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An elementary introduction to mathematical finance. 3rd ed. (English) Zbl 1221.91001
Cambridge: Cambridge University Press (ISBN 978-0-521-19253-8/hbk; 978-1-139-06510-8/ebook). xv, 305 p. (2011).
The book under review was first published in 1999 and a second edition in 2003; for the reviews of the previous editions see Zbl 0944.91024 and Zbl 1113.91025.
The beginning of the mathematics in modern finance lies in Louis Bachelier’s 1900 dissertation on speculation theory [L. Bachelier, Théorie de la spéculation. Ann. de l’Éc. Norm. (3) 17, 21–86. Paris: Gauthier-Villars (1900; JFM 31.0241.02)]. Bachelier’s work initiated the continuous-time stochastic processes in mathematics and the continuous-time derivative security pricing in economy. Kiyoshi Itô, influenced by Bachelier’s work, developed stochastic calculus that is the core of mathematical finance now. Before the 1950s, finance theory was more or less a collection of anecdotes, rules of thumb and accounting shuffling of data. By the early 1970s, finance theory in academics was considerably more involved with mathematics and statistics, many models of dynamic portfolio theory, intertemporal capital asset pricing, derivative security pricing with stochastic differential and integral equations, stochastic dynamic programming, game theory and stochastic optimization were adapted to financial research and empirical analysis. The milestone in terms of finance practice was the Black-Scholes model for option pricing [F. Black and M. Scholes, “The pricing of options and corporate liabilities”, J. Polit. Econ. 81, 637–654 (1973; Zbl 1092.91524)] and R. C. Merton’s model [“Theory of rational option pricing”, Bell J. Econ. 4, 141–183 (1973)]. By coincidence, the Chicago Board Options Exchange opened on April 26, 1973, with only 16 stocks traded on call options. In 1997, both Merton and Scholes won the Nobel Prize in Economics for their work on option pricing. Black was ineligible for the prize, having died in 1995. The Black-Scholes-Merton option pricing theory has enriched the finance, finance practice and mathematical finance. The comparison of the option pricing formulas between Black-Scholes-Merton and Bachelier was done by W. Schachermayer and J. Teichmann [“How close are the option pricing formulas of Bachelier and Black-Merton-Scholes?”, Math. Finance 18, No. 1, 155–170 (2008; Zbl 1138.91479)].
The book offers a clear presentation of the arbitrage theorem and the Black-Scholes option pricing formula without assuming much prior knowledge on mathematical and financial backgrounds. The book is accessible for both financial practicioners and undergraduates studying finance theory, and is a very good textbook to introduce these topics for students. The author provides many examples and exercises in each chapter in order to help the readers to digest the material. A few references are given in the end of each chapter for further study.
The first three chapters of the book give the basic mathematics on probability, random variables and Brownian motions, the next three chapters give the basic finance on interest rates, present value analysis and the arbitrage theorem (the first fundamental theorem of finance). The Chapters 7 and 8 give the essential part of mathematical finance, the Black-Scholes formulas and their variations and extensions. The author explains the Black-Scholes formula in detail and its Greek letters. Chapter 9 explains the valuing investment by expected utility, the portfolio selection problem and the capital asset pricing model. Chapter 10 is new in this edition, presents the first and second order stochastic dominance as well as likelihood ratio orderings. Chapter 11 on optimization models gives both deterministic optimization models and probabilistic optimization problems. Chapter 12 on stochastic dynamic programming is new in this edition, and shows optimal stopping problems in an interesting example. The last three chapters introduce Monte Carlo simulation and efficient estimators, models for the crude oil data as well as autoregressive models and mean reversion. Using this explicit example on the crude oil price, the author shows that not all security price data are consistent with the assumption that its price history follows a geometric Brownian motion. This relates to E. F. Fama’s efficient market hypothesis [“The behavior of stock market prices”, J. Business 38, 34–105 (1965)]. The predictability of future behavior of stock prices is a challenging puzzle that many financial economists attempted to solve.

The book is well-written in explaining basic concepts and deriving formulas. The Black-Scholes formula and its derivatives are derived elegantly. It would be nice to have further explanations on the Greeks found in Section 7.5 in hedging and risk management and the Black-Scholes-Merton equation. Chapter 9 is closely related to the Rubinstein theory on option pricing. Overall, the book is well-organized and an excellent textbook for a course.

91-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to game theory, economics, and finance
91Gxx Actuarial science and mathematical finance
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