Tuesday, May 29, 2012

Alternatives to Gaussian Curve


 As stated in my previous article about the problems of using Gaussian curve in events which have a low probability of occurrence and a very high impact, the Black Swan Events.So, while weight, height and calorie consumption are Gaussian, wealth is not. Nor are income, market returns, size of hedge funds, returns in the financial markets, earthquakes, number of deaths in wars or casualties in terrorist attacks. Almost all man-made variables are wild though Nassim Nicholas Taleb was the first person to construct the Black Swan Theory .Nassim Taleb is a  philosopher of randomness and a philosopher who praises erudition above all.Nassim is also a top Financial Math expert, philosopher, statistician and a Hedge fund manager,a guy who uses philosophy in decision making,model building and hedging the funds.He made the theory of Black Swans , the rare events that are massive in impactBlack Swan event or occurrence that deviates beyond what is normally expected of a situation would be extremely difficult to predict.For example, the previously successful hedge fund Long Term Capital Management (LTCM) was driven into the ground as a result of the ripple effect caused by the Russian government's debt default. The Russian government's default represents a Black Swan event because none of LTCM's computer models could have predicted this event and its subsequent effects.Most of the theories and events in economics, math ,finance, statistics, engineering and physics follow Gaussian curve and though some events are non-Gaussian and this misleads us to disaster.

Henri Poincare was the first one to state this problem , Poincare observed that a very small difference in the starting positions and velocities of the planets could actually grow to an enormous effect in the later motion .This led to proof that even if initial measurements could be specified with high precision the uncertainty will remain huge in few systems.Poincare was suspicious of Gaussian curve as he knew that high impact rare events in Quantum Physics and Solar Systems do not follow the patterns which Bell/Gaussian curve follows.Poincare suggested that as you project in the future you may need an increasing amount of precision ,near precision is not possible . Think of forecasting as in terms of tree branches, this grows in multiple ways and doubling every time so such increasing amount requires a lot of precision.This high impact and uncertainty is what is defined as a Black Swan event.Fractals distributions do better than Bell curve in avoiding the Big Black Swans. Sometimes a Fractal can make you believe it is Gaussian. Normally extreme events fit into Fractal category , fractals thus have very high standard deviation
Poincaré was one of France's most renowned mathematician, and contributed important works in physics, astronomy, mathematics and philosophy of science. He is often described as a polymath, and in mathematics as "The Last Universalist".

Nassim Taleb , the creator of Black Swan Theory.Nassim is called by many as the "Modern Nietzsche" or even "Modern Renaissance Man".

Benoit Mandelbrot developed the field of fractal geometry between 1970 and 1980 with books
 such as Fractals: Forms, Chance and Dimensions (1977) and Fractal Geometry of Nature (1982).
The importance of this science is that objects are not reduced to a few perfect symmetrical shapes
 as in Euclidean geometry. Instead it explores the asymmetry, roughness and fractal structures of nature.With fractal geometry: “clouds are not spheres,mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lighting travel in straight lines” (Mandelbrot 2004 p 124).
In this way, as Euclidean Geometry served as a descriptive language for the classical mechanics of motion, fractal geometry is being used for the patterns produced by chaos.High impact events follow the pattern that can be modelled on fractal theory or even Chaos theory.Fractals are examples of non-linear, chaotic systems. These are systems that start out simple then over time, as rules are recursively applied, evolve into something much more complex. Such systems are sensitive to initial conditions. A small change at the outset results in major changes later. A simple fractal shape can quickly become a complex shape (that still contains the seed of the starting shape).An example of fractal is "butterfly effect" , butterfly flapping its wings and creating tiny wind patterns in Africa can result in a hurricane in Florida.These sort of events which are rare, hard to predict and carry a massive impact are perfect examples of "Black Swans" and correspond to "Mandelbrotian Randomness"

