Modeling and analysis of financial markets

Much effort has gone into the study of financial markets and how prices vary with time. Charles Dow, one of the founders of Dow Jones & Company and The Wall Street Journal, enunciated a set of ideas on the subject which are now called Dow Theory. This is the basis of the so-called technical analysis method of attempting to predict future changes. One of the tenets of "technical analysis" is that market trends give an indication of the future, at least in the short term. The claims of the technical analysts are disputed by many academics, who claim that the evidence points rather to the random walk hypothesis, which states that the next change is not correlated to the last change.

The scale of changes in price over some unit of time is called the volatility. In 1900, Louis Bachelier modeled the time series of changes in the logarithm of stock prices as a random walk in which the short-term changes had a finite variance. This causes longer-term changes to follow a Gaussian distribution.

Modeling the changes by distributions with finite variance is now known to be inappropriate. In the 1960s it was discovered by Benoît Mandelbrot that changes in prices do not follow a Gaussian distribution, but are rather modeled better by Lévy stable distributions. The scale of change, or volatility, depends on the length of the time interval to a power a bit more than 1/2. Large changes up or down are more likely than what one would calculate using a Gaussian distribution with an estimated standard deviation.