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.  

HW3
HW4
HW5
HW6
HW7
HW8
HW9

Bibliography

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)