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In the context of science and technology we are used to employing sophisticated mathematical methods. Even more so, many concepts owe their very existence to applications in these areas. And thankfully, this has the effect that the graduates of what is perhaps the most abstract of all sciences find enough employers who are willing to afford them a decent living in the real world.

Many business administration students are painfully aware of the fact that you should also have a certain affinity for numbers in the realm of finance and business. However, we are not usually talking of difficult maths here. This fact perhaps explains the hype surrounding the much more demanding area of modern mathematical finance, whose triumphal march began in the early 1970s, together with the establishment of derivatives exchanges. And even though a certain degree of disillusion has taken hold in the wake of various financial crises, sums that exceed Germany’s annual gross domestic product are still being turned over every day on the basis of complex mathematical models.

Louis Bachelier’s doctoral thesis from 1900 can rightly be viewed as the birth of both mathematical finance and Brownian motion. But it initially received little attention. Itō’s work, on the other hand, played a greater role in the further development. Although he regarded himself as a pure mathematician, his stochastic integral, conceived in seclusion in Japan, became indispensable for modern financial mathematics. It still amazes me how well prepared the thriving theory of stochastic processes proved to be for the upcoming requirements of financial market modelling. The reverse development, i.e. the stimulation of stochastic analysis by financial applications, only set in quite late. Ironically, it was largely driven by Soviet-socialised Eastern European mathematicians and their partly leftist Western peers who were rather critical of capitalism. Due to its enormous practical importance, modern mathematical finance found its way into the curriculum, which led to challenges of a completely different kind. P. A. Meyer, the founder of the Strasbourg school of the so-called general theory of stochastic processes in France, once complained that it takes a whole semester to introduce the stochastic integral.

If you do not want to invest this amount of time, it is a good idea to explain the basic concepts of financial mathematics in a finite discrete-time framework in which both stochastic integrals and abstract conditional expectations can be expressed by sums. This is the approach taken in the present textbook, where seven of the eight chapters are set in this framework. However, since in practice one cannot avoid the much more elegant, powerful and ultimately handier continuous-time models, the author devotes 70 pages to the latter at the end of the book.

One of the main topics of modern mathematical finance and this textbook in particular is the valuation and hedging of derivative securities. It is based on arbitrage theory, which in turn rests on the seemingly modest assumption that entirely risk-free profits cannot be realised in real financial markets. In so-called complete market models, this has astonishingly far-reaching and practically relevant consequences, which explains the success of the theory. Completeness here means that the random uncertainty of any contingent payment obligation in the future (say, an option, forward, or futures contract) can be offset entirely by dynamic trading in liquid securities (called hedging). In a sense, hedging plays the same role in finance as the law of large numbers does in the insurance business: it contributes to the general welfare by eliminating randomness that we fear as risk-averse beings. In discrete time only models of binomial type are complete, which is why they form the focus of the book. In this case, stock price changes in any single period can attain only two possible values. This is of course unrealistic but if noting else it simplifies both computations and illustrations. These models are used to explain the standard relevant concepts such as self-financing portfolios, arbitrage, European and American options, forwards, futures, risk-free valuation, hedging and martingale measures. The author also takes a brief excursion into trinomial models, touching on the subject of incompleteness. The continuous-time part is naturally devoted primarily to the Black-Scholes model, where the logarithmic stock price follows a Brownian motion. Indeed, this is by far the most natural choice in the very small class of complete models. Moreover, it allows for explicit results and it is used as a benchmark and starting point for more realistic models in theory and practice. Numerical aspects such as the use of Monte Carlo and in particular the control variate method are also discussed in the book.

The first thing to emphasise about the text is that it is aimed specifically at beginners who only have a rudimentary knowledge of mathematics and in particular probability theory. A large number of concrete examples, exercises and short Python programmes help you to get to grips with what you have learnt. Naturally, the author must reach his limits with such a project. As in ordinary real analysis, the calculus only becomes handy after moving from differences to differentials, from sums to integrals. Not all phenomena are easy to understand in a discrete-time framework. Nevertheless, the concept of the stochastic integral with respect to Brownian motion \(W\) is initially avoided in the continuous-time part of the textbook. The author applies the trick of expressing it by a simple Riemann integral, using the identity

$$ \int _{0}^{t} f(W_{s})dW_{s} = F(W_{t})-F(W_{0})-{\frac{1}{2}}\int _{0}^{t} f'(W_{s})ds $$

for \(f=F'\). This avoids a complex mathematical notion, but at the cost of making concrete models appear somewhat unmotivated.

In my opinion, also some economic concepts could be explained a little better or presented more clearly, for example the seemingly paradoxical fact that the fair price of an option does not depend on the trend of the underlying stock. Morally speaking one would expect the latter to have the greatest impact on the price. Some of the more complex models seem to be motivated by the honourable intention of explaining interest rate theory in a complete discrete-time setup. In my view, however, the resulting deterministic relationship between stock prices and random interest rate trends is not entirely convincing.

All in all, the author has produced an instructive textbook with minor weaknesses. The real challenge for the text, however, is the fact that there are already a large number of introductory books that approach the squaring of the circle in a similar way. In particular, it may be difficult to compete with the well established texts by Björk, Hull or Shreve [1–4]. The closest counterpart to Calogero’s textbook is probably part I of the two-volume introduction by Shreve, who is not only a proven and extremely meticulous expert, but also has many years of teaching experience in this field. Since the better is the enemy of the good, I would prefer these classics and recommend Calogero’s textbook as a supplement.