BOOK REVIEW

# Statistical Phynance

*The Statistical Mechanics of Financial Markets*. Johannes Voit. xii + 220 pp. Springer, 2001. $44.95.

Some investors buy stocks on the basis of the number of articles published about a given company. Taking this approach, one should buy "econophysics": Scores of research papers on the subject can be found in physics journals, the Los Alamos National Laboratory Condensed Matter e-Print Archive (http://arXiv.org/archive/cond-mat), and new journals such as *International Journal of Theoretical and Applied Finance* and *Quantitative Finance*; articles have appeared in *New Scientist*, *Risk* and *Nature*; Web sites and radio programs are devoted to this new field; and physicists have written half a dozen books on it in less than three years.

Johannes Voit's *The Statistical Mechanics of Financial Markets* is one of them; it's an interesting example of how econophysics has invaded the traditional quarters of statistical and solid-state physics. For those trained in modern statistical physics, which has focused in recent decades on complex, strongly interacting systems exhibiting subtle collective effects, the issues at stake in finance and economics taste vaguely familiar and ignite curiosity. This is especially true now that empirical data are relatively easy to access?thousands of time series containing the intimate details of the frantic human activity registered by financial markets beg for quantitative models or more qualitative understanding. Analogies between financial signals and turbulent flows, avalanches, earthquakes, critical phenomena or anomalous diffusion have been mentioned, enhancing further the feeling that the cross-fertilization between statistical physics and finance could give birth to a really exciting discipline?one that is closer to lay concerns than are more traditional fields of physics.

There are different ways to enter a new field, and Voit thought of an original one: learning by teaching. He planned a long series of lectures on the still-babbling new field. Careful notes for his students soon became an interesting book that blends textbook explanations of well-established concepts in finance and in physics with a review of more recent ideas, some of which are still rough and sometimes rather controversial.

The first chapters introduce basic concepts in mathematical finance, defining various financial instruments, the classic random walk model of Louis Bachelier, and the famous Black-Scholes theory of option prices. The chapter on Bachelier is very welcome: His remarkable Ph.D. work completed in 1900 is often cited, but very few have actually read it as carefully as Voit. He convinces the reader that Bachelier was a precursor: He invented the theory of Brownian motion (in most clearly stated terms) and founded theoretical finance, which later built on his ideas; and he also analyzed financial data and compared the results with his theory. This mix (and feedback) between models and data analysis is characteristic of the econophysics endeavor.

The chapters that follow discuss the empirical properties of financial time series from the point of view of scaling and self-similarity. The theory of L?vy flights and many classical examples of anomalous diffusion in physics are explained and put into perspective. Voit rightly emphasizes the well-established "stylized facts" of financial data: fat tails and volatility fluctuations. Volatility fluctuations are important because they govern the progressive deformation of the distribution of price returns from high frequencies (where the distribution is strongly non-Gaussian) to low frequencies (where it becomes Gaussian). There are strong analogies with turbulence here, and the development of so-called multifractal models (interpolating, in a sense, between the too-mild Brownian walk and the too-wild L?vy flights) is currently the focus of intense work. Voit usefully introduces us to the basic ideas in the theory of turbulence and guides us through the analogies with financial markets.

With a faithful model of financial markets, one can (and must) address the problems of risk control and option pricing. Fat tails and volatility fluctuations jeopardize some of the important pillars of financial mathematics, such as the possibility of getting rid of risk altogether in the Black-Scholes world. This "zero risk" property of the Black-Scholes model would hold if financial time series were continuous-time random walks: no tails, no unexpected fluctuations and infinitely fast trading. What if this is not the case? In a section called "Derivative Pricing Beyond Black-Scholes," Voit reviews some of the recent ideas developed in the econophysics literature to deal with risky options and to explain the so-called volatility smile (that is, the deviations from the constant-volatility Black-Scholes model), observed in option markets, in terms of fat tails and volatility fluctuations.

There are many examples of "universality" in physics, where a wide variety of different microscopic models give rise to exactly the same macroscopic, collective behavior. Is something similar taking place in financial markets? What are the fundamental mechanisms by which all financial data look alike, showing similar qualitative and sometimes quantitative features? How deeply does one have to enter into the actual *financial* side of things in order to understand the curious statistical anomalies of financial price changes? Here again, the art of numerical simulation developed by physicists to understand fluid flows, magnets, forest fires, traffic jams or bird flocks turns out to be very valuable. Several "plausible" models have been proposed and analyzed, some of which (most notably the one developed by Thomas Lux and Michelle Marchesi) have been successful indeed in their predictions. Many of them contain a part of the story, but none is fully satisfactory. Nevertheless, they are valuable food for thought on the winding path of research. This is why Voit deliberately chooses to present many of them, without prejudice, in order to paint a faithful picture of the present state of confusion and excitement.

In the final chapter, Voit discusses the most fascinating phenomenon in stock markets (or any market for that matter), the one with potentially scathing human consequences: crashes. How can all U.S. companies lose more than 20 percent of their value in a few hours in the absence of any major economic catastrophe, as was the case in October 1987? It is tempting to think that the crash of 1987 was a genuine collective phenomenon in which herding and autoamplification mechanisms led to a generalized panic. Does this strange, irrational collective state emerge out of the blue, or are there hidden signals that would detect a slow, gradual increase of stress before the point of no return is reached?

The very same question appears in the context of earthquakes or material failures. Didier Sornette proposed that all of these systems reveal symptomatic "log periodic" oscillations before failure: The signal, be it the price of an asset or the acoustic emission of a strained material, starts to oscillate at an accelerating frequency. The extrapolation forward in time to the point where this frequency would diverge to infinity predicts the date of the earthquake or crash. These peculiar oscillations would be the trace of some scale-invariant organization of the crust, or of the pessimism of traders. Voit explains these arguments and reproduces the suggestive graphs provided by the proponents of log-periodic crashes. On the other hand, many question marks remain, and the log-periodic story has been fiercely challenged by some: The statistical evidence is questionable, and there is no compelling theoretical argument to explain why these oscillations should exist in the first place. Voit rightly considers crashes to be sufficiently important to spend a whole chapter discussing the matter, but, cautiously, he does not take a side. One can, however, read between the lines that he hopes that something like this could be true!

On the whole, the book is well balanced between the new and the old, truth and hope, lore and lure. It has some significant overlap with books such as Rosario N. Mantegna and H. Eugene Stanley's *An Introduction to Econophysics* and the book I coauthored with Marc Potters, *Theory of Financial Risks*; however, it is broader in scope and contains a lot more physics and speculative ideas. The useful references and Web links provided at the end of the book might make it an easier step to climb for physicists who want to know where the action is.-*Jean-Philippe Bouchaud, Service de Physique de l'?tat Condens?, CEA-Saclay, France*

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