Syllabus for Math 351, Section F1 (Financial Mathematics)
topic: Financial Mathematics
Text: The Mathematics of Finance by Goodman and Stampfli
Outline: This is an undergraduate course on some mathematical
aspects of finance. There are roughly two questions we will study.
The tone of the class will, as much as possible, be exploratory; I would
like to forego the standard Socratic method for a more conversational framework.
Ideally, I would like to present the basic ideas of some material and then
have students present some material which can expand upon the basic ideas.
Although it would help if you have had Math
361, it is not formally a prerequisite. In other words, although I
would like mathematical maturity, I will settle for mathematical adolescence.
This means that the students backgrounds will wildly vary and that we will
need to proceed slowly (in contrast to the standard courses which need
to proceed at a fixed pace due to curricular considerations); both sophomores,
juniors, and seniors will hopefully find the material accessible. I am
also investigating the possibility of using Mathematica.
Grading policy: There will be three exams, final, and either some
homeworks or some class presentations. The relative weights will be:
Financial Instruments. Most of the course will be dedicated
to exploring the implications of a very basic model (the binomial models)
involving financial derivatives (e.g., options). We will go very slowly
with this since it illustrates a central idea: arbitrage pricing. We will
use the binomial model and more generally the tree model to understand
a number of topics: Black-Scholes, incomplete markets, American options,
interest-rate derivatives, bond pricing, futures, etc. Once
we understand the tree model, we will understand how to take limits of
the tree model to find appropriate partial differential equations for the
appropriate financial products.
How should we invest. We will then consider mean-variance
analysis and the Capital Asset Pricing Model (CAPM). Once again, we will
build up a theory from very simple examples. Roughly, the question we will
investigate is how one should invest, under certain assumptions on volatility
and rates of return. This will involve a bit of probability and a bit of
linear programming. We will take this as slowly as needed.
We will also consider immunization, a way of reducing modelling risk.
Time permitting, we will also address some other issues, e.g. insurance.