chosen by the student in consultation with the instructor. The project will be submitted as a polished written paper and also as an oral presentation in class. In addition, students will complete a group project leading to a group in-class presentation, and complete other smaller assignments as they may arise during the course. An important part of the course is for students to practice critically listening to other projects and participating in constructive questions and discussion.

Text Materials:

Readings from the popular and financial presses.

Grading:

Letter graded.

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Financial Mathematics Proseminar, MAT 5939r (1, Fall and Spring)
Prerequisites:

Graduate standing in the Financial Mathematics program, or permission of the instructor.

This Professional Seminar meets weekly and covers topics relevant to professional practice that are not covered in the regular courses. Visiting speakers, class discussions, workshops, group presentations, and occasional written assignments cover such topics as resume writing, researching internships and jobs, interviewing skills, ethics, and other supplemental topics relevant to the quant profession.

Grading:

S/U.

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Advanced Seminar in Financial Mathematics, MAP 6939r (1, Fall and Spring)
Prerequisites:

Graduate standing in the Financial Mathematics program, or permission of the instructor.

This is a weekly research seminar where students, faculty, and visitors present current research in Financial Mathematics. Graduate students intending the PhD in Financial Math normally register for this seminar every semester after the first year.

Grading:

S/U.

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Scientific Programming, ISC 5305 (3, Fall)
Graduate Bulletin Description:

Object-oriented coding in C++, Java, and Fortran 90 with applications to scientific programming. Discussion of class hierarchies, pointers, function and operator overloading and portability. Examples include computational grids and multidimensional arrays.

Course Objective:

This course proposes to teach students the basic elements of Fortran 90, Java, and C++ to enable them to perform simple to intermediate object oriented programming tasks. The course assumes basic familiarity with a programming language, but not necessarily some of its advanced aspects such as structures, pointers, etc. A student that completes this course will be comfortable thinking in objected oriented terms, know how to manipulate objects with dynamically changing information, know how to access elements of one language from another, learn to construct software libraries and more. Emphasis is placed on the commonalities between the three languages with discussion of some of their features that make them unique.

Prerequisites:

Working knowledge of one programming language (preferably C++, Fortran, Java), or instructor permission.

Grading:

Based on weekly homework, which will consist of either articles to read and summarize (online), or programming homeworks.

Course Outline
I. Introduction to Scientific Programming
a. Compiling, linking, making, debugging
b. Code Structure
c. Libraries
d. Mixed coding
II. Object Oriented Programming and C++
a. Pointers and References
b. OOP
c. Standard Template Library
III. Java
IV. Fortran
V. Projects
a. Streamtube generation (C++)
b. Java Grid
c. Heat equation (Fortran)
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Financial and Monetary Economics I, ECO 5281 (3, Spring)
Texts:
  • Eichberger, Jürgen and Ian R. Harper. 1997. Financial Economics. Oxford University Press.
  • Altüg, Sumru and Pamela Labadie. 1994. Dynamic Choice and Asset Markets. Academic Press.
Prerequisites:

Calculus III.

Course Description & Objectives:

This course is intended to provide a comprehensive introduction to the field of financial economics. We will introduce the necessary mathematical tools at a leisurely pace, as needed. In this class we will focus on static and dynamic consumption based asset pricing models and a few elementary applications. The class is designed to set up the framework for models with production, financial institutions and monetary policy issues that will be the focus of Financial Economics II. By the end of this course students should have a clear understanding of the dynamic models used in modern finance along with the empirical implications of some simple applications of these models. The second course extends these models and addresses various policy issues and implications.

The course has three main units: static equilibrium models; dynamic programming; and consumption based asset pricing models. In each unit we develop some general theory and work towards an empirical application.

In the first unit we study general equilibrium in a single-period, pure exchange economy. First we develop the general theory of the consumer and then we introduce contingent claims and securities. Next we study incomplete markets, arbitrage, risk aversion, and risk-sharing. As applications we develop the Capital Asset Pricing Model and the Arbitrage Pricing Theory as specific partial equilibrium examples. We close this unit with a critical empirical evaluation of the CAPM and APT pricing theories.

In the second unit we introduce the essentials of dynamic decision making. We begin with a two period inter-temporal decision model with a simple one-period bond asset. We will then spend some time with an elementary introduction to dynamic programming. Within this context we will study optimal consumption and savings choices in a dynamic setting. We close this unit with an empirical evaluation of the consumption smoothing and asset volatility assumptions implied by these models.