With Einstein, and his Relativity theory, the physics world added time as the fourth dimension.
However in fractal science, the dimension depends on the point of view of the observer. “The same object can have more than one dimension, depending on how you measure it and what you can do with it. A dimension needs not to be a whole number; it can be fractional. Now and ancient concept, dimension, becomes thoroughly modern” .The fractal dimension is important because it recognizes that a process can be somewhere between deterministic or random. In spite of this, fractal geometry is in fact a simplifying and logical tool.In mathematics, fractal functions work like chaotic systems where random changes in the starting values can modify the value of the function in unpredictable ways within the system boundaries.The famous Mandelbrot set demonstrates this connection between fractals and chaos theory. Benoit Mandelbrot, father of fractal geometry, indeed first discovered the distinctive characteristics of fractals in financial time series.In 1963 the famous mathematician Mandelbrot produced a paper on "Stable Paretian Distribution" pointing out that the tails of security price distributions are far fatter than those of normal distributions (what he called the “Noah effect” in reference to the deluge in biblical times) and recommending instead a class of independent and identically distributed “alpha-stable” Paretian distributions with infinite variance.The importance of Mandelbrot’s discovery is that it highlights that under the apparent disorder of capital markets, there are some “stylized facts” that can describe the behavior of capital markets and that the very heart of finance is "fractal".He also checked his models on Computers using Monte Carlo Methods.

Taleb's sometime co-author Benoit Mandelbrot has been trying to sell the world on the big idea of fractals in finance for several decades.  James Gleick’s Chaos outlined the essence of Benoit Mandelbrot’s fractals, which takes a simple few lines of inputs to create graphics of insane complexity yet also beautiful recursive symmetry, in many cases eerily similar to nature (eg, ferns, snowflakes).  In dynamic systems, you have chaotic systems that are purely deterministic though sufficiently complex that they appear random.  These systems have large jumps, or phase shifts, reminiscent of market crashes or sudden bankruptcies; they have butterfly effects where small changes produce big differences in outcomes.  Mandelbrot and others have been trying to apply these ideas to financial markets for many decades now (since 1962!), and the effort has not gained any traction, in spite of many papers applying this concept (search skew or kurtosis in any financial journal and you will see many papers).  Mandelbrot’s big idea in finance is that finance relies on a profoundly flawed assumption, mainly that market prices are normally distributed.

 Mandelbrot argues market prices have much fatter distributions described by Cauchy distributions, as evidenced by the high number of 5+ standard deviation moves in financial markets.The result of these mistaken assumptions is to understate risk, according to Mandelbrot, and so overprice stocks and underprice options, and also understate the capital cushion financial institutions need to withstand market risk.   Mandelbrot’s alternative approach is based on new parameters that would replace the mean and standard deviation.  His first parameter is Alpha, derived from Pareto's Law, is an exponent that measures how wildly prices vary. It defines how fat the tails of the price change curve are. The second one, the H Coefficient, is an exponent that measures the dependence of price changes upon past changes. Unfortunately, Mandelbrot himself acknowledges in The Misbehavior of Markets that no two individuals calculate the same Alpha and H Coefficient when using the exact same historical data: there is no unique way to calculate these two parameters. Thus, using one method, you could derive Alpha and H coefficients that suggest a stock is risky, using another method you would reach the opposite conclusion. This flaw probably has some bearing on its lack of practitioner popularity.

Mandelbrot the "Poet of Randomness" and father of Fractal Geometry.Mandelbrot used his fractal theory to explain the presence of extreme events in Wall Street.

For a financial theory this will mean a great contribution for the improvement of financial models,
as it would generalize Gaussian assumptions to account for volatility clustering and path dependence. Thus, it allows having a better understanding of how risk drives markets, and possible outcomes in the short-term.Therefore, Mandelbrot’s legacy to finance theory is that he proposes new assumptions, to develop models that are based on the observed behavior of capital markets. After all, no matter how sophisticated is the model for financial analysis; it will be limited by the accuracy and reliability of the underlying assumptions describing the market dynamics.

Though many academics even used the methods from Quantum Physics/Mechanics and Statistical Physics to forecast markets , as the events in Quantum Mechanics have wild randomness and have massive impact so these models fit our financial world. But again most of the equations like Path Integrals(derived by Richard Feynman ,which are are used to price derivatives)  and use of Brownian Motion or even the methods of Quantum Mechanics i.e Wave Functions and Heat/Thermodynamics Equations are all rooted from Gaussian or either are special form of Gaussian distribution. Levy Distributions(special case of inverse-Gamma distribution) which are used in Geomagnetic reversal  and even the Brownian Motion with the time of hitting a single point \alpha  ,this distribution is considered a useful alternative and some consider it better than Bell/Gaussian Curve.But again Random Walks, Brownian motion, and Wiener processes ,used in Quantum Physics all stem from
the Bell/Gaussian curve.


