Introduction:
In this course we will study mathematical finance. Mathematical finance is not
notabout 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 marketin 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: -fields, conditional expectations, martingales. The secondcategory 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
2
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 Techniquesin Analysis, Springer, New York, 1995.I would also like to thank Evarist Gin´e who pointed out a number of errors.
larger than the marketin 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: -fields, conditional expectations, martingales. The secondcategory 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
2
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 Techniquesin Analysis, Springer, New York, 1995.I would also like to thank Evarist Gin´e who pointed out a number of errors.
-fields, conditional expectations, martingales. The secondcategory 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
2
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 Techniquesin Analysis, Springer, New York, 1995.I would also like to thank Evarist Gin´e who pointed out a number of errors.
Probabilistic Techniquesin Analysis, Springer, New York, 1995.I would also like to thank Evarist Gin´e who pointed out a number of errors.
, Springer, New York, 1995.I would also like to thank Evarist Gin´e who pointed out a number of errors.
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