Paolo Brandimarte’s *An Introduction to Financial Markets: A Quantitative Approac*h (Wiley, 2018) is an imposing 750-page book. It is meant as a textbook for students who want a thorough grounding in the mathematics and statistics of finance. It arose out of courses the author offered at Politecnico di Torino, where he is a professor in the department of mathematical sciences, to graduate students in mathematical engineering. Like most textbooks, at the each of each chapter it has a set of problems (the answers to which can be found on the book’s website) and a bibliography.

After an overview of markets and an outline of basic problems in quantitative finance, the author analyzes fixed-income assets, equity portfolios, derivatives, and advanced optimization models. Devoting nearly 150 pages to optimization models may seem a bit eccentric, but Brandimarte is particularly interested in this topic and has done extensive research on it. For instance, in 1995 he co-authored a book titled *Optimization Models and Concepts in Production Management*. And in 2013 he co-authored a book on distribution logistics, which is essentially an optimization problem.

Zeroing in on the third part of the book, on equity portfolios, we find four chapters that deal with (1) decision-making under uncertainty: the static case, (2) mean-variance efficient portfolios, (3) factor models, and (4) equilibrium models: CAPM and APT. Here I’ll look very briefly, and somewhat telegraphically, at the first problem—decision-making under uncertainty—to give some sense of the book’s approach.

Brandimarte distinguishes between a static decision model and a multistage decision model. In a static model, we assume that we are not “adjusting our decisions along the way, when we observe the actual unfolding of uncertain risk factors.” In a multistage model, we can update our decisions, “depending on the incoming information flow over time.” He is simplifying his discussion by considering only the static case.

If we are trying to choose among a set of lotteries, let’s say, we might use a utility function. Brandimarte spends ten pages on the math involved in explicating and applying these functions. But utility functions have been severely criticized over the years, most notably for mixing objective risk measurement and subjective risk aversion in decision-making. Therefore, academics and practitioners have proposed mean-risk models, using risk measures such as standard deviation and variance and quantile-based risk measures such as V@R and CV@R. (More math.) These “may provide us with a partial ordering of alternatives, as well as a set of efficient portfolios.”

A third alternative framework is the stochastic dominance framework, “resulting in a partial ordering that may be related to broad families of utility functions.” (Math.) Brandimarte finds stochastic dominance “an interesting concept, allowing us to establish a partial ordering between portfolios, which applies to a large range of sensible utility functions. Unfortunately, it is not quite trivial to translate the concept into computational terms, in order to make it suitable to portfolio optimization. Nevertheless, it is possible to build optimization models using stochastic dominance constraints with respect to a benchmark portfolio.” (Two theorem proofs, problems, bibliography, end of chapter.)