**Graduate Studies in Mathematics**

Volume: 70;
2013;
305 pp;
Hardcover

MSC: Primary 60; 91;
**Print ISBN: 978-0-8218-9490-3
Product Code: GSM/70.R**

List Price: $65.00

Individual Member Price: $52.00

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# Probability Theory in Finance: A Mathematical Guide to the Black-Scholes Formula, Second Edition

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*Seán Dineen*

The use of the Black-Scholes model and formula is pervasive in financial markets. There are very few undergraduate textbooks available on the subject and, until now, almost none written by mathematicians. Based on a course given by the author, the goal of this book is to introduce advanced undergraduates and beginning graduate students studying the mathematics of finance to the Black-Scholes formula. The author uses a first-principles approach, developing only the minimum background necessary to justify mathematical concepts and placing mathematical developments in context.

The book skillfully draws the reader toward the art of thinking mathematically and then proceeds to lay the foundations in analysis and probability theory underlying modern financial mathematics. It rigorously reveals the mathematical secrets of topics such as abstract measure theory, conditional expectations, martingales, Wiener processes, the Itô calculus, and other ingredients of the Black-Scholes formula. In explaining these topics, the author uses examples drawn from the universe of finance. The book also contains many exercises, some included to clarify simple points of exposition, others to introduce new ideas and techniques, and a few containing relatively deep mathematical results.

The second edition contains numerous revisions and additional material designed to enhance the book's usability as a classroom text. These changes include insights gleaned by the author after teaching from the text, as well as comments and suggestions made by others who used the book. Whereas the revised edition maintains the original approach, format, and list of topics, most chapters are modified to some extent; in addition, the rearrangement of material resulted in a new chapter (Chapter 9).

With the modest prerequisite of a first course in calculus, the book is suitable for undergraduates and graduate students in mathematics, finance, and economics and can be read, using appropriate selections, at a number of levels.

#### Table of Contents

# Table of Contents

## Probability Theory in Finance: A Mathematical Guide to the Black-Scholes Formula, Second Edition

- Cover Cover11 free
- Title page iii4 free
- Contents v6 free
- Preface ix10 free
- Money and markets 116 free
- Fair games 1732
- Set theory 2944
- Measurable functions 5974
- Probability spaces 7792
- Expected values 107122
- Continuity and integrability 143158
- Conditional expectation 165180
- Lebesgue measure 189204
- Martingales 209224
- The Black-Scholes formula 227242
- Stochastic integration 243258
- Solutions 281296
- Bibliography 299314
- Index 301316 free
- Back Cover Back Cover1322

#### Readership

Undergraduate and graduate students interested in probability and financial mathematics.

#### Reviews

Dineen is doing something valuable by trying to find ways to communicate mathematics in a serious way to an audience that often gets little more than recipes and rules. It's a project definitely worth supporting.

-- MAA Reviews

In addition to the usual improvements in response to comments and suggestions, the new edition reflects the experience of teaching real analysis. ... Dineen is doing something valuable by trying to find ways to communicate mathematics in a serious way to an audience that often gets little more than recipes and rules. It's a project definitely worth supporting.

-- Fernando Q. Gouvêa, MAA Reviews

This book provides a thorough presentation of the mathematics behind the derivatives pricing formula. It is written for mathematicians as well as finance students or professionals ... very lucid ... an excellent choice for graduate students in mathematical finance as well as practitioners with the required background.

-- Ita Cirovic Donev, MAA Reviews

A neatly and lively written text that may be warmly recommended to undergraduates and graduate students in mathematics and finance.

-- EMS Newsletter

The author has done a very good job presenting the abstract probability theory needed to understand the derivation of the Black-Scholes formula, as well as the intuitive ideas, often simple, behind it. By starting with simple models in which the financial principles used can be clearly seen and then building smoothly toward the more complex model used in the Black-Scholes formula, the author allows the reader to appreciate the financial principles used even in the more complex settings. Furthermore, the exercises included after each chapter help the reader understand the material presented. ... Finally, the many historical footnotes often add insight and help bring the mathematics to life.

-- Errol Caby, AT&T Laboratories

... an excellent choice ... This book provides a thorough presentation of the mathematics behind the derivatives pricing formula. It is written for mathematicians as well as finance students or professionals.

-- MAA Reviews

... one of the author's aims is to entice the students by the beauty of mathematics, which he illustrates very well. If a course instructor teaching from the book succeeds in this, my guess is that the course will be a great success.

It outshines most of its competitors ... variety of nice exercises and solutions ...

-- Mathematical Reviews