Basics of Financial Mathematics

2003 by Richard F. Bass

Introduction

In this course we will study mathematical finance. Mathematical finance is not about predicting the price of a stock. What it is about is figuring out the price of options and derivatives. The most familiar type of option is the option to buy a stock at a given price at a given time. For example, suppose Microsoft is currently selling today at $40 per share. A European call option is something I can buy that gives me the right to buy a share of Microsoft at some future date. To make up an example, suppose I have an option that allows me to buy a share of Microsoft for $50 in three months time, but does not compel me to do so. If Microsoft happens to be selling at $45 in three months time, the option is worthless. I would be silly to buy a share for $50 when I could call my broker and buy it for $45. So I would choose not to exercise the option. On the other hand, if Microsoft is selling for $60 three months from now, the option would be quite valuable. I could exercise the option and buy a share for $50. I could then turn around and sell the share on the open market for $60 and make a profit of $10 per share. Therefore this stock option I possess has some value. There is some chance it is worthless and some chance that it will lead me to a profit. The basic question is: how much is the option worth today? The huge impetus in financial derivatives was the seminal paper of Black and Scholes in 1973. Although many researchers had studied this question, Black and Scholes gave a definitive answer, and a great deal of research has been done since. These are not just academic questions; today the market in financial derivatives is larger than the market in stock securities. In other words, more money is invested in options on stocks than in stocks themselves. Options have been around for a long time. The earliest ones were used by manufacturers and food producers to hedge their risk. A farmer might agree to sell a bushel of wheat at a fixed price six months from now rather than take a chance on the vagaries of market prices. Similarly a steel refinery might want to lock in the price of iron ore at a fixed price. The sections of these notes can be grouped into five categories. The first is elementary probability. Although someone who has had a course in undergraduate probability will be familiar with some of this, we will talk about a number of topics that are not usually covered in such a course: σ-fields, conditional expectations, martingales. The second category is the binomial asset pricing model. This is just about the simplest model of a stock that one can imagine, and this will provide a case where we can see most of the major ideas of mathematical finance, but in a very simple setting. Then we will turn to advanced probability, that is, ideas such as Brownian motion, stochastic integrals, stochastic differential equations, Girsanov transformation. Although to do this rigorously requires measure theory, we can still learn enough to understand and work with these concepts. We then return to finance and work with the continuous model. We will derive the Black-Scholes formula, see the Fundamental Theorem of Asset Pricing, work with equivalent martingale measures, and the like. The fifth main category is term structure models, which means models of interest rate behavior. I found some unpublished notes of Steve Shreve extremely useful in preparing these notes. I hope that he has turned them into a book and that this book is now available. The stochastic calculus part of these notes is from my own book: Probabilistic Techniques in Analysis, Springer, New York, 1995.