In this model, any variations in price would correspond to the arrival of new information. Some of the information is predictable, and some isn't. If the market works effectively, predictable information is already accounted for in the price. In other words, a rise or fall has already taken place on the basis of predictions. When the news finally arrives, it can no longer affect the price.
Hence, price fluctuations occur only in response to truly new pieces of information—the parts that aren't predictable based on available data. It makes sense, then, to model the fluctuations in stock prices as a Markov process, which proceeds in independent increments (see "Walking Wild I").
This picture of a stock market, however, isn't completely realistic, and Brownian motion generally fails as a full measure of stock-price fluctuations. Human ingenuity, fear, intuition, fallibility, and incompetence conspire to complicate the situation and put it outside the realm of pure randomness.
Interestingly, random-walk and Brownian-motion models have made a comeback in recent decades in finance theory, specifically in methods of evaluating stock options.
An option is a contract by which a seller commits himself to selling specified shares within a certain time period at a price determined today. The buyer has the right not to exercise the option if this price turns out to be higher than the price at the date the option falls due.
Such contracts allow holders of a large number of shares to hedge their bets. They protect themselves against a fall in price by buying the corresponding options with the idea of not using them if prices rise or remain stable. The person who sells the option takes a risk and must be compensated.
A technique for determining a fair price for such options was initially established in 1970 by mathematician Fischer Black (1938-1995) and economist Myron Scholes, who worked out a formula that takes into account a stock's tendency to oscillate in price.
According to their model, it's not necessary to know whether a stock will go up or down in price. The direction of the price change doesn't matter. Instead, the only thing that matters is how much the stock price is likely to vary during the period of the option.
According to the model, a company whose stock price fluctuates a great deal over a wide range presents a bigger investment risk than one whose stock price is expected to remain relatively stable.
The Black-Scholes equation and model were so innovative when they were first proposed that the authors had difficulty getting their paper published. Eventually they prevailed, and soon after its publication in 1973 traders were successfully applying their pricing formula to real markets. Nonetheless, the formula represented an idealization of the behavior of markets, and researchers have since considerably refined and extended the basic model.
Finance theory has become one of the most active fields of research in modern applied mathematics. At the heart of these developments is the constant need to evaluate assets by determining fair, rational market prices and predicting how those prices will change in concert with the prices of other goods and financial instruments.
As a result, many of the mathematical tools and computational techniques already familiar to mathematicians, scientists, and engineers are finding new homes in the world of finance.
One concept that may prove of great value is the notion of scaling—the way phenomena look on different scales. The motion of a simple pendulum, for example, has an intrinsic scale because the pendulum oscillates at a particular, fixed frequency. Avalanches, on the other hand, appear to be totally random events, and they have no intrinsic scale. They come in all sizes, from just a few pebbles rolling down a hill to massive rock slides roaring and crashing down a mountainside.
In the case of avalanches, earthquakes, and even stock-market crashes, it's possible to show that the number of events goes down as the size of the events increases. Big events occur less frequently than small events, a relationship that can in many cases be expressed by a simple mathematical formula.
Discovering such relationships in experimental data provides potentially valuable information about apparently random events. Similarly, fractals, which look the same on all scales, also make an appearance as geometric models of phenomena that occur on a wide range of scales.
"We don't predict when an event like an earthquake will happen because it is random; all we predict is the probability that it will happen," said physicist H. Eugene Stanley, a pioneer in this field. "There are a zillion problems like this, ranging from the stock market to lots of other things that are scale-free phenomena."
Next: Galaxies and Coffee Cups
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