The main purpose of *An Introduction to Mathematical Finance with Applications* by Arlie O. Petters and Xiaoying Dong is to bridge the gap between books that give a theoretical treatment without many applications and books that present and apply formulas without deriving them. As pointed out in the preface, given the intricacies and interconnectedness of financial firms future financial engineers need to have a more diverse knowledge base of mathematical financial models. In dealing with challenging and complex modern securities, more sophisticated mathematical tools are needed, which is creating a knowledge gap between the qualitative and quantitative approaches to the world of finance.

As this is an advanced text, the reader should have a solid understanding of multivariable calculus, probability, and linear algebra. Along the way, the student will also need to know measure theory and binomial trees (page 213), which are introduced from scratch in the book. The beauty of the book is that no financial background is needed. The book is organized into four main parts:

- Chapters 1–2: Introduction to securities markets and the time value of money.
- Chapters 3–4: Markowitz portfolio theory, capital market theory, and portfolio risk measures.
- Chapters 5–6: Modeling underlying securities using binomial trees and stochastic calculus.
- Chapters 7–8: Derivative securities, BSM module, and Merton jump-diffusion module.

The topics in the book were tested with undergraduate and masters-bound students and can be used in a one semester or one-year course.

My background is not in finance, so I was very interested, intrigued, and eager to pick up this book and to learn new applications. I was not disappointed. The book is easy to follow and the topics are clearly explained in detail. From the very first example on page 3, which talks about how the fractional reserve banking system works to establish an intuition of the concepts of money, credit/debt, and leverage, the use of a geometric series is used to find that the original $10,000 could generate $100,000 by using \[ 10\,000\sum_{n=0}^\infty (90\%)^n =\frac{10\,000}{1-0.9} = 100\,000.\] The various formulas for interest rates are all derived in the book. For example, the basic formulas of simple interest, compound interest, and continuous compound interest covered in a traditional College Algebra course are covered in full. As an example, from the derived compound interest formula on page 29, \(F_{kt}=\left(1+\frac{r}{k}\right)^{kt} F_0\), we have the derived formula for continuously compounding interest on page 31, \[ F_{cts} = \lim_{k\to\infty} F_{kt}= F_0\lim_{k\to \infty} \left(1+\frac{r}{k}\right)^{kt} = F_0e^{rt},\] so that the return rate is \(R(\tau)=e^{r\tau}-1\), where \(\tau\) is time in years.

Anyone who has taught a basic financial mathematics course knows how hard it is to justify the formulas and definitions. Present and Future Value of a Simple Ordinary Annuity is one of those topics. Pages 49–52 present a clear derivation and justification of these critical formulas. Each section has a set of two types of exercises for the students, conceptual exercises and application exercises. These problems challenge the student to make connections between the topics covered in a given section so they are properly trained to think not only about the mathematics they need for finance, but the strategies needed to be effective financial engineers. One such question I found to be of particular interest was in Chapter 3, on page 143: Can you find five examples of pairs of stocks in the USA that are negatively correlated? Are such occurrences common? Not only this is a good group project question, I see it as a spin off point to a deeper research project a professor can do with a single or group of students.

Towards the later chapters of the book, the Itô Integral, The Geometric Brownian Motion, Forward, Futures, Swaps, Options, and the S&P Index Daily Log Returns, more complex applications and theory, are introduced. Each of these topics involves knowledge of integration, differentiation, probability, finding partial derivatives, and the Fundamental Theorem of Calculus. They are applications that prepare students for the financial arena and give them a strong preview of what is to come once employed. On page 301, the famous example of calculating the integral \[\int_0^t B_s\, dB_s \] of Brownian motion is presented; it uses the same principle as in calculus: dividing up an interval \([0, t]\) and using Riemann sums. Studying geometric Brownian motion requires knowledge of stochastic processes, differential equations, and probability. Forward, futures, swaps, and options use derivatives, and the S&P Index Daily Log Returns involve the normal distribution bell curve.

Overall, the book is thick with rich with applications and solid justifications of all concepts. Even if you are a pure mathematician, you can use this book for application problems in a calculus, linear algebra, probability, or differential equations class for further enrichment for your students. If you are a financial mathematics instructor, this book is for you. It progresses through easier topics to more advanced topics very well and is practical, meaningful, and more importantly, relevant to the 21st century financial student. I strongly believe this book can be used as the one standard book for your class but can also be a springboard to research projects with your more advanced and curious students. In short, not considering this book would be a great disservice to you and your students.

Peter Olszewski is a Mathematics Lecturer at The Pennsylvania State University, The Behrend College, an editor for Larson Texts, Inc. in Erie, PA, and is the 362nd Chapter Advisor of the Pennsylvania Alpha Beta Chapter of Pi Mu Epsilon. His Research fields are in mathematics education, Cayley Color Graphs, Markov Chains, and mathematical textbooks. He can be reached at pto2@psu.edu. Outside of teaching and textbook editing, he enjoys playing golf, playing guitar and bass, reading, gardening, traveling, and painting landscapes.