Rosario N. Mantegna and H. Eugene Stanley, An Introduction to Econophysics: Correlations and Complexity in Finance
Under the
Efficient Market Hypothesis,
the market accurately reflects all knowledge available about all stocks at all times. Consequently, the movement of stock prices follows a
Brownian motion — specifically
a
geometric Brownian motion, meaning
that the logarithm of the price at any given time is a Brownian-motion process.
It’s long been known that this model doesn’t really work: the “tails are too heavy,” in the jargon, meaning that there are more large and small stock prices than the Brownian model would predict. The Taylor and Karlin book that we used in my first stochastic-processes class proposed that this is because the number of stock trades is itself a random variable — specifically a Poisson process. At least at a first pass, this doesn’t really provide a satisfactory explanation of the heavy tails. It seems like it lacks “microfoundations,” in the economists’ lingo: it doesn’t explain the higher-level process (stock-price motion) in terms of the aggregated behaviors of economic actors.
It’s not clear from Mantegna and Stanley’s book that these microfoundations are really in the offing. They proceed through a wide variety of stock-price models, each of which explains more of the stock market’s behavior. For starters, they ask whether individual stock-price changes can be best modeled by a random variable with a finite or an infinite variance. If the stock-price changes have a finite variance, then the trajectory of the stock price — which is the sum of the price changes — will follow the Central Limit Theorem, and the trajectory will be your standard, friendly Gaussian random variable.
If the variance is infinite, on the other hand — meaning that stock prices can be much more volatile — then the Central Limit Theorem breaks down: the trajectory converges instead to a Lévy stable process. So Mantegna and Stanley converge on a set of models called a “truncated Lévy flight,” which assigns probability densities like a Lévy process within a symmetric interval around the origin, and no probability outside. By adjusting the width of that interval, we can bring the Lévy closer to a Gaussian.
So the truncated Lévy flight is a convenient modeling device, in that a suitable adjustment of parameters can explain one part of the distribution of stock prices. But does it actually provide understanding? Do we understand why this particular distribution looks more like the actual distribution of stock prices than does a Gaussian? Here we would have understanding if we could explain why variance is either finite or unbounded. Unless I misread Mantegna and Stanley, there’s no good theoretical reason to expect either finite or infinite variance, and professional opinion is only just coalescing around finite variance.
The book contains other related concepts, among them ARCH and GARCH models that explain both heavy tails and a long-term dependence in stock volatility. But again, these models merely postulate a certain form of relationship, then conclude that various parameters are set in certain ways; they don’t seem to confer understanding.
Mantegna and Stanley end with a couple quick related topics: how to identify stocks whose prices move together, and the Black-Scholes options-pricing model. Understanding synchronized stock-price movements is important in the construction of diversified portfolios — “diversity” here being basically synonymous with “unsynchronized” (if you buy a share of Proctor and Gamble and a share of RJR, your portfolio is not diversified).
Throughout, they are as brisk as they could be, but no brisker; they give the reader a great breadth of understanding with decent depth, which is quite a trick in a book that’s not much more than 100 pages long. This book would actually make a fantastic first course in stochastic processes: students wishing to move beyond the initial sketches of geometric Brownian motion can easily do so using the bibliography. If this book were a bit cheaper on the used market, I would snap it up in a heartbeat.
P.S. (5 March 2008): I think Farmer’s paper “Market Force, Ecology and Evolution” may provide the long-sought behavioral foundations that would underlie Mantegna and Stanley’s statistical findings. I’ve intended to read it for a while; I’ll move it up in the queue now. (Paper via Cosma Shalizi.)
While I’m at it, I should note that it’s not clear to me why stock-price movements ought to be geometric Brownian; Mantegna and Stanley don’t explain why the EMH implies Brownian motion specifically. The proof of that claim is supposed to lie in a few papers: “Brownian Motion In The Stock Market”, Samuelson’s “Proof That Properly Anticipated Prices Fluctuate Randomly”, and finally Fama’s “The Behavior of Stock-Market Prices”. All of them are in my to-read queue. The first of those three somehow uses the Weber-Fechner Law, which is a model of human sensory perception, to justify geometric Brownian motion over simple Brownian motion. I don’t yet understand why that would be.