**Graduate Studies in Mathematics**

Volume: 72;
2006;
150 pp;
Hardcover

MSC: Primary 60; 62; 91;
Secondary 58

**Print ISBN: 978-0-8218-3903-4
Product Code: GSM/72**

List Price: $45.00

AMS Member Price: $36.00

MAA Member Price: $40.50

**Electronic ISBN: 978-1-4704-1805-2
Product Code: GSM/72.E**

List Price: $42.00

AMS Member Price: $33.60

MAA Member Price: $37.80

#### Supplemental Materials

# Introduction to the Mathematics of Finance

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*R. J. Williams*

The modern subject of mathematical finance has undergone
considerable development, both in theory and practice, since the seminal work
of Black and Scholes appeared a third of a century ago. This book is intended
as an introduction to some elements of the theory that will enable students and
researchers to go on to read more advanced texts and research papers.

The book begins with the development of the basic ideas of hedging and
pricing of European and American derivatives in the discrete (i.e., discrete
time and discrete state) setting of binomial tree models. Then a general
discrete finite market model is introduced, and the fundamental theorems of
asset pricing are proved in this setting. Tools from probability such as
conditional expectation, filtration, (super)martingale, equivalent martingale
measure, and martingale representation are all used first in this simple
discrete framework. This provides a bridge to the continuous (time and state)
setting, which requires the additional concepts of Brownian motion and
stochastic calculus. The simplest model in the continuous setting is the famous
Black-Scholes model, for which pricing and hedging of European and American
derivatives are developed. The book concludes with a description of the
fundamental theorems for a continuous market model that generalizes the simple
Black-Scholes model in several directions.

#### Readership

Graduate students interested in financial mathematics.

#### Reviews & Endorsements

This monograph gives a far-reaching and easily readable advanced introduction to the mathematical modelling of the absence of riskless financial profits, as well as to the connected topic of pricing and risk-protecting-replication/hedging of securities whose value depend on an underlying asset. ...The book's style is pragmatic, precise, concise, with smoothly and fast increasing technical level including the quotation of mathematical subtleties.

-- Wolfgang Stummer

The text is clearly written and well-arranged and most of the results are proved in detail. Each chapter is completed with exercises, which makes the textbook very comprehensive.

-- EMS Newsletter

#### Table of Contents

# Table of Contents

## Introduction to the Mathematics of Finance

- Cover Cover11 free
- Title page iii4 free
- Contents v6 free
- Preface vii8 free
- Financial markets and derivatives 110 free
- Binomial model 716
- Finite market model 3140
- Black-Scholes model 5564
- Multi-dimensional Black-Scholes model 8998
- Conditional expectation and 𝐿^{𝑝}-spaces 123132
- Discrete time stochastic processes 127136
- Continuous time stochastic processes 131140
- Brownian motion and stochastic integration 135144
- Bibliography 145154
- Index 149158 free
- Back Cover Back Cover1162