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Measure, Probability, and Mathematical Finance: A Problem-Oriented Approach

Book Description

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.

    Table of Contents

    1. Cover
    2. Half Title page
    3. Title page
    4. Copyright page
    5. Dedication
    6. Preface
    7. Financial Glossary
    8. Part I: Measure Theory
      1. Chapter 1: Sets and Sequences
        1. 1.1 Basic Concepts and Facts
        2. 1.2 Problems
        3. 1.3 Hints
        4. 1.4 Solutions
        5. 1.5 Bibliographic Notes
      2. Chapter 2: Measures
        1. 2.1 Basic Concepts and Facts
        2. 2.2 Problems
        3. 2.3 Hints
        4. 2.4 Solutions
        5. 2.5 Bibliographic Notes
      3. Chapter 3: Extension of Measures
        1. 3.1 Basic Concepts and Facts
        2. 3.2 Problems
        3. 3.3 Hints
        4. 3.4 Solutions
        5. 3.5 Bibliographic Notes
      4. Chapter 4: Lebesgue-Stieltjes Measures
        1. 4.1 Basic Concepts and Facts
        2. 4.2 Problems
        3. 4.3 Hints
        4. 4.4 Solutions
        5. 4.5 Bibliographic Notes
      5. Chapter 5: Measurable Functions
        1. 5.1 Basic Concepts and Facts
        2. 5.2 Problems
        3. 5.3 Hints
        4. 5.4 Solutions
        5. 5.5 Bibliographic Notes
      6. Chapter 6: Lebesgue Integration
        1. 6.1 Basic Concepts and Facts
        2. 6.2 Problems
        3. 6.3 Hints
        4. 6.4 Solutions
        5. 6.5 Bibliographic Notes
      7. Chapter 7: The Radon-Nikodym Theorem
        1. 7.1 Basic Concepts and Facts
        2. 7.2 Problems
        3. 7.3 Hints
        4. 7.4 Solutions
        5. 7.5 Bibliographic Notes
      8. Chapter 8: LP Spaces
        1. 8.1 Basic Concepts and Facts
        2. 8.2 Problems
        3. 8.3 Hints
        4. 8.4 Solutions
        5. 8.5 Bibliographic Notes
      9. Chapter 9: Convergence
        1. 9.1 Basic Concepts and Facts
        2. 9.2 Problems
        3. 9.3 Hints
        4. 9.4 Solutions
        5. 9.5 Bibliographic Notes
      10. Chapter 10: Product Measures
        1. 10.1 Basic Concepts and Facts
        2. 10.2 Problems
        3. 10.3 Hints
        4. 10.4 Solutions
        5. 10.5 Bibliographic Notes
    9. Part II: Probability Theory
      1. Chapter 11: Events and Random Variables
        1. 11.1 Basic Concepts and Facts
        2. 11.2 Problems
        3. 11.3 Hints
        4. 11.4 Solutions
        5. 11.5 Bibliographic Notes
      2. Chapter 12: Independence
        1. 12.1 Basic Concepts and Facts
        2. 12.2 Problems
        3. 12.3 Hints
        4. 12.4 Solutions
        5. 12.5 Bibliographic Notes
      3. Chapter 13: Expectation
        1. 13.1 Basic Concepts and Facts
        2. 13.2 Problems
        3. 13.3 Hints
        4. 13.4 Solutions
        5. 13.5 Bibliographic Notes
      4. Chapter 14: Conditional Expectation
        1. 14.1 Basic Concepts and Facts
        2. 14.2 Problems
        3. 14.3 Hints
        4. 14.4 Solutions
        5. 14.5 Bibliographic Notes
      5. Chapter 15: Inequalities
        1. 15.1 Basic Concepts and Facts
        2. 15.2 Problems
        3. 15.3 Hints
        4. 15.4 Solutions
