Introduction to the Mathematics of Finance: Arbitrage and Option Pricing

This quick tour of option pricing is concise and well-written. It begins with the basics of options, spends most of its time with discrete-time models, and then culminates in the Black-Scholes formula.

As the author points out in the preface, the mathematics is not watered down but kept at a level appropriate for the intended audience, namely advanced undergraduates and beginning graduate students. For example, throughout the discussion of discrete-time models, students will need to be comfortable with an abundance of structures (sets, sets of sets, functions defined on these, etc.) along with the blizzard of symbols denoting them. In this longest portion of the text, students will not require a command of many deep mathematical results, but they will almost certainly need to have had experience with mathematical abstraction, such as comes from, say, a fairly rigorous undergraduate probability course.

The biggest strength of the book, especially in view of the intended audience, is its attention to pedagogy. Authors who attempt to cover a lot of ground in relatively few pages often end up producing a mere compendium of results. This book avoids that danger. The author has crafted something that is more a tutorial than a reference. Many results are paraphrased either before or after they are formally stated and the examples peppered throughout the text are extremely helpful. So, although a definition-theorem-proof framework still prevails, there is enough connective tissue to aid the student in following the discussion.

Here is a taste of the author’s approach. When defining what it means for a stochastic process to be adapted to a filtration, he first motivates the definition with a very simple example. Then he states the definition formally. Immediately after follow two short paragraphs. The first begins, “Thus, in loose terms,…” Here the author provides an informal restatement of the definition, one of his many paraphrases of definitions, equations, and theorems. The second paragraph reads, “We strongly urge the reader to ponder this concept until it seems very clear before continuing. To help put this notion in perspective, we define a related notion that will be important in the next chapter.” (He next defines what it means for a stochastic process to be predictable.)

The second quote above hints at another of the book’s strengths: Much thought has been put into how topics are ordered. Let the structure of Chapter 1 serve as an example. The chapter begins by describing some basic properties of options, information that will be welcomed by finance novices. Next follows a discussion of payoffs for calls and puts, complete with graphs showing payoff curves. Next, time premium is introduced, illustrated with graphs of option premium curves. One more graph then allows a concise yet intuitive introduction to delta: it is simply the slope of the tangent line to the option premium curve. The text flows well and is at just the right level for a first chapter. Later chapters are more mathematically challenging, of course, but the order of topics is still carefully crafted.

A quick summary of the book: After Chapter 1 there follow three chapters on arbitrage, discrete probability, and stochastic processes. Chapter 5 then uses all of this to discuss discrete-time pricing models. The examples toward the end of the chapter are extremely helpful, especially since the risk-free interest rate is assumed to be zero. (My only quibble here concerns the example on p. 127, in which the author uses a simpler notation than is being used in the rest of the chapter, a chapter especially replete with structures and notation. Granted, the simpler notation is a good choice for the clarity of the example per se. But it would be nice to at least remind the reader what the more complicated symbols are, especially since getting used  to the notation constitutes a significant part of the learning curve in this portion of the book.) The discussion of the binomial model in Chapter 6 allows students to apply much of the work of Chapter 5 in a useful yet relatively simple model. It will be used later to derive the Black-Scholes formula.

Chapters 7 and 8 are more difficult. Until now, the mathematical sophistication has consisted primarily in the patience required to become comfortable with the abundance of structures, functions, and symbols. Chapter 7, however, on pricing nonattainable alternatives in an incomplete market, ratchets up the level with a discussion of linear functionals. The chapter is almost devoid of examples. Chapter 8, on stopping times and American options, at least features a binary model example that is repeatedly referenced.

Many students will find Chapter 9, on continuous probability, easier than much of the earlier material on discrete-time models. They will have already covered most of the material in a probability course, and there is a substantially smaller quantity of symbols to juggle. The book culminates in Chapter 10, on the Black-Scholes formula. The author admirably refrains from putting the formula on too high a pedestal, pointing out some imperfections, including its normality assumption. The author is clearly having fun in this capstone chapter, as indicated by a little joking and a more personal tone here and there. The light-heartedness befits the end of the journey.

The book is largely self-contained, with the necessary probability explained as needed. (Nevertheless, prior exposure to probability is recommended.) Two appendices include the more advanced mathematics needed in two of the chapters. There are 150 end-of-chapter exercises, of which 30 have solutions provided. (Over a third of the exercises are in the two chapters on discrete probability and continuous probability.) A prefatory chapter succinctly yet deftly leads from the basics of what an option is to the No-arbitrage Pricing Principle and the idea of a replicating portfolio — all in a few pages. This preview of some of the book’s main ideas provides motivation for students who will labor slowly through later portions of the text.

Intrepid advanced undergraduates with no prior knowledge of the subject matter can work their way through the book, perhaps omitting Chapters 7 and 8. They will find the text demanding, but not unreasonably so, especially if they are interested in studying the mathematics of finance at the graduate level, where the mathematical sophistication will be greater than in this book. (For example, measure theory does not appear at all in this text.)

Perhaps, however, the most satisfied readers will already have had at least a little exposure to the material, either from the mathematical or financial side. The excellent development of the topics and the frequent paraphrasing will provide such readers several “ah-ha” moments.

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