Measure, Probability, and Mathematical Finance

PREFACE

Mathematical finance, a new branch of mathematics concerned with financial markets, is experiencing rapid growth. During the last three decades, many books and papers in the area of mathematical finance have been published. However, understanding the literature requires that the reader have a good background in measure-theoretic probability, stochastic processes, and stochastic calculus. The purpose of this book is to provide the reader with an introduction to the mathematical theory underlying the financial models being used and developed on Wall Street. To this end, this book covers important concepts and results in measure theory, probability theory, stochastic processes, and stochastic calculus so that the reader will be in a position to understand these financial models. Problems as well as solutions are included to help the reader learn the concepts and results quickly.

In this book, we adopted the definitions and theorems from various books and presented them in a mathematically rigorous way. We tried to cover the most of the basic concepts and the important theorems. We selected the problems in this book in such a way that the problems will help readers understand and know how to apply the concepts and theorems. This book includes 516 problems, most of which are not difficult and can be solved by applying the definitions, theorems, and the results of previous problems.

This book is organized into five parts, each of which is further organized into several chapters. Each chapter is divided into five sections. The first section presents the definitions of important concepts and theorems. The second, third, and fourth sections present the problems, hints on how to solve the problems, and the full solutions to the problems, respectively. The last section contains bibliographic notes. Interdependencies between all chapters are shown in Table 0.1.

Table 0.1: Interdependencies between Chapters.

Chapter Related to Chapter(s) 1. Sets and Sequences   2. Measures 1 3. Extension of Measures 1;2 4. Lebesgue-Stieltjes Measures 2;3 5. Measurable Functions 2 6. Lebesgue Integration 1;2;5 7. The Radon-Nikodym Theorem 2;6 8. Lp Spaces 2;6 9. Convergence 1;2;6;8 10. Product Measures 2;3;5;6 11. Events and Random Variables 1;2;4;5 12. Independence 2;3;5;11 13. Expectation 2;6;8;10;11;12 14. Conditional Expectation 1;2;5;6;7;8;10;11;12;13 15. Inequalities 8;11;14 16. Law of Large Numbers 2;8;9;10;12;13;15 17. Characteristic Functions 5;6;8;11;12;13;15 18. Discrete Distributions 12;14;17 19. Continuous Distributions 6;10;12;13;17 20. Central Limit Theorems 6;9;11 21. Stochastic Processes 2;5;10;11;12;19 22. Martingales 2;5;11;13;14;15 23. Stopping Times 2;5;9;11;14;21;22 24. Martingale Inequalities 2;6;8;13;14;15;23 25. Martingale Convergence Theorems 1;6;9;11;14;15;22 26. Random Walks 8;9;13;14;15;19;20;22;23;24 27. Poisson Processes 11;12;14;17;21;22 28. Brownian Motion 8;9;11;12;14;15;16;17;19 29. Markov Processes 2;6;11;14;21 30. Lévy Processes 1;5;6;11;12;14;17;19;22;27;28;29 31. The Wiener Integral 6;9;15;19;28 32. The Itô Integral 5;6;8;10;14;15;22;24;28 33. Extension of the Itô Integrals 9;10;14;22;23;32 34. Martingale Stochastic Integrals 14;15;19;27;32 35. The Itô Formula 6;8;9;22;24;32;34 36. Martingale Representation Theorem 9;14;25;28;32;33;35 37. Change of Measure 7;14;32;34;35 38. Stochastic Differential Equations 8;11;13;32;34;35 39. Diffusion 6;9;11;14;19;21;24;32;35;38 40. The Feynman-Kac Formula 6;14;32;35;38;39 41. Discrete-Time Models 7;12;14;22;23 42. Black-Scholes Option Pricing Models 9;14;19;24;32;33;35;36;37;38;41 43. Path-Dependent Options 10; 14;19;28;37;38;42 44. American Options 14; 15;21;22;23;32;35;36;37;42;43 45. Short Rate Models 11; 14;19;29;32;35;37;38;39;40 46. Instantaneous Forward Rate Models 10; 14;19;32;34;35;37;38;40;45 47. LIBOR Market Models 14; 32;37;45;46

In Part I, we present measure theory, which is indispensable to the rigorous development of probability theory. Measure theory is also necessary for us to discuss recently developed theories and models in finance, such as the martingale measures, the change of numeraire theory, and the London interbank offered rate (LIBOR) market models.

In Part II, we present probability theory in a measure-theoretic mathematical framework, which was introduced by A.N. Kolmogorov in 1937 in order to deal with David Hilbert’s sixth problem. The material presented in this part was selected to facilitate the development of stochastic processes in Part III.

In Part III, we present stochastic processes, which include martingales and Brownian motion. In Part IV, we discuss stochastic calculus. Both stochastic processes and stochastic calculus are important to modern mathematical finance as they are used to model asset prices and develop derivative pricing models.

In Part V, we present some classic models in mathematical finance. Many pricing models have been developed and published since the seminal work of Black and Scholes. This part covers only a small portion of many models.

In this book, we tried to use a uniform set of symbols and notation. For example, we used N, R, and to denote the set of natural numbers (i.e., nonnegative integers), the set of real numbers, and the empty set, respectively. A comprehensive list of symbols is also provided at the end of this book.

We have taken great pains to ensure the accuracy of the formulas and statements in this book. However, a few errors are inevitable in almost every book of this size. Please feel free to contact us if you spot errors or have any other constructive suggestions.

How to Use This Book

This book can be used by individuals in various ways:

(a) It can be used as a self-study book on mathematical finance. The prerequisite is linear algebra and calculus at the undergraduate level. This book will provide you with a series of concepts, facts, and problems. You should explore each problem and write out your solution in such a way that it can be shared with others. By doing this you will be able to actively develop an in-depth and comprehensive understanding of the concepts and principles that cannot be archived by passively reading or listening to comments of others. (b) It can be used as a reference book. This book contains the most important concepts and theorems from mathematical finance. The reader can find the definition of a concept or the statement of a theorem in the book through the index at the end of this book. (c) It can be used as a supplementary book for individuals who take advanced courses in mathematical finance. This book starts with measure theory and builds up to stochastic financial models. It provides necessary prerequisites for students who take advanced courses in mathematical finance without completing background courses.

Acknowledgments

We would like to thank all the academics and practitioners who have contributed to the knowledge of mathematical finance. In particular, we would like to thank the following academics and practitioners whose work constitutes the backbone of this book: Robert B. Ash, Krishna B. Athreya, Rabi Bhattacharya, Patrick Billingsley, Tomas Björk, Fischer Sheffey Black, Kai Lai Chung, Erhan Çinlar, Catherine A. Doléans-Dade, Darrell Duffie, Richard Durrett, Robert J. Elliott, Damir Filipović, Allan Gut, John Hull, Ioannis Karatzas, Fima C. Klebaner, P. Ekkehard Kopp, Hui-Hsiung Kuo, Soumendra N. Lahiri, Damien Lamberton, Bernard Lapeyre, Gregory F. Lawler, Robert C. Merton, Marek Musiela, Bernt Oksendal, Andrea Pascucci, Jeffrey S. Rosenthal, Sheldon M. Ross, Marek Rutkowski, Myron Scholes, Steven Shreve,...