        5. 15.5 Bibliographic Notes
      6. Chapter 16: Law of Large Numbers
        1. 16.1 Basic Concepts and Facts
        2. 16.2 Problems
        3. 16.3 Hints
        4. 16.4 Solutions
        5. 16.5 Bibliographic Notes
      7. Chapter 17: Characteristic Functions
        1. 17.1 Basic Concepts and Facts
        2. 17.2 Problems
        3. 17.3 Hints
        4. 17.4 Solutions
        5. 17.5 Bibliographic Notes
      8. Chapter 18: Discrete Distributions
        1. 18.1 Basic Concepts and Facts
        2. 18.2 Problems
        3. 18.3 Hints
        4. 18.4 Solutions
        5. 18.5 Bibliographic Notes
      9. Chapter 19: Continuous Distributions
        1. 19.1 Basic Concepts and Facts
        2. 19.2 Problems
        3. 19.3 Hints
        4. 19.4 Solutions
        5. 19.5 Bibliographic Notes
      10. Chapter 20: Central Limit Theorems
        1. 20.1 Basic Concepts and Facts
        2. 20.2 Problems
        3. 20.3 Hints
        4. 20.4 Solutions
        5. 20.5 Bibliographic Notes
    10. Part III: Stochastic Processes
      1. Chapter 21: Stochastic Processes
        1. 21.1 Basic Concepts and Facts
        2. 21.2 Problems
        3. 21.3 Hints
        4. 21.4 Solutions
        5. 21.5 Bibliographic Notes
      2. Chapter 22: Martingales
        1. 22.1 Basic Concepts and Facts
        2. 22.2 Problems
        3. 22.3 Hints
        4. 22.4 Solutions
        5. 22.5 Bibliographic Notes
      3. Chapter 23: Stopping Times
        1. 23.1 Basic Concepts and Facts
        2. 23.2 Problems
        3. 23.3 Hints
        4. 23.4 Solutions
        5. 23.5 Bibliographic Notes
      4. Chapter 24: Martingale Inequalities
        1. 24.1 Basic Concepts and Facts
        2. 24.2 Problems
        3. 24.3 Hints
        4. 24.4 Solutions
        5. 24.5 Bibliographic Notes
      5. Chapter 25: Martingale Convergence Theorems
        1. 25.1 Basic Concepts and Facts
        2. 25.2 Problems
        3. 25.3 Hints
        4. 25.4 Solutions
        5. 25.5 Bibliographic Notes
      6. Chapter 26: Random Walks
        1. 26.1 Basic Concepts and Facts
        2. 26.2 Problems
        3. 26.3 Hints
        4. 26.4 Solutions
        5. 26.5 Bibliographic Notes
      7. Chapter 27: Poisson Processes
        1. 27.1 Basic Concepts and Facts
        2. 27.2 Problems
        3. 27.3 Hints
        4. 27.4 Solutions
        5. 27.5 Bibliographic Notes
      8. Chapter 28: Brownian Motion
        1. 28.1 Basic Concepts and Facts
        2. 28.2 Problems
        3. 28.3 Hints
        4. 28.4 Solutions
        5. 28.5 Bibliographic Notes
      9. Chapter 29: Markov Processes
        1. 29.1 Basic Concepts and Facts
        2. 29.2 Problems
        3. 29.3 Hints
        4. 29.4 Solutions
        5. 29.5 Bibliographic Notes
      10. Chapter 30: Lévy Processes
        1. 30.1 Basic Concepts and Facts
        2. 30.2 Problems
        3. 30.3 Hints
        4. 30.4 Solutions
        5. 30.5 Bibliographic Notes
    11. Part IV: Stochastic Calculus
      1. Chapter 31: The Wiener Integral
        1. 31.1 Basic Concepts and Facts
        2. 31.2 Problems
        3. 31.3 Hints
        4. 31.4 Solutions
        5. 31.5 Bibliographic Notes
      2. Chapter 32: The Itô Integral