In the third unit we introduce the consumption based asset pricing model. Within this framework we will examine discount bonds and the yield curve, derivative securities and contingent claims contracts. We will derive the Black-Scholes option pricing formula and introduce the notion of Martingale pricing theory. We will close this unit by examining several empirical issues including: volatility and variance bounds; the equity premium puzzle; conditional volatility in asset prices; and the pricing of “real-world” securities such as American options, exotic options and other derivatives.

Grading:

Grades will be determined from homework, a midterm exam, and a final exam.

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Distribution Theory, STA 5326 (3, Fall)
Prerequisites:

Three semesters of calculus and an undergraduate course in probability (or some exposure to probability plus a sufficiently strong math background).

Text:

Statistical Inference by Casella and Berger

Grading:

Based on three exams (equally weighted). The exams are in-class, but with an open-ended time limit.

Topics Covered:
Axioms and basic properties of probability.
Combinatorial probability.
Conditional probability and independence.
Applications of the Law of Total Probability and Bayes Theorem.
Random variables.
Cumulative distribution, density, and mass functions.
Distributions of functions of a random variable.
Expected values.
Computations using indicator random variables.
Moments and moment generating functions.
Common families of distributions.
Location and scale families. Exponential families.
Joint and conditional distributions.
Bivariate transformations.
Covariance and correlation.
Hierarchical Models. Variance and Conditional variance.
Introduction to Brownian Motion
Discrete Markov chains; Poisson processes.
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Topics in Stochastic Processes, STA 5807 (3, Summer)
Prerequisites:

Distribution Theory (STA 5326) or equivalent background in probability.

Text:

Stochastic Processes (2nd edition) by Sheldon Ross

Grading:

Based on two exams and four quizzes.

Topics Covered

Poisson process with constant rate. Nonhomogeneous and compound Poisson processes - Renewal theory. Wald's equation. Key renewal theorem; alternating renewal processes - Markov chains. Chapman-Kolmogorov equations. Classification of states. Limit theorems. Transitions among classes. Mean times in transient states. Gibbs sampler - Brownian motion. Reflection principle, hitting times, distribution of the maximum - Brownian motion with drift. Brownian bridge.

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Investment Management and Analysis, FIN 5515 (3,Fall)
Graduate Bulletin Description:

Analysis of financial assets with emphasis on the securities market, the valuation of individual securities, and portfolio management.

Prerequisites:

Background knowledge corresponding to material covered at FSU by the undergraduate Finance courses FIN 3403 and FIN 4504. This includes accounting statements and analysis, corporate finance and financial management, options and risk management, and the investment environment. Representative texts for this prerequisite material include:

  • Ross, Westerfield, and Jordan, Fundamentals of Corporate Finance, 5th edition (or later), Irwin McGraw-Hill, 2000; and
  • Brigham and Ehrhardt, Financial Management: Theory and Practice, 10th edition (or later), Southwestern, 2002.

Taken with the MBA students.

Text:

Graduate level investments text that includes theory and practice, e.g. Bodie, Kane, and Marcus, Investments", McGraw-Hill, Fourth or later edition.

Grading:

Based on two exams (equally weighted). The exams are in-class, but with an open-ended time limit.

Topics Covered
I. Investment Environment and Process.
II. Security Markets.
III. Return and Risk Concepts.
IV. Expected Returns, Risk, and Portfolio Analysis.
V. Equity Valuation.
VI. Market Efficiency.
VII. Technical Analysis.
VIII. Investment Companies.
IX. Interest Bearing Securities.
X. Options Markets, Trading, Returns, and Pricing.
XI. Futures Markets and Trading.
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Actuarial Models, MAP 5177 (3, Fall terms only)
Graduate Bulletin description:

Survival probabilities, mortality laws, table construction, and contingent payments and annuities; premium principles and reserves for continuous, discrete and semi-continuous insurance products; multiple decrement theory (competing risks) and application to pension plans; pricing and nonforfeiture models.

Prerequisites:

MAP 4170, STA 4322 or equivalents.

Resources and Texts:
  • Bowers et al, Actuarial Mathematics, Second Edition, Society of Actuaries 1997;
  • Gauger, Survival Models, Contingent Payment Models, Vol. I, Actex Manual Course 3 Society of Actuaries;
  • also Vol. III of preceding (sample exams).
Course Objectives & Content:

Theory and applications of survival and life contingency models. Understanding of and ability to use actuarial contingency models as presented in the Bowers text; there, it is framed in terms of human lives but is equally applicable to the design of a machine or a bridge... This material may be useful in a wide range of educational, business and government applications and is necessary for one actuarial examination each of the SOA and the CAS. Secondary Objectives include various concerns for a career in actuarial science. Testing follows that typical in mathematics courses, with added concerns due to this latter component. Student presentations and other activities contributing to professional development are included; additionally there is consideration of professional standards of conduct.

