Introduction to Quantitative Finance A Math Tool Kit

by Reitano, Robert R.

• ISBN13: 9780262013697
• ISBN10: 026201369X
• Format: Hardcover
• Copyright: 2010-01-29
• Publisher: The MIT Press

Summary

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.

Author Biography

Robert R. Reitano is Professor of the Practice in Finance at Brandeis University's International Business School. He was formerly Executive Vice President and Chief Investment Strategist of John Hancock/Manulife.

Table of Contents

Complete Table of Contents
List of Figures and Tables	p. xix
Introduction	p. xxi
Mathematical Logic	p. 1
Introduction	p. 1
Axiomatic Theory	p. 4
Inferences	p. 6
Paradoxes	p. 7
Propositional Logic	p. 10
Mathematical Logic	p. 23
Applications to Finance	p. 24
Exercises	p. 27
Number Systems and Functions	p. 31
Numbers: Properties and Structures	p. 31
Functions	p. 49
Applications to Finance	p. 51
Exercises	p. 64
Euclidean and Other Spaces	p. 71
Euclidean Space	p. 71
Metric Spaces	p. 82
Applications to Finance	p. 93
Exercises	p. 112
Set Theory and Topology	p. 117
Set Theory	p. 117
Open, Closed, and Other Sets	p. 122
Applications to Finance	p. 134
Exercises	p. 139
Sequences and Their Convergence	p. 145
Numerical Sequences	p. 145
Limits Superior and Inferior	p. 152
General Metric Space Sequences	p. 157
Cauchy Sequences	p. 162
Applications to Finance	p. 167
Exercises	p. 172
Series and Their Convergence	p. 177
Numerical Series	p. 177
The lp-Spaces	p. 196
Power Series	p. 206
Applications to Finance	p. 215
Exercises	p. 224
Discrete Probability Theory	p. 231
The Notion of Randomness	p. 231
Sample Spaces	p. 233
Combinatorics	p. 247
Random Variables	p. 252
Expectations of Discrete Distributions	p. 264
Discrete Probability Density Functions	p. 287
Generating Random Samples	p. 301
Applications to Finance	p. 307
Exercises	p. 337
Fundamental Probability Theorems	p. 347
Uniqueness of the m.g.f. and c.f.	p. 347
Chebyshev's Inequality	p. 349
Weak Law of Large Numbers	p. 352
Strong Law of Large Numbers	p. 357
De Moivre-Laplace Theorem	p. 368
The Normal Distribution	p. 377
The Central Limit Theorem	p. 381
Applications to Finance	p. 386
Exercises	p. 411
Calculus I: Differentiation	p. 417
Approximating Smooth Functions	p. 417
Functions and Continuity	p. 418
Derivatives and Taylor Series	p. 450
Convergence of a Sequence of Derivatives	p. 478
Critical Point Analysis	p. 488
Concave and Convex Functions	p. 494
Approximating Derivatives	p. 504
Applications to Finance	p. 505
Exercises	p. 549
Calculus II: Integration	p. 559
Summing Smooth Functions	p. 559
Riemann Integration of Functions	p. 560
Examples of the Riemann Integral	p. 574
Mean Value Theorem for Integrals	p. 579
Integrals and Derivatives	p. 581
Improper Integrals	p. 587
Formulaic Integration Tricks	p. 592
Taylor Series with Integral Remainder	p. 598
Convergence of a Sequence of Integrals	p. 602
Numerical Integration	p. 609
Continuous Probability Theory	p. 613
Applications to Finance	p. 641
Exercises	p. 675
References	p. 685
Index	p. 689
Table of Contents provided by Publisher. All Rights Reserved.