Math 456, Fall 2008
Topics in Financial Mathematics: Pricing
Options
Instructor
John E. McCarthy
Class
Tu-Th 1.00-2.30 in TBA
Office
105 Cupples I
Office Hours
M 3:00-4:00, Tu 2:30-3:30, Th 3:00-4:00, or by appointment
Phone
935-6753
Text The Mathematics of Financial Derivatives P. Wilmott, S. Howison and J. DeWynne
Exams There will be two exams in the course:
1) Exam 1 In class, on Thursday, Oct 23rd
2) Exam 2 Final exam, on Wednesday, Dec
17, 1-3 pm
Prerequisites Math 318, and either 3200 or a strong performance in 2200 (320) and permission of the instructor.
Content
The main purpose of this course is to understand how options should be priced.
An example of an option is if I agree, in exchange for $10 now, that on December 31st of this year I will sell you one bushel of apples for $50, though you are not obliged to make the purchase. If the market price of apples in December is less than $50, you will not exercise the option, and I will have made a profit of $10. If the price is between $50 and $60, you will exercise the option, and I will have made a smaller profit. If the market price is over $60, I will have made a loss.
Is $10 a fair price for this option? We do not know what price apples will be in December, so we have to guess at a probability distribution. Getting this right is the difference between being written about in the newspaper for your huge philanthropic gifts, and making the front page as the person who bankrupt the Never Fail Hedge Fund.
In the course, we will study in particular the Black-Scholes equation, which gives a method for pricing options if one assumes that the underlying asset price has a Gaussian white noise component superimposed on a linear trend.
Setting up the Black-Scholes equation requires some understanding of stochastic processes, which we shall develop. After some transformations, solving it is equivalent to solving the partial differential equation called the Heat equation or Diffusion equation, which is used in physics to model the diffusion of heat (or gas particles). We shall develop the theory necessary to solve the equation, both analytically and numerically.
Basis for Grading
The midterm and home work will each be 20% of your grade, the final will be 60%.
I expect you to come to class every day, and to participate in class discussions. I also expect you to stay abreast of the material we are covering, and may call on you at any time to answer a question.
Homework Solutions
Here are some model solutions.
HW3Bibliography
The following is a brief bibliography you may find useful.
S. Dineen Probability
Theory in Finance: A mathematical guide to the Black-Scholes Formula
(Assumes
no knowledge of probability, and builds up to using measure theory)
J. Stampfli and V. Goodman The
Mathematics of Finance: Modeling and Hedging (The
most elementary treatment among the listed books)
S.M. Ross An
elementary introduction to mathematical finance: Options and other
topics
J.C. Hull Options,
futures and other derivatives
(Lots about applications, less mathematical)
P.Wilmott Paul
Wilmott introduces Quantitative Finance (A friendlier, but more
expensive, version
of the assigned text)
P.Wilmott Paul
Wilmott on Quantitative Finance (This is a three volume encyclopedic
treatment, on
reserve in the Business Library)