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Discrete Time Series, Processes, and Applications in Finance

Gilles Zumbach
Springer Finance
Publication Date: 
Number of Pages: 
Springer Finance
[Reviewed by
Omar Rojas
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The study and description of financial time series goes back to 1900, when Louis Bachelier, a student of Poincaré, defended his thesis Théorie de la Spéculation at La Sorbonne. In his thesis, Bachelier presented the first model of what is now known as Brownian motion as a tool to evaluate stock options. Many years went by until the sixties, when B. Mandelbrot and E. Fama, independently, showed how the probability density function of several asset returns did not follow a normal distribution, but a fat-tailed one. The description of financial time series by various models had explosive growth after the recognition given to R. F. Engle with the 2003 Nobel Memorial Prize in Economical Science “for methods of analyzing economic time series with time-varying volatility (ARCH)”, presented for the first time in 1982.

The book aims to synthesize the present status of the field, but it also represents a subjective snapshot of the current situation, as viewed by the author. It is written in a very concise and elegant way, explaining the notation used as it is required. It provides many examples of different financial processes that share some of the statistical characteristics termed stylized facts (general properties that are expected to be present in any set of returns).

As for the mathematics used in the book, the focus is not on formal proofs or detailed formula derivations, but on the practical side of things. As the author clearly states, the goal is to offer an original and comprehensive statistical analysis of empirical time series, and to apply the same tools to a wide range of theoretical processes. I find that this goal is achieved and, furthermore, the book invites the reader to explore and apply the statistical tools presented to empirical time series of the reader´s own interest.

In order to achieve the construction of mathematical processes describing financial time series, the author describes, first of all, the statistical empirical stylized facts exhibited by the financial time series. For this purpose, the most relevant graphs are presented in a series of mug shots, which constitute a collection of graphs summarizing the most important statistical facts of a given empirical time series or process. Such a name comes from the mug shot, which, in police slang, is the pair of pictures of someone´s face, one front, and one profile. After presenting such facts in full color graphs, the author begins constructing mathematical processes that can reproduce some, possibly all, of these facts.

These empirical mug shots consist of eight pictures in full color that characterize the heteroscedasticity, the convergence toward a Gaussian distribution, the volatility clustering, the dependency structure between time horizons, the asymmetry with respect to time reversal invariance and the dynamics of the volatility evolution. All these graphs, using foreign exchange data of some main currencies, can be obtained from the companion website ( However, even if the reader appreciates the graphs, some computer code in Matlab or R would be of better use. However, as the author kindly told me by email when asked about it, the mug shots were created with a large C++ program and then, using scripts, the plots were created using the commercial software Tecplot.

The shortest chapter of the book (only three pages) deals with regime-switching processes. The idea behind a regime-switching model is that the market can take on different states, such as periods of high and low volatility. The mug shot of a three-state regime-switching process, with Student innovations and dynamics specified by a Markov chain with given transition probabilities is presented. The author concludes that the process is not a good candidate as a model for the empirical price time series, since the properties of the process are quite far from those of the empirical time series. I believe more research should be done before discrediting such processes. If a chapter is to be devoted to the subject, Threshold Auto-Regressive models were available; they have proved very fruitful in modeling non-linear financial time series of returns and volatility.

The final chapter of the book rummages through some topics like Multi-time horizon analysis, Slow decay for the lagged correlation, Definition of volatility, the Importance of heteroscedasticity, Fat-tailed distributions, Convergence towards a Gaussian distribution, Temporal aggregation and Mean reversion and Ornstein-Uhlenbeck processes. This chapter opens the door to some questions, both from the research and the philosophical point of view.

This book is definitely recommended to anyone (practitioners, quants, academics or graduate students) interested in attaining a deeper understanding of the dynamics of prices, as well as the corresponding stylized facts, which are central to many applications like portfolio optimizations, risk evaluations, or the valuation of contingent claims.

Omar Rojas ( is a mathematician turned to finance, working as a research professor at the School of Business and Economy at Panamerican University, Guadalajara, Mexico. His areas of interest are Mathematical Finance, Operations Research and Applied Dynamical Systems.