Math 628 (MathFinance) - Syllabus (Spring'01)

Professor: Thomas Schlumprecht
Office: Graduateoffice of Mathematics in Blocker Tel:845-8840 Hours: M 10:30 - 11:30 (in Blocker) and by appointment E-mail: schlump@math.tamu.edu
Homepage: http://www.math.tamu.edu/~thomas.schlumprecht

This course tries to introduce graduate students of Mathematics and Statistics as well as graduate students of Business and Economics to the mathematical theories necessary for solving problems in finance. These theories originate in various areas of mathematics, for example: linear algebra, convex geometry, numerical analysis, stochastic processes, and partial differential equations. The variety of these tools and the wide spectrum of the audience necessitates the introduction of mathematical concepts as needed and only at a rather intuitive level. The central question we want to address is the following:

What is a fair price for an option/future, given the spotprice of the underlying security?

We will use different models to approach this problem, but our main starting point will be always the principle of absence of arbitrage possibilities, the fundamental axiom in the theory of asset pricing.

In more detail, we intend to cover the following subjects.

Introduction

Basic asset classes and their derivatives
Basic arbitrage ideas, first consequences

Part I: discrete models

Introduction to the Arrow-Debreu model with finite state space and discrete time
State-contingent securities, their prices, options and futures represented as linear combinations of state- contingent securities
Binomial and multinomial models

Part II: continuous models

Short introduction to basic ideas of probability theory; in particular: probability spaces, expeceted values, independence, conditional probabilities and conditional expectations.
Modeling the stockmarket as a stochastic process, i.e. viewing the stock prize as a solution of stochastic differential equations
The Ito formula
Derivation of the Black-Scholes equation
Basics on difffusion equations
Solution of the Black-Scholes equation
Variation on Black-Scholes
American options; pricing; when should one sell an American option?

Prerequisites

A graduate course in Probability theory or Real Analysis (M606, M607, M618) or a concurrent enrolement in M625.
Beyond these prerequisites it is intended to introduce all necessary concepts, nevertheless ``a mathematical mind'' and ``some mathematical experience'' acquired in higher level courses are indispensable to grasp the rather deep ideas.

Literature

Unfortunately no single text satisfies completely the purpose of this course.
We therefore provide Course Notes
The following text is easy to read but is does not contain proofs. Financial Calculus, Martin Baxter and Andrew Rennie, Cambridge University Press. ISBN 0-521-55289-3.

The following texts cover the whole content of this course (and much more) but are written in a very demanding and abstract style (good for further studies):
Martingal Methods in Finacial Modelling, Marek Musiela and Marek Rutkowski,Sprnger Verlag,ISBN: 3-540-61477-X.
Darrell Duffie, Dynamic asset pricing theory, second edition, Princeton University Press.

Grading

Grades will be based on homework asigned every two to three weeks. Furthermore students may be given short research projects based on individual interests.