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Fall 2006 course
Title: Topics in Mathematical Finance
Time: Tuesdays & Thursdays, 12:251:55PM
Location: WBB 517
Instructor: Jingyi Zhu
Catalog number: 49501
Credits: 3
Text:
No standard text is assigned. References and relevant materials will be given over the course of the semester.
Prerequisites:
We expect backgrounds beyond calculus and basic probability theory
(such as differential equations, basic statistics, or stochastic processes). However, due to
the variety of topics available in the class, we do not have a uniform set of requirements.
Instead, each prospective student is required to visit the instructor to determine the suitability
before registering for the class.
Course Outline:
This course is part of the REU (Research Experiences for Undergraduates) program, as an
important component of the VIGRE grant funded by NSF. The subject varies from one semester
to another, in which a faculty member, a graduate assistant, and up to 10 students explore a topic of significant mathematical interest. The students help to present the material or the results of their
own investigations, and write a report on their findings. This fall, we are offering the subject of mathematical finance, with several possible topics listed below.
The format of this course is rather flexible,
with the class divided into groups, each with 23 students and a particular topic to study
over the semester. Available topics are listed and described in the following.
Tuition Benefit: The NSF VIGRE program provides a small tuition benefit for US students (US citizens
and permanent residents). Details will be provided once total enrollment has been determined.
Topics:
 The myth of technical analysis
Many financial advisors tell you about the magic power of technical analysis. Do they have
solid mathematical grounds to be taken seriously? We will do a short literature search and
use simple examples to test for ourselves.
 Random walks in stock price models
Brownian motion (a mathematical object derived from random walks) is ubiquitous in our
daily lives, and the earliest mathematical theory was usually attributed to Einstein
(which earned him a Nobel prize, not because of the relativity theory though).
However, it is not widely known that a French mathematician named
Bachelier actually did some ground breaking work in Brownian motion
in 1900 (before Einstein) and his motivation was to
analyze the Paris stock market. We will study his PhD thesis and follow the developments
since then in the last hundred years.
 BlackScholes formula deconstructed
The BlackScholes formula for option pricing is available in most financial calculators. Do you
wonder how the formula is derived and what the shortcomings are? We will see to it that you
derive the formula by the end of the semester, and explain to your business friends why the
formula is over pricing or under pricing a particular stock option in a particular situation.
 Does the stock market have a memory?
If the answer is yes, then Brownian motion (Markov processes for this matter) is not the right tool
for analyzing the stock market. We shall explore several ramifications beyond the standard
Brownian motion model which are currently in use.
 Bond yield vs. bond price
Every time you hear in the evening news about the bond market, the announcer will make the
comment that
the yield and price move in opposite directions. You often wonder why and we will have
it explained in five minutes. However, the exact relation and other consequent matters are
rather complicated,
and sophisticated hedge funds have actually made billions of dollars by exploring the mathematical
subtleties
that are not understood by common investors and many professionals without the proper mathematical
training. We will look into the basics of bond mathematics, which contains, as it turns out, a wonderful
collection of mathematical theories.
Course grade: There will be no exams in the course. Grades are to be determined by the
presentations during the course, and the final
project report submitted by each group.
