The goal of this textbook is to provide a mathematically rigorous, yet essentially self-contained introduction to option pricing based on arbitrage theory. The book is aimed at advanced undergraduate and beginning graduate students of economics, finance, and mathematics, and no background knowledge of probability theory is assumed. The key mathematical prerequisite is a solid grasp of introductory-level linear algebra.
The first six chapters of the book develop arbitrage pricing theory in discrete-time models on finite probability spaces and introduce the necessary probabilistic concepts (such as probability spaces, martingales, and filtrations). The discussion begins from the basic properties of equity options and culminates to the proofs of the first and second fundamental theorems of asset pricing. The binomial model is then analyzed in detail, setting the stage for the asymptotic derivation of the Black-Scholes formula in later chapters. Chapter 7 touches upon asset pricing in incomplete market models and Chapter 8 provides a concise introduction to optimal stopping and American options. Finally, the remaining two chapters discuss briefly probability theory on general probability spaces and establish the convergence of the binomial model to the Black-Scholes model. These chapters are, for obvious reasons, more informal than the rest of the book. The first edition of the book was not available to the reviewer, but according to the author, this second edition is completely rewritten and the list of covered topics has been changed slightly.
The pedagogical approach employed in this book, which entails building the arbitrage pricing theory without any probabilistic prerequisites, is rather unique, but somewhat similar to the one taken by [D. Sondermann, Introduction to stochastic calculus for finance. A new didactic approach. Lecture Notes in Economics and Mathematical Systems 579. Berlin: Springer. (2006; Zbl 1136.91014)] in the context of continuous-time models. It is commendable that, in contrast to many conventional finance textbooks, in this book the theory is not developed at the expense of mathematical rigor, but instead by carefully limiting the mathematical apparatus to the bare essentials.
Students who are already familiar with measure-theoretic probability might also want to consider the textbook by [H. Föllmer and A. Schied, Stochastic finance. An introduction in discrete time. 3rd revised and extended ed. de Gruyter Graduate. Berlin: de Gruyter. (2011; Zbl 1213.91006)], which has a similar scope, but develops the theory at a more advanced level.