Grading:

Hour Tests, a Comprehensive Final, and a Presentations and Professionalism component. Letter graded.

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Loss and Stochastic Models, MAP 5178 (3, Spring)
Graduate Bulletin description:

Topics may include loss distributions, frequency distributions, the individual and collective risk models, Markov chains, Brownian motion, ruin models, and simulation.

Course Objectives:

This course is intended as a sequel to Actuarial Models (MAP 5177). The topics in this course parallel the topics tested on Course 3 of the Society of Actuaries exam sequence. The primary objective of the course is to increase students’ understanding of the topics covered, and a secondary objective is to prepare students for a career in actuarial science.

Prerequisite:

MAP 5177.

Resources:
  • Bowers et al., Actuarial Mathematics, Second Edition, Society of Actuaries 1997;
  • Ross, S. M., Introduction to Probability Models, Eighth Edition, Academic Press 2003;
  • Klugman et al., Loss Models: From Data to Decisions, John Wiley & Sons 1998;
  • Ross, S. M., Simulation, Third Edition, Academic Press, 2002;
  • Vol. II, Actex Manual Course 3 Society of Actuaries;
  • also Vol. III of preceding (sample exams).
Grading:

Hour Tests, a Comprehensive Final, and a Presentations and Professionalism component (the latter about 15%). Letter graded.

Course Outline:
I. Aggregate Loss Models
a. Loss Distributions
i. Mixed Distributions
ii. Modeling Inflation, Coinsurance, Deductibles, and Policy Limits
b. Frequency (Counting) Distributions
i. (a,b,0) and (a,b,1) Families of Distributions
ii. Compound Counting Distributions
c. Individual and Collective Risk Models
i. Double Expectation Theorem
ii. Compound Poisson Collective Model
iii. Stop Loss Reinsurance
II. Stochastic Process Models
a. Markov Chains
i. Discrete Time with Discrete State Spaces
ii. Continuous Time with Discrete State Spaces
b. Brownian Motion
i. Standard Brownian Motion
ii. Brownian Motion with Drift
c. Ruin Models
i. Surplus Models and Ruin Calculations
ii. The Compound-Poisson Surplus Process
III. Simulation
a. Simulation of Discrete Random Variables via a Standard Search Algorithm
b. Simulation of Continuous Random Variables via Inverse Functions
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Topics in Stochastic Calculus, MAA 6416 (3)
Prerequisites:

Measure and Integration II or co-enrollment in M&I II. Knowledge of probability through square-integrable martingales is suggested. Enrollment in Financial Mathematics program or permission of the professor and the FinMath Director.

Texts & Resources:
  • Protter, P., Stochastic Integration and Differential Equations, Springer-Verlag, Berlin (1990);
  • Billingsley, P., Probability and Measure, John Wiley & Sons, New York (1995);
  • Karatzas, I. and Shreve S. E., Brownian Motion and Stochastic Calculus, Springer-Verlag, Berlin (1991);
  • Chung, K. L. & Williams, R.J., Introduction to Stochastic Integration, Birkhäuser, Boston (1983);
  • Elliot, R. J., Stochastic Calculus and Applications, Springer-Verlag, New York (1982);
  • Ikeda, N. & Watanabe, S., Stochastic Differential Equations and Diffusion Processes, North Holland, Amsterdam (1981);
  • Kunita, H. & Watanabe, S., "On square-integrable martingales," Nagoya Math J., 30, 209-245 (1967).
Description and Topics:

General semimartingale processes are now widespread in financial mathematics. The stochastic calculus of semimartingales includes the majority of financial models.

Grading:

There are hour tests and a final examination, TBA.

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Numerical Optimization, MAD 5420 (3)
Description and Aims:

The course intends to provide the students a thorough understanding of methods for unconstrained and constrained non-linear programming as well as modern methods of global optimization by combining recent theory with concrete practical experience based on analysis and comparison of efficient up-to-date algorithms for solving real life optimization problems and their implementation on supercomputers, taught by an instructor active in research in optimization. The material will be presented following a new text book reflecting advances in the field in the last 15 years along with adequate software.

Application to problems arising in modeling of various fields of computational mathematics, atmospheric and oceanographic sciences, variational data assimilation, finance and economics, optimal control and engineering applications will be emphasized along with new optimization software and its implementation for multidisciplinary computational science and engineering oriented research projects.

