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Mathematics for Finance: An Introduction to Financial Engineering

Marek Capiński and Tomasz Zastawniak
Publication Date: 
Number of Pages: 
Springer Undergraduate Mathematics Series
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The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
David Huckaby
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There is a lot of good stuff packed into this book. As the authors point out in the preface, undergraduate textbooks rarely feature two Nobel Prize winning theories.

The reader should know enough calculus to handle partial derivatives and Lagrange multipliers, enough linear algebra to invert matrices and compute determinants, and enough probability to understand random variables, distributions, expectation, and variance and covariance. A short appendix provides a quick review of most of this.

Assuming these modest preliminaries, the authors show how to price stocks, bonds, and derivatives. Starting with such basics as simple and compound interest, readers are led — in less than 300 pages — to understand some of the big, overarching ideas of financial engineering. The authors show to price options and other derivatives using Black-Scholes. They cover Markowitz portfolio optimization and the Capital Asset Pricing Model. Along the way, readers will become adept at applying the no-arbitrage principle, an assumption stated on page five that serves as a powerful tool in many arguments throughout the book.

A glance at chapter 1 provides a feel for the exposition. In a mere twenty-four pages the authors cover the basics of stocks, bonds, forward contracts, options, and foreign exchange. On the very first page the notation S(T) is introduced for the price of a stock at (final) time T. Return is then defined as K= [S(T) – S(0)]/S(0). For the price of a bond at time T, the notation A(T) is used. An investor having x shares of stock and y bonds is said to have the portfolio (x, y), its value V(T) at time T clearly given by V(T) = xS(T) + yA(T). This simple framework is a satisfying way to codify some basics while getting the discussion started.

Later in the chapter, when option pricing is being discussed, the following result appears: “If the option can be replicated by investing in a portfolio (x, y) of stock and bonds, then C(0) = xS(0) + yA(0), or else an arbitrage opportunity would exist.” Note that C(T) is the price of the option at time T, and that replicating an option means constructing a portfolio (x,y) such that C(T) = xS(T) + yA(T). The context here is a binomial model with one time step after which the stock price S(T) will be one of two possible values. The result is proved the same way as many others in the book: assume a contradiction, and then devise an arbitrage opportunity via a sequence of buying and selling different securities. Working through such a proof, the novice acquires a better feel for the properties of the various securities and how they interrelate.

Throughout the text, the authors invite active reader participation. One way is by opening and closing each chapter with a case study. The portion of the case study at the start of the chapter is a description of a financial scenario. A mere one paragraph, it contains no mention of potential problems or solutions. After reading the paragraph, the exceptionally astute reader will anticipate issues that might arise from the scenario and will seek out resolution while studying the chapter. When the case study is revisited at chapter’s end, potential problems are disclosed and their solutions discussed in light of the tools and ideas learned in the chapter.

To further promote active learning, the authors have embedded all of the exercises in the discussion. Many of the exercises closely resemble immediately preceding examples and hence provide practice and serve to cement the ideas. Others require more effort, such as providing a proof or derivation. Solutions to all exercises appear in an appendix. This makes the book excellent for self-study.

The exposition is very clear, yet in most places compact (for an undergraduate text). One example of the book’s conciseness and expectation of reader maturity: Near the start of chapter three, the formula for return K = [S(T) – S(0)]/S(0) is redisplayed. The authors next mention E(K) = μ as notation for expected return and then state, “By the linearity of the mathematical expectation, μ = [E(S(T)) – S(0)]/S(0).” The reader with experience in probability or linear algebra will have no problem following this discussion; a reader without the background will find this a mysterious leap. Needless to say, the level of sophistication demanded is much higher by the time the reader arrives at continuous time models in chapter eight and finds pages full of integral signs and partial derivatives.

The authors excel at situating results and at conveying essentials. For example, prior to stating and proving the first fundamental theorem of mathematical finance, they characterize the theorem as a generalization of earlier results that gave equivalent conditions to no arbitrage. Thus the reader is explicitly alerted to the result’s importance and its connection to previous material. Frequently the authors indicate when one quantity is independent of others. For example, after developing the formula to price a European call option under a binomial model for the underlying stock (whose price moves up or down with probability p and 1 – p, respectively), they remark that the formula is independent of p, which is unknown, and instead uses the risk-free probability, which is easily computable.

Though compact, the book features plenty of examples, and tables and graphs enhance the discussion. For instance, when the authors want to motivate the concept of duration of a bond, they provide a long example investigating the values of 3-year bonds (with annual coupons) under various fluctuations in interest rates over the 3-year period. The tables supporting this example take up a page and a half, and a brief paragraph points out the important features of each table. With this example thoroughly analyzed, the authors ask what would happen if all annual coupons were a fixed dollar amount. The perhaps surprising result motivates the definition of duration.

Some of the reviewer’s favorite parts of the book feature graphs. Three examples: 1) While treating a trinomial model, the authors provide discussion and plots of two very nice geometric interpretations of the risk-neutral probabilities. 2) The discussion of convexity of the option price as a function of the underlying stock is well done, and the accompanying graphs drive home the point. 3) In the discussion of portfolio management in chapter three, the various graphs of the μσ plane are one of the highlights of the text.

If studied slowly, this book provides an excellent introduction to financial engineering. For such a short text, the authors display impressive dexterity in ushering the reader from basics to an understanding of some of the deepest and most far-reaching ideas in the discipline.

David A. Huckaby  is an associate professor of mathematics at Angelo State University.