Math
5725 provides an introduction to some of the basic ideas from
stochastic
processes that are used in constructing mathematical models of
financial
markets, and the no-arbitrage principle upon which many of the
derivative
pricing formulas are based. In Math 5726 (Spring 2008) we will
change
the focus from stochastic models to basic methods for numerical
solutions
of partial differential equations. It is possible to take 5726
without having taken 5725, provided you are willing to just accept the
significance of the PDEs we will consider.
In brief, a stochastic model leads to a partial differential
equation
(PDE) which describes the market price of an option or
derivative.
In some cases (like the Black-Scholes formula for European puts and
calls)
a formula can be produced which gives the solution to this PDE and
therefore
the predicted market price. But usually this is not possible, and
one must resort to some sort of numerical method to compute values of
the
solution of the PDE. Our goal is to provide a basic
understanding
of these methods as they are used for financial problems, including the
pricing of American and path-dependent, such as "Asian" options.
Students
will be expected to implement the methods we discuss as homework
assignments using Matlab. We will discuss details of using Matlab
as needed. There will be no exams. Grades will be assigned
based on homework scores.
Like 5725, this will not be a mathematically rigorous treatment of numerical methods (for that you should take Math 5474 and 5484). Rather it is more of a practical introduction that is intended to be accessible to graduate students from other departments as well as interested mathematics students.
The lectures will be essentially self-contained, with
a short set of printed notes provided. There will not be a required
textbook. However if you want to look at something to see what
the course will be about, I would suggest The Mathematics of
Financial Derivatives:
A Student Introduction, by P. Wilmott, S. Howison and J. Dewynne,
(Cambridge
Univ. Press, 1995). You can find a copy in the library. The book
discusses a fair amount of what we will cover.
Please feel free to come by or send me a message if you have questions about the course.