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Inroduction to the Mathematics of Finance

R. J. Williams
American Mathematical Society
Publication Date: 
Number of Pages: 
Graduate Studies in Mathematics 72
[Reviewed by
Ita Cirovic Donev
, on

Introduction to the Mathematics of Finance is yet another "introduction" to the subject of pricing financial derivatives. There are now quite a few of books that will teach you the basics of pricing the European and American style derivatives, some more technical than others. This book can be considered as an introduction to the subject by means of mathematical proficiency, as it requires a good knowledge of graduate probability theory and stochastic processes. However, it can by no means be considered an introduction from the point of view of the subject of finance and financial derivatives. The author points you to other references on the second page. The explanation of what derivatives are is given in two pages with one example. This constitutes chapter 1. Chapter 2 begins discrete time modeling and binomial models.  Fundamental Theorem of Asset Pricing is presented in chapter 3.  The rest of the book deals with continuous time modeling and the famous Black-Scholes model. Overall, the book gives a good presentation of the pricing of financial derivatives, going from the discrete to the continuous and presenting the most famous continuous model.

The writing of the book is at the level of a graduate probability/stochastic processes/measure theory course, i.e. the results are given in theorem-proof style. The proofs are easy to follow. Exercises are given at the end of each chapter (even the first one). Naturally, most exercises are proof based but there are also some practical exercises. Personally, I would not use this book as the sole text for a class, but perhaps as an additional reference.  I see it more as a "study group" text on pricing financial derivatives and from that point of view it is excellent.

I agree with the author that the book should serve well as an introduction and a starting point to reading some more advanced texts. However, I would still add that it requires some familiarity with the theory of finance, i.e., financial derivatives should be understood prior to reading this book. The results will just be more clear.  I would recommend reading sectons of J. Hull's book on Options, Futures and Other Derivatives or Cvitanic's and Zapatero's Introduction to the Economics and Mathematics of Financial Markets before attempting Williams' book.

Ita Cirovic Donev is a PhD candidate at the University of Zagreb. She hold a Masters degree in statistics from Rice University. Her main research areas are in mathematical finance; more precisely, statistical mehods of credit and market risk. Apart from the academic work she does consulting work for financial institutions.

Preface vii

Chapter 1. Financial Markets and Derivatives 1

1.1. Financial Markets 1

1.2. Derivatives 2

1.3. Exercise 5

Chapter 2. Binomial Model 7

2.1. Binomial or CRR Model 7

2.2. Pricing a European Contingent Claim 10

2.3. Pricing an American Contingent Claim 19

2.4. Exercises 28

Chapter 3. Finite Market Model 31

3.1. Definition of the Finite Market Model 32

3.2. First Fundamental Theorem of Asset Pricing 34

3.3. Second Fundamental Theorem of Asset Pricing 39

3.4. Pricing European Contingent Claims 44

3.5. Incomplete Markets 47

3.6. Separating Hyperplane Theorem 51

3.7. Exercises 52

Chapter 4. Black-Scholes Model 55

4.1. Preliminaries 56

4.2. Black-Scholes Model 57


vi Contents

4.3. Equivalent Martingale Measure 61

4.4. European Contingent Claims 63

4.5. Pricing European Contingent Claims 65

4.6. European Call Option — Black-Scholes Formula 69

4.7. American Contingent Claims 74

4.8. American Call Option 80

4.9. American Put Option 83

4.10. Exercises 86

Chapter 5. Multi-dimensional Black-Scholes Model 89

5.1. Preliminaries 91

5.2. Multi-dimensional Black-Scholes Model 92

5.3. First Fundamental Theorem of Asset Pricing 99

5.4. Form of Equivalent Local Martingale Measures 101

5.5. Second Fundamental Theorem of Asset Pricing 110

5.6. Pricing European Contingent Claims 116

5.7. Incomplete Markets 120

5.8. Exercises 121

Appendix A. Conditional Expectation and Lp-Spaces 123

Appendix B. Discrete Time Stochastic Processes 127

Appendix C. Continuous Time Stochastic Processes 131

Appendix D. Brownian Motion and Stochastic Integration 135

D.1. Brownian Motion 135

D.2. Stochastic Integrals (with respect to Brownian motion) 136

D.3. Itˆo Process 139

D.4. Itˆo Formula 141

D.5. Girsanov Transformation 142

D.6. Martingale Representation Theorem 143

Bibliography 145

Index 149