Seven states of randomness
Encyclopedia
The seven states of randomness in probability theory
, fractals and risk analysis
are extensions of the concept of normal distribution. These seven states were first introduced in by Benoît Mandelbrot
in his 1997 book Fractals and scaling in finance which applied fractal analysis
to the study of risk and randomness.
These seven states build on earlier work of Mandelbrot in 1964, titled The two stages of indeterminism in which he argued that most statistical models approached only a first stage of dealing with indeterminism in science, and that they ignored many aspects of real world turbulence, in particular, most cases of financial modeling
.
Intuitively speaking, Mandelbrot argued that the traditional normal curves do not properly capture empirical and "real world" distributions and there are other forms of randomness that can be used to model extreme changes in risk and randomness. He observed that randomness can become quite "wild" if the requirements regarding finite mean and variance are abandoned. Wild randomness corresponds to situations in which a single observation, or a particular outcome can impact the total in a very disproportionate way.
The seven states are:
Traditional normal distributions are at the mild end of the scale within this categorization.
Using elements of this theory, in March 2006, a year before the Financial crisis of 2007–2010, and four years before the Flash crash of May 2010, during which the Dow Jones Industrial Average
had a 1,000 point intraday swing within minutes, Mandelbrot and Nassim Taleb
published an article in the Financial Times
arguing that the traditional "bell curves" that have been in use for over a century are inadequate for measuring risk in financial markets, given that such curves disregard the possibility of sharp jumps or discontinuities. Contrasting this approach with the traditional approaches based on random walk
s, they stated:
Mandelbrot and Taleb pointed out that although one can assume that the odds of finding a person who is several miles tall are extremely low, similar excessive observations can not be excluded in other areas of application. They argued that while traditional bell curves may provide a satisfactory representation of height and weight in the population, they do not provide a suitable modeling mechanism for market risks or returns, where just ten trading days represent 63 per cent of the returns of the past 50 years.
Wild randomness has applications outside financial markets, e.g. it has been used in the analysis of turbulent situations such as wild forest fires.
Probability theory
Probability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single...
, fractals and risk analysis
Risk analysis
Risk Analysis may refer to:*Quantitative risk analysis*Risk analysis **Probabilistic risk assessment, an engineering safety analysis*Risk analysis *Risk Management*Risk management tools* Certified Risk Analyst...
are extensions of the concept of normal distribution. These seven states were first introduced in by Benoît Mandelbrot
Benoît Mandelbrot
Benoît B. Mandelbrot was a French American mathematician. Born in Poland, he moved to France with his family when he was a child...
in his 1997 book Fractals and scaling in finance which applied fractal analysis
Fractal analysis
Fractal analysis is the modelling of data by fractals.It consists of methods to assign a fractal dimension and other fractal characteristics to a signal, dataset or object which may be sound, images, molecules, networks or other data....
to the study of risk and randomness.
These seven states build on earlier work of Mandelbrot in 1964, titled The two stages of indeterminism in which he argued that most statistical models approached only a first stage of dealing with indeterminism in science, and that they ignored many aspects of real world turbulence, in particular, most cases of financial modeling
Financial modeling
Financial modeling is the task of building an abstract representation of a financial decision making situation. This is a mathematical model designed to represent the performance of a financial asset or a portfolio, of a business, a project, or any other investment...
.
Intuitively speaking, Mandelbrot argued that the traditional normal curves do not properly capture empirical and "real world" distributions and there are other forms of randomness that can be used to model extreme changes in risk and randomness. He observed that randomness can become quite "wild" if the requirements regarding finite mean and variance are abandoned. Wild randomness corresponds to situations in which a single observation, or a particular outcome can impact the total in a very disproportionate way.
The seven states are:
- Proper mild randomness
- Borderline mild randomness
- Slow randomness with finite delocalized moments
- Slow randomness with finite and localized moments
- Pre-wild randomness
- Wild randomness
- Extreme randomness
Traditional normal distributions are at the mild end of the scale within this categorization.
Using elements of this theory, in March 2006, a year before the Financial crisis of 2007–2010, and four years before the Flash crash of May 2010, during which the Dow Jones Industrial Average
Dow Jones Industrial Average
The Dow Jones Industrial Average , also called the Industrial Average, the Dow Jones, the Dow 30, or simply the Dow, is a stock market index, and one of several indices created by Wall Street Journal editor and Dow Jones & Company co-founder Charles Dow...
had a 1,000 point intraday swing within minutes, Mandelbrot and Nassim Taleb
Nassim Taleb
Nassim Nicholas Taleb is a Lebanese American essayist whose work focuses on problems of randomness and probability. His 2007 book The Black Swan was described in a review by Sunday Times as one of the twelve most influential books since World War II....
published an article in the Financial Times
Financial Times
The Financial Times is an international business newspaper. It is a morning daily newspaper published in London and printed in 24 cities around the world. Its primary rival is the Wall Street Journal, published in New York City....
arguing that the traditional "bell curves" that have been in use for over a century are inadequate for measuring risk in financial markets, given that such curves disregard the possibility of sharp jumps or discontinuities. Contrasting this approach with the traditional approaches based on random walk
Random walk
A random walk, sometimes denoted RW, is a mathematical formalisation of a trajectory that consists of taking successive random steps. For example, the path traced by a molecule as it travels in a liquid or a gas, the search path of a foraging animal, the price of a fluctuating stock and the...
s, they stated:
We live in a world primarily driven by random jumps, and tools designed for random walks address the wrong problem.
Mandelbrot and Taleb pointed out that although one can assume that the odds of finding a person who is several miles tall are extremely low, similar excessive observations can not be excluded in other areas of application. They argued that while traditional bell curves may provide a satisfactory representation of height and weight in the population, they do not provide a suitable modeling mechanism for market risks or returns, where just ten trading days represent 63 per cent of the returns of the past 50 years.
Wild randomness has applications outside financial markets, e.g. it has been used in the analysis of turbulent situations such as wild forest fires.