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 graduate 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.

• A Scholar's Introduction to Stocks, Bonds and Derivatives.  For the benefit of student's with no prior understanding of stocks, bonds and financial derivatives this provides a brief explanation.
  
• A Primer on Stochastic Processes.  Bjork's text offers little introduction to probability and stochastic processes.  This is intended as a quick and dirty introduction, ending with the basics of manipulating stochastic differentials (Ito calculus).
• Cross reference chart for editions 1, 2, and 3 of Björk's text.
  

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. 

• Black-Scholes Option Pricing: PDEs, Probability, and Matlab.  These are the lecture notes.
  
• Matlab files used in the notes.
  

Disclaimers

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.