Introduction to Quantitative Finance

A Math Tool Kit

by Reitano

ISBN: 9780262333139 | Copyright 2010

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This text offers an accessible yet rigorous development of many of the fields of mathematics necessary for success in investment and quantitative finance, covering topics applicable to portfolio theory, investment banking, option pricing, investment, and insurance risk management. The approach emphasizes the mathematical framework provided by each mathematical discipline, and the application of each framework to the solution of finance problems. It emphasizes the thought process and mathematical approach taken to develop each result instead of the memorization of formulas to be applied (or misapplied) automatically. The objective is to provide a deep level of understanding of the relevant mathematical theory and tools that can then be effectively used in practice, to teach students how to "think in mathematics" rather than simply to do mathematics by rote.Each chapter covers an area of mathematics such as mathematical logic, Euclidean and other spaces, set theory and topology, sequences and series, probability theory, and calculus, in each case presenting only material that is most important and relevant for quantitative finance. Each chapter includes finance applications that demonstrate the relevance of the material presented. Problem sets are offered on both the mathematical theory and the finance applications sections of each chapter. The logical organization of the book and the judicious selection of topics make the text customizable for a number of courses. The development is self-contained and carefully explained to support disciplined independent study as well. A solutions manual for students provides solutions to the book's Practice Exercises; an instructor's manual offers solutions to the Assignment Exercises as well as other materials.
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Contents (pg. vii)
List of Figures and Tables (pg. xix)
Introduction (pg. xxi)
1 Mathematical Logic (pg. 1)
1.1 Introduction (pg. 1)
1.2 Axiomatic Theory (pg. 4)
1.3 Inferences (pg. 6)
1.4 Paradoxes (pg. 7)
1.5 Propositional Logic (pg. 10)
1.6 Mathematical Logic (pg. 23)
1.7 Applications to Finance (pg. 24)
2 Number Systems and Functions (pg. 31)
2.1 Numbers: Properties and Structures (pg. 31)
2.2 Functions (pg. 49)
2.3 Applications to Finance (pg. 51)
3 Euclidean and Other Spaces (pg. 71)
3.1 Euclidean Space (pg. 71)
3.2 Metric Spaces (pg. 82)
3.3 Applications to Finance (pg. 93)
4 Set Theory and Topology (pg. 117)
4.1 Set Theory (pg. 117)
4.2 Open, Closed, and Other Sets (pg. 122)
4.3 Applications to Finance (pg. 134)
5 Sequences and Their Convergence (pg. 145)
5.1 Numerical Sequences (pg. 145)
5.2 Limits Superior and Inferior (pg. 152)
5.3 General Metric Space Sequences (pg. 157)
5.4 Cauchy Sequences (pg. 162)
5.5 Applications to Finance (pg. 167)
6 Series and Their Convergence (pg. 177)
6.1 Numerical Series (pg. 177)
6.2 The lp-Spaces (pg. 196)
6.3 Power Series (pg. 206)
6.4 Applications to Finance (pg. 215)
7 Discrete Probability Theory (pg. 231)
7.1 The Notion of Randomness (pg. 231)
7.2 Sample Spaces (pg. 233)
7.3 Combinatorics (pg. 247)
7.4 Random Variables (pg. 252)
7.5 Expectations of Discrete Distributions (pg. 264)
7.6 Discrete Probability Density Functions (pg. 287)
7.7 Generating Random Samples (pg. 301)
7.8 Applications to Finance (pg. 307)
8 Fundamental Probability Theorems (pg. 347)
8.1 Uniqueness of the m.g.f. and c.f. (pg. 347)
8.2 Chebyshev’s Inequality (pg. 349)
8.3 Weak Law of Large Numbers (pg. 352)
8.4 Strong Law of Large Numbers (pg. 357)
8.5 De Moivre–Laplace Theorem (pg. 368)
8.6 The Normal Distribution (pg. 377)
8.7 The Central Limit Theorem (pg. 381)
8.8 Applications to Finance (pg. 386)
9 Calculus I: Differentiation (pg. 417)
