## The Market is Neither “Random” nor “Unpredictable”

In discussions regarding the effect of the presidential election on the stock market, the favorite line of argument appears to be that: there is much more randomness in the markets than most people recognize: and, people have a tendency to become comfortable with the perceived patterns even though the patterns are nothing more than random noise. Some have even suggested that the works of Mr. Nassim Nicholas Taleb are definitive on the issue.

I have carefully read the volume “The Black Swan” by Nassim Nicholas Taleb, the seminal work “Statistical Mechanics of Financial Markets” by Johannes Voit as well as many other works on the mathematical theory of the stock market. I also have also studied statistical mechanics in a number of disciplines for more than 20 years. Based upon my understanding of statistical mechanics, the aforementioned works, and several years of studying the stock market data, it is my observation that the two fatal flaws in the works by both Taleb and Voit, as well as the plethora of others who claim that the market is “random”, are embedded in the assumption that the incremental change in the either the share price of a company or the stated level of an index is either a stationary levy process or a quadratic brownian process. Both of these processes lead to a normal probability density function for the incremental price change, which has been demonstrated to be incorrect. From the flawed basic assumption, the standard argument arises that the market is “random” and, therefore, unpredictable.

If the market were to be truly “random” process, then it will return to its starting point, because it is a one-dimensional random walk in time, an infinite number of times and the recurrence time diverges to infinity. Before diverging to infinity, the infinity minus one times that the random walk returns to the starting part has a finite and predictable recurrence time. That means that the market will return to precisely zero, or any other number that you choose, an infinite number of times and will do so with a finite recurrence time until the recurrence time diverges to infinity.

Now, let us look at the data: never, in the history of the stock market, has the market returned to zero. Over the history of the stock market, the average annual return has been greater than zero and a significant number of large incremental price changes have happened. Even a cursory examination of the data leads to the conclusion that the stock market, as measured by any index you might wish, is a stochastic process with a non-zero drift and “fat-tail” price changes in continuous time. If that were not true, then there would not be a single sane investor in the market and the market would not exist. The immediate response is whether the price of a share can be predicted at any moment in the future. The answer is no unless all the factors that affect the share price can be determined with any certainty. In the absence of complete knowledge of the factors affecting the price, it is sufficient and necessary to predict the distribution of the share price and hedge against the probability of of being incorrect about the selected price.

I hope that this helps.

Nathan A. Busch

25, August 2012 at 11:02

I don’t believe Taleb argues that the market as a whole is random. He does argue that changes in the price of a stock are explained more by noise over very short time-periods. More specifically, the shorter the time-period in which a stock is analyzed, the greater the influence noise has. Over long time-periods this noise will cancel out, making true trends more apparent.

Movements in CocaCola’s stock price over the course of a day is mostly just noise, but over the course of a year, it better reflects true changes in the value of the company.

Taleb’s analysis is very useful when considering the riskiness (volatility) of investments. It’s surprising how much of an impact unexpected events can have on financial prices, and really life in general. In finance there are opportunities to profit from unexpected events and there are opportunities to protect yourself from them.

25, August 2012 at 12:37

Mr. Slocum:

Thank you for your kind and well-considered response.

I would first like to reply to your comment that: “over the course of a year, it better reflects true changes in the value of the company.” My interpretation of your comment is that drift in the underlying stochastic process reflects the “true changes in the value of the company.” With regard to that philosophy, I am in agreement with Phillip Fisher in that the drift in the price reflects only the

perceptionof the market in regards to the value of the company. The true value of the company will often not change over the course of a year, or longer, as rapidly as the change in price might suggest. It is only for this reason that “value investing” can possibly yield exceptional returns.It is not possible for Taleb or Voit to present the theory of a process that is an Ito process for very short time periods yet exhibit drift in the long term without the possibility of recurrence to zero. Once the fundamental assumption of an Ito process is adopted, the long-term trajectory is well defined independent of whether one employs a standard normal, or log-normal, probability density function for the price increment or a fat-tail probability density function. To propose a theory for the fluctuations in the financial markets it is necessary to provide a theory that explains both the stochastic short-term price increments and the long-term drift. To my way of thinking, superposition of a deterministic long-term drift upon a short-term Ito process is neither mathematically rigorous nor satisfactory.

Protecting against the downside risk is merely a derivative function of the assumed process for the price changes in the underlying stock. That protection also depends upon the deviation between the price of the underlying security and the price of the derivative being an Ito process. My preliminary examination of option prices indicates that that deviation is not an Ito process.

An example of this is found when examining the price of integrated oil companies and the companies in the Philadelphia Oil Service Index along with the spot price of West Texas Intermediate Crude. It is my hypothesis that the price of shares in these companies is a derivative of the spot price of WTI. As I examined in a sister comment to this one,

see Conoco and Phillips,1st May 2012, over a five to six week period of time, the price of shares in Chevron (NYSE:CVX) was highly correlated to the spot price of WTI: however, that correlation was not a constant function in time. This means that a drift existed in the stochastic derivative function between the price of WTI and the price of CVX. It appears that, over the same period of time, such behaviour was also found in Marathon Petroleum, Marathon Oil, Phillips 66, and Total. Even the drift appears to be a stochastic process but not an Ito process. This, of course, demands closer inspection and further examination.I hope that this helps.

Nathan A. Busch