The Statistical Mechanics of Financial Markets
DOI: 10.1063/1.1510282
Physicists’ study of financial markets has deep historical roots, beginning with the now-famous 1900 thesis “Theory of Speculation” by Henri Poincaré’s student Louis Bachelier. In modern times, physicists studying financial markets have seized the data-analysis opportunities presented by the availability of large, high-frequency data sets for price movements on many markets. Such analyses are attractive for several reasons: Physics, especially statistical physics, has long been concerned with the appearance of universal features in seemingly unrelated systems; financial markets represent, in some sense, the most complex of many-body problems, as each interacting body is a sentient being; and it is natural to ask whether markets exhibit any statistical structures akin to those in more physical systems. Finally, there is the pragmatic, but hardly unattractive, goal of profiting from insights into market mechanics.
Johannes Voit’s The Statistical Mechanics of Financial Markets provides an excellent introduction for physicists interested in the statistical properties of financial markets. Appropriately early in the book, the basic financial terms and concepts such as shorts, limit orders, puts, calls, and others are clearly defined. Examples, often with graphs, augment the reader’s understanding of what may be a plethora of new terms and ideas.
The random walk, probably the central concept of the book, is discussed in detail. The seminal work of Bachelier is well described and appropriately acknowledged (which other authors are not always careful to do). Bachelier’s analysis is applied to the concrete example of a bond future. Connection to Brownian motion (including Albert Einstein’s work), and to diffusion processes is then established. Indeed, a general strength of this book is that a corresponding, presumably familiar physical system is analyzed in conjunction with a financial one. The reader is then introduced to more general stochastic processes and geometric Brownian motion, and to the derivation of the log-normal distribution of price moves. The results on geometric Brownian motion are used to obtain the classic result in option pricing, the Black-Scholes equation. These results, and others that Voit presents, hold under somewhat unrealistic, idealized assumptions. While treatments of this kind are entirely appropriate in an introductory text, a somewhat more extensive critique of these assumptions would have been welcome, especially in light of the “long-term capital management” debacle in 1998.
The chapter on Scaling in Financial Data and in Physics may be the most interesting to a cursory reader. Beginning with a brief discussion of the statistical properties of time series (stationarity, correlation, autocorrelation), Voit concentrates on the scaling properties of price movements. He demonstrates that price movements on time scales that vary by several orders of magnitude may be collapsed onto a universal curve by a simple rescaling of the time interval and the observed probability density. His observation leads to the consideration of Levy-stable probability densities, of which the familiar Gaussian is a special case. A key feature of these general Levy distributions is their heavy tails—a considerably higher probability of large (positive or negative) values than in a Gaussian. The financial implication is obvious: If you model the world as Gaussian, you will have a significantly greater probability of striking it rich (or going completely bust!) than your model leads you to believe. The reader interested in a more extensive treatment of scaling and correlations in markets should consult An Introduction to Econophysics: Correlations and Complexity in Finance by Rosario M. Mantegna and Eugene Stanley (Cambridge U. Press, 2000).
Voit examines the interesting comparison that can be drawn between the energy cascade over different length scales in turbulent flow and the information cascade over varying time scales in the foreign exchange (FX) market. No other market generates the wealth of high-frequency data that FX does, with its round-the-clock trading of some 1012 dollars per day. Voit demonstrates that certain features scale similarly in these two cases. It is hard to know if the similarity is more than accidental. While a microscopic description of turbulent flow is exceedingly ambitious theoretically and computationally, it seems achievable in principle. In contrast, a similar level of description of markets would require a model for each participant, and this is well beyond our grasp.
The Statistical Mechanics of Financial Markets concludes with a somewhat strained analogy between market crashes and earthquakes. The paucity of data on crashes makes the evaluation of the apparent log-periodic precursors to the crash difficult. Although definite physical mechanisms for earthquakes have been identified, considerable disagreement remains on what causes market crashes. Obviously, psychological variables play a centrol role. Of course there would be a great benefit to participants in financial markets if crashes could be reliably predicted. No doubt there will be continuing intense research in this area.
In conclusion, The Statistical Mechanics of Financial Markets is an excellent starting point for the physicist interested in the subject. Some of the book’s strongest features are its careful definitions, its detailed examples, and the connections it establishes to physical systems. The mathematics are at the level of upper undergraduate statistics and statistical physics, making the book suitable for students as well as practicing physicists. A more serious student would need to augment this text with something closer to a traditional approach to time series analysis.
More about the Authors
Robert W. Lourie, director of futures research at Renaissance Technologies Corp, is a physicist working on financial modeling. He formerly was a professor of physics at the University of Virginia.
Robert W. Lourie. Renaissance Technologies Corp, US .