Prerequisites:

The course is intended primarily to graduate students and senior undergraduate students with some background in linear algebra, and programming skills in a high-level language as well as familiarity with one of the operating computer systems at FSU.

Required Texts:
  • Linear and Nonlinear Programming, by Stephen G. Nash and Ariela Sofer, McGraw-Hill, 1996, 782pp.
  • Optimization Software Guide, by Jorge J. More and Stephen J. Wright, SIAM Frontiers in Applied Mathematics, Vol. 14, 1993, 154pp.
Details:

In the last few years the topic of numerical optimization has become the focal point for applications in fields as diverse as atmospheric sciences, in particular variational data assimilation, oceanography, engineering, finance and economics. The desire to solve a problem in an optimal way is so common that optimization models arise in almost every area of application. The last few decades have witnessed astonishing improvements in computer hardware and these advances have made optimization models a practical tool in business, science and engineering. It is now possible to solve problems with millions of variables. The theory and algorithms that make this possible form a large portion of this course.

In the first part of this course we start by presenting basics of unconstrained minimization followed by methods of unconstrained minimization. Then we introduce low-storage methods for large-scale unconstrained minimization as well as nonlinear least-squares data fitting. Each method will be computationally illustrated with adequate software and examples as well as hands on experience.

In the second part of this course we start with optimality conditions for constrained optimization followed by feasible point methods such as Sequential Quadratic Programming and Reduced Gradient Methods. We then proceed to penalty and barrier methods as well as multiplier based methods such as Augmented Lagrangian method.

Finally we dedicate some attention to global minimization methods dealing in particular with simulated annealing and genetic algorithms and illustrate them with applications.

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Financial Economics II, ECO 5282 (3, Fall)
Texts:
  • Altug, Sumru and Pamela Labadie (1994). Dynamic Choice and Asset Markets. San Diego: Academic Press.
  • Freixas, Xavier and Jean-Charles Rochet. The Microeconomics of Banking.
Prerequisite:

Financial Economics I

Course Content:

This course is the second in the financial economics course sequence. It focuses on three broad areas: production-based asset pricing theory; financial inter-mediation; and monetary theory and policy. There will be no additional mathematical requirements beyond what was needed to complete Financial Economics I.

The course will begin by developing the basic general equilibrium growth model, which is an extension of the exchange economy studied in Financial Economics I to an economy with production. This basic model will be used to study asset pricing dynamics as they relate to the nature of risk and diversification, and the roles of capital structure, inflation, and taxation. The second third of the course will focus on financial intermediation, with particular emphasis given to the economic role played by commercial banks in private information economies, including delegated monitoring of risky borrowers, the provision of liquidity as insurance against income shocks, and bank activities within the overall macroeconomy. The course will conclude with topics in monetary theory and policy. These topics will include the financial markets that are directly affected by Federal Reserve policy, and the impact of policy decisions on overall economic activity. The requisite institutional backdrop for these topics will be provided.

Course Objective:

Students should develop a theoretical understanding of: the relationship between production and financing decisions of firms and the absolute pricing of capital assets within a general macroeconomic setting; why financial intermediaries exist, and the particular role of banks in the financial markets; and how and to what end monetary policy is conducted.

Grading:

Grades will be based on a midterm and a final.

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Internship in Financial Mathematics, (1-3, any term)
Graduate Bulletin description:

Prerequisite: Instructor approval. Supervised internships individually assigned to accommodate a student’s professional development in the field of financial mathematics. May be repeated for a maximum of three (3) semester hours.

Rationale:

Many of the entering students have their bachelor’s degrees in the mathematical sciences where the tradition is to go to graduate school directly from undergraduate. For such students the experience and knowledge of the work environment gained through an internship is strongly desirable. In some related fields (e.g., Actuarial Science), internships are an established feature.

At FSU:

The student may arrange an internship with a financial, banking, commodity, or exchange organization in the U.S. for any term of the year. There is also the possibility that a student may arrange an internship in Tallahassee with a firm or government agency, and this might be spread over a longer time period on a part-time basis. Information is available from the Graduate Advisor concerning proper reporting requirements. 1 hour is normal credit.

FSU London Study Center:

The FSU London Center is available for students who can arrange an Internship at the London stock exchange or in a banking or financial institution. These are individually arranged in accordance with the student’s interests, up to 3 hours credit.

Students with this interest may explore the London Study Center website, http://www.fsu.edu/~london. The 12 week summer internship program is early May to late July.

Up to 3 hours of internship credit from London can count toward your degree in Financial Mathematics.

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