        1. 32.1 Basic Concepts and Facts
        2. 32.2 Problems
        3. 32.3 Hints
        4. 32.4 Solutions
        5. 32.5 Bibliographic Notes
      3. Chapter 33: Extension of the Itô Integral
        1. 33.1 Basic Concepts and Facts
        2. 33.2 Problems
        3. 33.3 Hints
        4. 33.4 Solutions
        5. 33.5 Bibliographic Notes
      4. Chapter 34: Martingale Stochastic Integrals
        1. 34.1 Basic Concepts and Facts
        2. 34.2 Problems
        3. 34.3 Hints
        4. 34.4 Solutions
        5. 34.5 Bibliographic Notes
      5. Chapter 35: The Itô Formula
        1. 35.1 Basic Concepts and Facts
        2. 35.2 Problems
        3. 35.3 Hints
        4. 35.4 Solutions
        5. 35.5 Bibliographic Notes
      6. Chapter 36: Martingale Representation Theorem
        1. 36.1 Basic Concepts and Facts
        2. 36.2 Problems
        3. 36.3 Hints
        4. 36.4 Solutions
        5. 36.5 Bibliographic Notes
      7. Chapter 37: Change of Measure
        1. 37.1 Basic Concepts and Facts
        2. 37.2 Problems
        3. 37.3 Hints
        4. 37.4 Solutions
        5. 37.5 Bibliographic Notes
      8. Chapter 38: Stochastic Differential Equations
        1. 38.1 Basic Concepts and Facts
        2. 38.2 Problems
        3. 38.3 Hints
        4. 38.4 Solutions
        5. 38.5 Bibliographic Notes
      9. Chapter 39: Diffusion
        1. 39.1 Basic Concepts and Facts
        2. 39.2 Problems
        3. 39.3 Hints
        4. 39.4 Solutions
        5. 39.5 Bibliographic Notes
      10. Chapter 40: The Feynman-Kac Formula
        1. 40.1 Basic Concepts and Facts
        2. 40.2 Problems
        3. 40.3 Hints
        4. 40.4 Solutions
        5. 40.5 Bibliographic Notes
    12. Part V: Stochastic Financial Models
      1. Chapter 41: Discrete-Time Models
        1. 41.1 Basic Concepts and Facts
        2. 41.2 Problems
        3. 41.3 Hints
        4. 41.4 Solutions
        5. 41.5 Bibliographic Notes
      2. Chapter 42: Black-Scholes Option Pricing Models
        1. 42.1 Basic Concepts and Facts
        2. 42.2 Problems
        3. 42.3 Hints
        4. 42.4 Solutions
        5. 42.5 Bibliographic Notes
      3. Chapter 43: Path-Dependent Options
        1. 43.1 Basic Concepts and Facts
        2. 43.2 Problems
        3. 43.3 Hints
        4. 43.4 Solutions
        5. 43.5 Bibliographic Notes
      4. Chapter 44: American Options
        1. 44.1 Basic Concepts and Facts
        2. 44.2 Problems
        3. 44.3 Hints
        4. 44.4 Solutions
        5. 44.5 Bibliographic Notes
      5. Chapter 45: Short Rate Models
        1. 45.1 Basic Concepts and Facts
        2. 45.2 Problems
        3. 45.3 Hints
        4. 45.4 Solutions
        5. 45.5 Bibliographic Notes
      6. Chapter 46: Instantaneous Forward Rate Models
        1. 46.1 Basic Concepts and Facts
        2. 46.2 Problems
        3. 46.3 Hints
        4. 46.4 Solutions
        5. 46.5 Bibliographic Notes
      7. Chapter 47: Libor Market Models
        1. 47.1 Basic Concepts and Facts
        2. 47.2 Problems
        3. 47.3 Hints
        4. 47.4 Solutions
        5. 47.5 Bibliographic Notes
    13. References
    14. List of Symbols
    15. Subject Index