Chaos Theory and the Science of Fractals characterize financial markets as systems sensitive to initial conditions that progress in a non-linear behaviour due to feedback mechanisms. In addition,
it conceptualizes agents as having limited cognition capabilities,and most important, behaving irrationally in the market. For risk management this is important because it describes markets not as efficient and stable, but turbulent and volatile.Essentially, it recognizes the risky nature of financial markets.As nowadays the risk in complex financial markets is very high.Wherever you look either in Physics , Economics , Finance , Statistics or Math you will notice that all random events are studied with the assumption of Gaussian Distribution so you cannot use any of the method from these social and scientific fields.However, today’s methods to control and price risk are still based on the neoclassical assumptions of normal distributions and Brownian motions. This is probably one of the reasons that explains the failure of risk management systems in times of crisis.Price changes are assumed to move in a smooth and in continuous manner from one value to another, and extreme events or Black Swan events are just considered far outliers improbable to happen.As all the main assumptions for risk calculation in Portfolios and other important financial instruments are based on Bell/Gaussian curve so this leads us to disasters.

  • Within the framework of Chaos Theory and The Science of Fractals, Stable Paretian Distributions are proposed as an alternative to model financial returns.  

The advantages of these distributions for risk management are the following:

1) Similar to normal distributions, they are also stable under addition and, thus, allow the generalization of the Central Limit theorem, which shows that if the finite variance assumption
is dropped, the sum of independent identical distributions converges to a stable distribution.

2) Stable distributions provide a good approximation of observed data as they can accommodate
 fat tails and high peaks better. Thus, they are able to reflect the real risk of large and abrupt changes that is not taken into account in a Gaussian market.

Stable Paretian Distribution with Theoretical Density plots of various α-stable distributions.

The α-stable distributions are intimately connected to Pareto (power law) distributions: the tails of α-stable distributions are asymptotically Pareto (power law) distributed.What Mandelbrot referred to in one of his papers as a "stable Paretian" distribution is a power law distribution with "fat tails".

Benoit Mandelbrot’s empirical analysis strongly suggested actual stock returns follow fat tailed Stable Paretian distributions with infinite variances ,he proposed to replace the Gaussian distribution throughout by L-stable (stable Paretian) laws.Also evidence demonstrates that in the 99% quantile VaR (Value at risk)estimates with stable Paretian distributions are more accurate than assuming normal distributions. This implies a significant improvement in risk management, as the risk in the extremes of the distributions can be measure more adequately. Therefore, by changing the assumption in risk model from Gaussian statistics to fractals statistics, it would be possible to control better risk, and even, ensure a safer financial system.

  • The other method is using Path Integrals(Quantum Physics) for pricing derivatives and for returns of stocks (non- Gaussian).

Path Integrals, which were invented by Richard Feynman in 1948 , Feynman used path integrals along with the probability methods designed by Norbert Wiener to reformulate methods of Quantum Physics.The path integral also relates quantum and stochastic processes, and this provided the basis for the grand synthesis of the 1970s which unified quantum field theory with the statistical field theory of a fluctuating field near a second-order phase transition.Feynman suggested that when considering the Quantum Mechanics of a moving particle, every conceivable path can be assigned a certain complex(probability amplitude) number called the probability amplitude for that path.
 Jan Dash a particle physicist later applied path integrals in Finance, the reason was simple ,the value of the financial derivative depends on the "path" followed by underlying asset.

Richard Feynman , the pioneer of Quantum Physics and Path Integrals

 Path Integrals also does not use the Brownian Motion approach(which is infact derived from Gaussian), also ARMA models from time series and even Weiner processes all use Gaussian distribution A path-integral description of the Black-Scholes model was recently developed by Belal Baaquie of the National University of Singapore and co-workers. From this, Baaquie and co-workers went on to devise a quantum-mechanical version of the Black-Scholes equation to describe the price of a simple, non-dividend-paying option

This book applies the mathematics and concepts of quantum mechanics and quantum field theory to the modelling of interest rates and the theory of options. Particular emphasis is placed on path integrals and Hamiltonians.

In The (Mis)behavior of Markets and Fractals and Scaling in Finance Mandelbrot argues that the Gaussian models for financial risk (like CAPM and efficient market hypothesis)used by economists like William Sharpe and Harry Markowitz should be discarded, since these models do not reflect reality. Mandelbrot argues that fractal techniques may provide a more powerful way to analyze risk. 

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