Introduction to the Mathematics of Finance
About this Title
R. J. Williams, University of California, San Diego, La Jolla, CA
Publication: Graduate Studies in Mathematics
Publication Year 2006: Volume 72
ISBNs: 978-0-8218-3903-4 (print); 978-1-4704-1805-2 (online)
MathSciNet review: MR2218734
MSC: Primary 91-01; Secondary 60G44, 60H10, 60H30, 91B28
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.
Graduate students interested in financial mathematics.
Table of Contents
- Chapter 1. Financial markets and derivatives
- Chapter 2. Binomial model
- Chapter 3. Finite market model
- Chapter 4. Black-Scholes model
- Chapter 5. Multi-dimensional Black-Scholes model
- Appendix A. Conditional expectation and $L^p$-spaces
- Appendix B. Discrete time stochastic processes
- Appendix C. Continuous time stochastic processes
- Appendix D. Brownian motion and stochastic integration