1973 was a key year in the development of financial mathematics. The Chicago Board Options Exchange was founded and at the same time Fischer Black and Myron Scholes published the paper, “The Pricing of Options and Corporate Liabilities” [1], while Robert Merton published the article, “Theory of Rational Option Pricing” [2].

These works contained a new methodology for the valuation of derivatives (financial instruments whose payoffs depend upon the value of other instruments) and developed the famous Black-Scholes pricing formulae for calls and puts.

In 1997 the Royal Swedish Academy of Sciences awarded the Bank of Sweden Prize in Economic Sciences in Memory of Alfred Nobel (the Nobel Prize for Economics) to Merton and Scholes “for a new method to determine the value of derivatives” (Black had passed away some years before).

Black, Merton and Scholes’ framework rests upon two main pillars. The first is the modelling of stock prices with geometric Brownian motion (as Mandelbrojt and Nobel Prize winner Samuelson had done some years before). The second, and here lies their contribution, is a principle of equilibrium: the absence of arbitrage opportunities (and the corresponding replicating portfolio arguments). Nowadays, their arguments might sound natural, yet it is worth noting that Black and Scholes’ paper, which has become one of the most cited papers in the scientific literature of this field, was rejected by some prestigious economics journals for some years.

The history of derivatives is not as recent a topic as one might think. There are references to the use of these financial contracts (mainly futures) throughout history. It is said that Thales of Miletus made a fortune trading with the rights of use of oil mills. In the 17th century there was a speculative boom in the tulip futures market in Amsterdam. However, the first mathematical attempt to model the financial markets was made in 1900, in the doctoral thesis “Théorie de la Spéculation” by Louis Bachelier, one of Poincaré’s students.

Black, Merton and Scholes’ ideas and valuation formulae were immediately understood and accepted by the markets. This is an amazing example of vertiginous transference of knowledge between the academic world and the ‘real’ world; as they allowed risks to be translated into prices and different derivatives to be compared, the formulae actually worked!

Since then, the range of applications of stochastic processes and partial differential equations in finance (modelling and designing of financial instruments, pricing, hedging, risk management, etc.) has been increasing day by day. Thus, it has become a standard topic in undergraduate courses in mathematics and economics. The book, “Mathematics for Finance: An Introduction to Financial Engineering”, is a textbook for an introductory course on three basic questions: the non-arbitrage option pricing theory, the Markowitz portfolio theory and the modelling of interest rates.

The book

The book is divided into eleven chapters. The first four serve as an initial approach to financial markets. The first chapter describes some of the basic hypotheses, illustrated with the one-step binomial model. A detailed account of riskless assets (time value of the money, bonds, etc.) is included in the second chapter. The third is devoted to the dynamics of risky assets, with particular emphasis on the binomial tree model, although the trinomial tree model is also analysed and a brief discussion on the continuous time limit is included as well. The fourth chapter ties together the previous concepts in the general framework of discrete models. Questions relating to portfolio management (Markowitz theory, efficient frontiers, CAPM) are treated in detail in the fifth chapter. Chapters six and seven are dedicated to explain some basic characteristics of derivatives such as futures, forward contracts and options.

Chapter eight deals with option pricing. The Cox-Ross-Rubinstein binomial model is used to price European and American options. A brief outline of the arguments that lead to the Black-Scholes formula is also included. Issues dealing with the use of derivatives in risk management are described in the ninth chapter: hedging, risk measures as value-at-risk (VaR), speculating strategies with options, etc. These questions are treated by making use of a case study approach. Finally, chapters ten and eleven are devoted to interest rates including term structure, the binomial model and a brief account of interest rates derivatives (swaps, caps, floors).

A textbook on financial mathematics like the one we are reviewing unavoidably faces a dilemma, namely the balance between mathematical rigor and financial concepts and their practical implementation.

In most chapters, Capiński and Zastawniak’s work succeeds in moving away from some advanced mathematical language, such as stochastic calculus and partial differential equations, and focuses on discrete time models, so that only basic notions of calculus, linear algebra and probability are needed to read it. At the same time, despite its mathematically formal prose, the text is very readable. The relevant concepts are introduced gradually; often these concepts are preceded by numerical examples and in the end they are all given their corresponding formal definitions. Almost all the main results are accompanied by corresponding proofs. A good deal of worked examples and remarks elucidate the associated financial concepts. Each chapter contains a collection of useful exercises (interspersed in the text). Some of the quite detailed solutions are included in an appendix. The reader will find other solutions (in Excel files) on the web page www-users.york.ac.uk/~tz506/m4f/index.html (see also www.springeronline.com/1-85233-330-8). Some typos are listed on the aforementioned web page.

Thus, we can say that the book strikes a balance between mathematical rigor and practical application and we recommend it as a textbook for an introductory course on financial mathematics.