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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…mehr
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.
Inhaltsangabe
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
