Measure, Probability, and Mathematical Finance

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

Part I Measure Theory
Chapter 1 Sets And Sequences
Chapter 2 Measures
Chapter 3 Extension Of Measures
Chapter 4 Lebesgue-Stielt Jes Measures
Chapter 5 Measurable Functions
Chapter 6 Lebesgue Integration
Chapter 7 The Radon-Nikodym Theorem
Chapter 8 Lp Spaces
Chapter 9 Convergence
Chapter 10 Product Measures

Part II Probability Theory
Chapter 11 Events And Random Variables
Chapter 12 Independence
Chapter 13 Expectation
Chapter 14 Conditional Expectation
Chapter 15 Inequalities
Chapter 16 Law Of Large Numbers
Chapter 17 Characteristic Functions
Chapter 18 Discrete Distributions
Chapter 19 Continuous Distributions
Chapter 20 Central Limit Theorems

Part III Stochastic Processes
Chapter 21 Stochastic Processes
Chapter 22 Martingales
Chapter 23 Stopping Times
Chapter 24 Martingale Inequalities
Chapter 25 Martingale Convergence Theorems
Chapter 26 Random Walks
Chapter 27 Poisson Processes
Chapter 28 Brownian Motion
Chapter 29 Markov Processes
Chapter 30 Levy Processes

Part IV Stochastic Calculus
Chapter 31 The Wiener Integral
Chapter 32 The Ito Integral
Chapter 33 Extension Of The Ito Integral
Chapter 34 Martingale Stochastic Integrals
Chapter 35 The Ito Formula
Chapter 36 Martingale Representation Theorem
Chapter 37 Change Of Measure
Chapter 38 Stochastic Differential Equations
Chapter 39 Diffusion
Chapter 40 The Feynman-Kac Formula

Part V Stochastic Financial Models
Chapter 41 Discrete-Time Models
Chapter 42 Black-Scholes Option Pricing Models
Chapter 43 Path-Dependent Options
Chapter 44 American Options
Chapter 45 Short Rate Models
Chapter 46 Instantaneous Forward Ratemodels
Chapter 47 Libor Market Models