9.1 Approximating Smooth Functions (pg. 417)
9.2 Functions and Continuity (pg. 418)
9.3 Derivatives and Taylor Series (pg. 450)
9.4 Convergence of a Sequence of Derivatives (pg. 478)
9.5 Critical Point Analysis (pg. 488)
9.6 Concave and Convex Functions (pg. 494)
9.7 Approximating Derivatives (pg. 504)
9.8 Applications to Finance (pg. 505)
10 Calculus II: Integration (pg. 559)
10.1 Summing Smooth Functions (pg. 559)
10.2 Riemann Integration of Functions (pg. 560)
10.3 Examples of the Riemann Integral (pg. 574)
10.4 Mean Value Theorem for Integrals (pg. 579)
10.5 Integrals and Derivatives (pg. 581)
10.6 Improper Integrals (pg. 587)
10.7 Formulaic Integration Tricks (pg. 592)
10.8 Taylor Series with Integral Remainder (pg. 598)
10.9 Convergence of a Sequence of Integrals (pg. 602)
10.10 Numerical Integration (pg. 609)
10.11 Continuous Probability Theory (pg. 613)
10.12 Applications to Finance (pg. 641)
References (pg. 685)
Index (pg. 689)
Contents (pg. vii)
List of Figures and Tables (pg. xix)
Introduction (pg. xxi)
1 Mathematical Logic (pg. 1)
1.1 Introduction (pg. 1)
1.2 Axiomatic Theory (pg. 4)
1.3 Inferences (pg. 6)
1.4 Paradoxes (pg. 7)
1.5 Propositional Logic (pg. 10)
1.6 Mathematical Logic (pg. 23)
1.7 Applications to Finance (pg. 24)
2 Number Systems and Functions (pg. 31)
2.1 Numbers: Properties and Structures (pg. 31)
2.2 Functions (pg. 49)
2.3 Applications to Finance (pg. 51)
3 Euclidean and Other Spaces (pg. 71)
3.1 Euclidean Space (pg. 71)
3.2 Metric Spaces (pg. 82)
3.3 Applications to Finance (pg. 93)
4 Set Theory and Topology (pg. 117)
4.1 Set Theory (pg. 117)
4.2 Open, Closed, and Other Sets (pg. 122)
4.3 Applications to Finance (pg. 134)
5 Sequences and Their Convergence (pg. 145)
5.1 Numerical Sequences (pg. 145)
5.2 Limits Superior and Inferior (pg. 152)
5.3 General Metric Space Sequences (pg. 157)
5.4 Cauchy Sequences (pg. 162)
5.5 Applications to Finance (pg. 167)
6 Series and Their Convergence (pg. 177)
6.1 Numerical Series (pg. 177)
6.2 The lp-Spaces (pg. 196)
6.3 Power Series (pg. 206)
6.4 Applications to Finance (pg. 215)
7 Discrete Probability Theory (pg. 231)
7.1 The Notion of Randomness (pg. 231)
7.2 Sample Spaces (pg. 233)
7.3 Combinatorics (pg. 247)
7.4 Random Variables (pg. 252)
7.5 Expectations of Discrete Distributions (pg. 264)
7.6 Discrete Probability Density Functions (pg. 287)
7.7 Generating Random Samples (pg. 301)
7.8 Applications to Finance (pg. 307)
8 Fundamental Probability Theorems (pg. 347)
8.1 Uniqueness of the m.g.f. and c.f. (pg. 347)
8.2 Chebyshev’s Inequality (pg. 349)
8.3 Weak Law of Large Numbers (pg. 352)
8.4 Strong Law of Large Numbers (pg. 357)
8.5 De Moivre–Laplace Theorem (pg. 368)
8.6 The Normal Distribution (pg. 377)
8.7 The Central Limit Theorem (pg. 381)
8.8 Applications to Finance (pg. 386)
9 Calculus I: Differentiation (pg. 417)
9.1 Approximating Smooth Functions (pg. 417)
9.2 Functions and Continuity (pg. 418)
9.3 Derivatives and Taylor Series (pg. 450)
9.4 Convergence of a Sequence of Derivatives (pg. 478)
9.5 Critical Point Analysis (pg. 488)
9.6 Concave and Convex Functions (pg. 494)
9.7 Approximating Derivatives (pg. 504)
9.8 Applications to Finance (pg. 505)
10 Calculus II: Integration (pg. 559)
10.1 Summing Smooth Functions (pg. 559)
10.2 Riemann Integration of Functions (pg. 560)
10.3 Examples of the Riemann Integral (pg. 574)
10.4 Mean Value Theorem for Integrals (pg. 579)
10.5 Integrals and Derivatives (pg. 581)
10.6 Improper Integrals (pg. 587)
10.7 Formulaic Integration Tricks (pg. 592)
10.8 Taylor Series with Integral Remainder (pg. 598)
10.9 Convergence of a Sequence of Integrals (pg. 602)
10.10 Numerical Integration (pg. 609)
10.11 Continuous Probability Theory (pg. 613)
10.12 Applications to Finance (pg. 641)
References (pg. 685)
Index (pg. 689)
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