Introduction to financial risks, optimizaton in finance, probability of stochastic processes, binomial pricing model and arbitrage pricing theory, introduction to stochastic calculus and the Black-Scholes equation.

I. Introduction

a. Classification of financial risks

b. Derivatives

c. Optimization in finance

II. Binomial model

a. Fundamental theorem of asset pricing

b. Calibration

c. Evaluation and hedging ofEuropean, American and exotic options

III. Continuous-time models

a. Bachelier model

b. Black-Scholes model

c. European, American and exotic options

d. Calibrarion

e. Numerical methods

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.

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

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.

I. Portfolio Theory

a. portfolio construction

b. the efficient frontier

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

c. the fundamental theorem of asset pricing

III. Continuous time asset pricing

a. Ito calculus and the Black-Scholes model

b. Girsanov and Martingale Representation theorems

c. interest rate models: short rate, HJM, multifactor

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

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.

This Professional Seminar meets weekly with first year Financial Math students 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.

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 second year.

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

This course proposes to teach students the basic elements of Python 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.

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

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. Python

IV. Projects

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.

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.

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.

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

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.

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.

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.

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

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.

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

Selected advanced topics in stochastic calculus.

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.

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.

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.

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.

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.

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.