## MA 490-02: The Mathematics of Option and Derivative Pricing

#### Kurt Bryan, Winter 2007-08

Most people are familiar with traditional investments like stocks, bonds, and commodities. However, in the past few decades a huge market has arisen in the trading of "options" and other "financial derivatives", contracts in which payment is based on the value of some benchmark, e.g., the price of a given stock on a certain date. In short, the value of the contract is derived from the price of some underlying asset (hence the term "derivative').

As an example, suppose a contract is written in which I give you the option (but not the obligation) to buy from me one share of Microsoft stock for a guaranteed price of \$25 on March 1, 2008 (today, September 28, 2007, it's selling for about \$29). This is an example of a European Call Option, in which you have the right to buy some asset at a guaranteed price sometime in the future. How much should you pay to enter into such an agreement? Surprisingly, there is a very quantitative strategy for determining the value of this option contract.

In this class we'll examine the problem of option pricing. We'll start by looking at various types of options, and at probabilistic models for random fluctuations in asset prices. We'll also derive the celebrated Black-Scholes partial differential equation which shows how one can rationally determine option prices. This is the work for which Robert Merton and Myron Scholes won the 1997 Nobel Prize in economics.

A variety of different mathematics will appear in the course, including simple probability, differential equations, and some numerical techniques. We'll learn what we need as we need it. The only prerequisites are the first year calculus and second year differential equations sequence. The grading in this course will be based on homework assignments, in-class work, and a couple take-home exams.