Introduction to Financial Mathematics
Introduction to Financial Mathematics is ideal for an introductory undergraduate course. Unlike most textbooks aimed at more advanced courses, the text motivates students through a discussion of personal finances and portfolio management. The author then goes on to cover valuation of financial derivatives in discrete time, using all of closed form, recursive, and simulation methods.
The text covers nearly all of the syllabus topics of the Financial Mathematics Actuarial examination, providing students with the foundation they require for future studies and throughout their careers. It begins by covering standard material on the mathematics of interest, including compound interest, present value, annuities, loans, several versions of the rate of return on an investment, and interest in continuous time.
The text explains how to value bonds at their issue dates, at coupon times, between coupon times, and in cases where the bonds are terminated early. Next, it supplies a rapid-fire overview of the main ideas and techniques of discrete probability, including sample spaces and probability measures, random variables and distributions, expectation, conditional probability, and independence.
The author introduces the basic terminology of stocks and stock trading. He also explains how to derive the rate of return on a portfolio and how to use the idea of risk aversion to model the investor tradeoff between risk and return. The text also discusses the estimation of parameters of asset models from real data.
The text closes with a detailed discussion of how to value financial derivatives using anti-arbitrage assumptions. The one-step and multi-step cases are covered, and exotic options such as barrier options are also introduced, to which simulation methods are applied.
Many of the examples in the book involve numerical solution of complicated non-linear equations; others ask students to produce algorithms which beg to be implemented as programs. For maximum flexibility, the author has produced the text without adhering to any particular computational platform.
A digital version of this text is also available in the form of Mathematica notebooks that contain additional content.
Theory of Interest
Rate of Return and Present Value
Measuring Rate of Return
Continuous Time Interest Theory
More on Bonds
Term Structure of Interest Rates
Discrete Probability for Finance
Sample Spaces and Probability Measures
Random Variables and Distributions
Independence and Dependence
Estimation and Simulation
Portfolios of Risky Assets
Optimal Portfolio Selection
Valuation of Derivatives
Basic Terminology and Ideas
Valuation of Exotic Options and Simulation
Appendix: Answers to Selected Exercises
Hastings (mathematics, Knox College) perceived a need for an introductory text to financial mathematics, and the result is an excellent book that superbly fits that niche. This clearly written work serves as a bridge to more advanced texts, such as the second edition of Capiński and Zastawniak's Mathematics for Finance: An Introduction to Financial Engineering (CH, Jun'11, 48-5740). In addition to its clear explanations, this volume emphasizes real problem solving with examples and exercises that challenge students to apply knowledge of basic concepts to new situations. Another unique aspect is the application of discrete probability to finance; the author provides an overview and illustrates problems in which the rates of interest are random variables, instead of traditional problems that consider only known constants. Topics covered include the mathematics of interest, valuation of bonds, discrete probability for finance, portfolio selection, and derivatives. Though the problems in this book can be solved with an advanced calculator, the author suggests using a computational platform, such as Mathematica. There is an electronic version of this text, which can be obtained as a Mathematica notebook. This book is highly recommended for undergraduates and those preparing for actuarial credentialing and exams.
--S. J. Chapman Jr., Purdue University NorthWest