VIGRE2 Vertical Intergration of Research and Education Department of Mathematics, University of Utah

MATH 4950

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Fall 2006 course

Title: Topics in Mathematical Finance
Time: Tuesdays & Thursdays, 12:25-1:55PM
Location: WBB 517
Instructor: Jingyi Zhu
Catalog number: 4950-1
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 2-3 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:
  1. 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.

  2. 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.

  3. Black-Scholes formula deconstructed

    The Black-Scholes 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.

  4. 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.

  5. 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.