Materials from Math 5725, 5726: Mathematics for Financial Modeling
Beginning in the Fall of 1998 the Mathematics
Department at Virginia Tech began offering a pair of beginning level
courses introducing some topics in mathematical finance. The
courses no longer are offered at Virginia Tech, but for the benefit
of past students and any others who may be interested I am making
available here some supplemental materials and lecture notes that were
prepared for the courses.
From Math 5725
The first course was initially offered as Math 5415 (Fall semesters of
1998, 2000, 2002) and later as Math 5725 (Fall semesters of 2004, 2007
and 2010). It introduced the Black-Scholes model of a
stock price and the ideas of risk neutral or no-arbitrage pricing for
derivative securities based on it. The focus was on
stochastic modeling and martingale pricing, using various editions of
Tomas Björk's Arbitrage Theory in Continuous Time
as a text. The following two supplements were prepared to help students get started with the material.
From Math 5726
Begining in 2001 a second semester was added
emphasizing the use of finite difference methods to compute option
prices from the PDE characterization of the pricing function for
various types of standard and exotic options. This was initially Math 5416 (Spring semesters of 2001, 2003) and later
Math 5726 (Spring semesters of 2005 and 2008). Initially this was based on the text The Mathematics of Financial Derivatives: A Student Introduction
by P. Wilmott, S. Howison and J. Dewynne. But as the presentation
was elaborated and additional material added it evolved into a
self-contained set of lecture notes.
Some disclaimers are in order regarding the 5726 lecutre notes.
Many of the figures are hand-drawn. Here and there you may notice
notes to myself (in colored text) about changes I contemplated but
never implemented. Surely you will find typos and artifacts of
hasty editing from eariler editions. Some topics are not well
developed (for instance Ch. 6 on the connection between the finite
difference methods and a Markov chain interpretation). I have not
put any time into updating or completing the notes
since the course was last taught in 2008. They remain as they
were at the end of that semester. I am not an expert in
mathematical or computational finance. Please do not expect
these notes do not represent the state of the art in those
topics. They are simply what evolved through my efforts to teach
these introductory courses.