An introduction to the mathematical theory and financial models developed and used on Wall Street

Providing both a theoretical and practical approach to the underlying mathematical theory behind financial models, Measure, Probability, and Mathematical Finance: A Problem-Oriented Approach presents important concepts and results in measure theory, probability theory, stochastic processes, and stochastic calculus. Measure theory is indispensable to the rigorous development of probability theory and is also necessary to properly address martingale measures, the change of numeraire theory, and LIBOR market models. In addition, probability theory is presented to facilitate the development of stochastic processes, including martingales and Brownian motions, while stochastic processes and stochastic calculus are discussed to model asset prices and develop derivative pricing models.

The authors promote a problem-solving approach when applying mathematics in real-world situations, and readers are encouraged to address theorems and problems with mathematical rigor. In addition, Measure, Probability, and Mathematical Finance features:

• A comprehensive list of concepts and theorems from measure theory, probability theory, stochastic processes, and stochastic calculus


• Over 500 problems with hints and select solutions to reinforce basic concepts and important theorems


• Classic derivative pricing models in mathematical finance that have been developed and published since the seminal work of Black and Scholes

Measure, Probability, and Mathematical Finance: A Problem-Oriented Approach is an ideal textbook for introductory quantitative courses in business, economics, and mathematical finance at the upper-undergraduate and graduate levels. The book is also a useful reference for readers who need to build their mathematical skills in order to better understand the mathematical theory of derivative pricing models.

GUOJUN GAN, PHD, ASA, is Director of Quantitative Modeling and Model Efficiency at Manulife Financial, Canada. His research interests include empirical corporate finance, actuarial science, risk management, data mining, and big data analysis.

CHAOQUN MA, PHD, is Professor and Dean of the School of Business Administration at Hunan University, China. The recipient of First Prize in Outstanding Achievements in Teaching in 2009, Dr. Ma's research interests include financial engineering, risk management, and data mining.

HONG XIE, PHD, is Adjunct Professor in the Department of Mathematics and Statistics at York University as well as Vice President of Models and Analytics at Manulife Financial, Canada. Dr. Xie is on the Board of Directors for the Canadian-Chinese Finance Association, and his research interests include financial engineering, mathematical finance, and partial differential equations.

Preface xvii

Financial Glossary xxii

Part I Measure Theory

1 Sets and Sequences 3

2 Measures 15

3 Extension of Measures 29

4 Lebesgue-Stieltjes Measures 37

5 Measurable Functions 47

6 Lebesgue Integration 57

7 The Radon-Nikodym Theorem 77

8 L^P Spaces 85

9 Convergence 97

10 Product Measures 113

Part II Probability Theory

11 Events and Random Variables 127

12 Independence 141

13 Expectation 161

14 Conditional Expectation 173

15 Inequalities 189

16 Law of Large Numbers 199

17 Characteristic Functions 217

18 Discrete Distributions 227

19 Continuous Distributions 239

20 Central Limit Theorems 257

Part III Stochastic Processes

21 Stochastic Processes 271

22 Martingales 291

23 Stopping Times 301

24 Martingale Inequalities 321

25 Martingale Convergence Theorems 333

26 Random Walks 343

27 Poisson Processes 357

28 Brownian Motion 373

29 Markov Processes 389

30 Lévy Processes 401

Part IV Stochastic Calculus

31 The Wiener Integral 421

32 The Itô Integral 431

33 Extension of the Itô Integral 453

34 Martingale Stochastic Integrals 463

35 The Itô Formula 477

36 Martingale Representation Theorem 495

37 Change of Measure 503

38 Stochastic Differential Equations 515

39 Diffusion 531

40 The Feynman-Kac Formula 547

Part V Stochastic Financial Models

41 Discrete-Time Models 561

42 Black-Scholes Option Pricing Models 579

43 Path-Dependent Options 593

44 American Options 609

45 Short Rate Models 629

46 Instantaneous Forward Rate Models 647

47 LIBOR Market Models 667

References 687

List of Symbols 703

Subject Index 707